Stan Math Library  2.9.0
reverse mode automatic differentiation
Namespaces | Classes | Typedefs | Functions | Variables
stan::math Namespace Reference

Matrices and templated mathematical functions. More...

Namespaces

 detail
 

Classes

class  accumulator
 Class to accumulate values and eventually return their sum. More...
 
struct  array_builder
 Structure for building up arrays in an expression (rather than in statements) using an argumentchaining add() method and a getter method array() to return the result. More...
 
struct  AutodiffStackStorage
 
class  chainable_alloc
 A chainable_alloc is an object which is constructed and destructed normally but the memory lifespan is managed along with the arena allocator for the gradient calculation. More...
 
struct  child_type
 Primary template class for metaprogram to compute child type of T. More...
 
struct  child_type< T_struct< T_child > >
 Specialization for template classes / structs. More...
 
class  cholesky_decompose_v_vari
 
struct  common_type
 
struct  common_type< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > >
 
struct  common_type< std::vector< T1 >, std::vector< T2 > >
 
class  container_view
 Primary template class for container view of array y with same structure as T1 and size as x. More...
 
class  container_view< dummy, T2 >
 Dummy type specialization, used in conjunction with struct dummy as described above. More...
 
class  container_view< Eigen::Matrix< T1, R, C >, Eigen::Matrix< T2, R, C > >
 Template specialization for Eigen::Map view of array with scalar type T2 with size inferred from input Eigen::Matrix. More...
 
class  container_view< Eigen::Matrix< T1, R, C >, T2 >
 Template specialization for scalar view of array y with scalar type T2. More...
 
class  container_view< std::vector< Eigen::Matrix< T1, R, C > >, Eigen::Matrix< T2, R, C > >
 Template specialization for matrix view of array y with scalar type T2 with shape equal to x. More...
 
class  container_view< std::vector< T1 >, T2 >
 Template specialization for scalar view of array y with scalar type T2 with proper indexing inferred from input vector x of scalar type T1. More...
 
struct  coupled_ode_observer
 Observer for the coupled states. More...
 
struct  coupled_ode_system
 Base template class for a coupled ordinary differential equation system, which adds sensitivities to the base system. More...
 
struct  coupled_ode_system< F, double, double >
 The coupled ode system for known initial values and known parameters. More...
 
struct  coupled_ode_system< F, double, stan::math::var >
 The coupled ODE system for known initial values and unknown parameters. More...
 
struct  coupled_ode_system< F, stan::math::var, double >
 The coupled ODE system for unknown initial values and known parameters. More...
 
struct  coupled_ode_system< F, stan::math::var, stan::math::var >
 The coupled ode system for unknown intial values and unknown parameters. More...
 
struct  dummy
 Empty struct for use in boost::condtional<is_constant_struct<T1>::value, T1, dummy>::type as false condtion for safe indexing. More...
 
struct  fvar
 
class  gevv_vvv_vari
 
struct  include_summand
 Template metaprogram to calculate whether a summand needs to be included in a proportional (log) probability calculation. More...
 
struct  index_type
 Primary template class for the metaprogram to compute the index type of a container. More...
 
struct  index_type< const T >
 Template class for metaprogram to compute the type of indexes used in a constant container type. More...
 
struct  index_type< Eigen::Matrix< T, R, C > >
 Template metaprogram defining typedef for the type of index for an Eigen matrix, vector, or row vector. More...
 
struct  index_type< std::vector< T > >
 Template metaprogram class to compute the type of index for a standard vector. More...
 
class  LDLT_alloc
 This object stores the actual (double typed) LDLT factorization of an Eigen::Matrix<var> along with pointers to its vari's which allow the *_ldlt functions to save memory. More...
 
class  LDLT_factor
 
class  LDLT_factor< stan::math::var, R, C >
 A template specialization of src/stan/math/matrix/LDLT_factor.hpp for stan::math::var which can be used with all the *_ldlt functions. More...
 
class  LDLT_factor< T, R, C >
 LDLT_factor is a thin wrapper on Eigen::LDLT to allow for reusing factorizations and efficient autodiff of things like log determinants and solutions to linear systems. More...
 
class  op_ddv_vari
 
class  op_dv_vari
 
class  op_dvd_vari
 
class  op_dvv_vari
 
class  op_matrix_vari
 
class  op_v_vari
 
class  op_vd_vari
 
class  op_vdd_vari
 
class  op_vdv_vari
 
class  op_vector_vari
 
class  op_vv_vari
 
class  op_vvd_vari
 
class  op_vvv_vari
 
struct  OperandsAndPartials
 A variable implementation that stores operands and derivatives with respect to the variable. More...
 
class  partials_vari
 
struct  pass_type
 
struct  pass_type< double >
 
struct  pass_type< int >
 
class  precomp_v_vari
 
class  precomp_vv_vari
 
class  precomp_vvv_vari
 
class  precomputed_gradients_vari
 A variable implementation taking a sequence of operands and partial derivatives with respect to the operands. More...
 
struct  promote_scalar_struct
 General struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_struct< T, Eigen::Matrix< S, 1,-1 > >
 Struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_struct< T, Eigen::Matrix< S,-1, 1 > >
 Struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_struct< T, Eigen::Matrix< S,-1,-1 > >
 Struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_struct< T, std::vector< S > >
 Struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_struct< T, T >
 Struct to hold static function for promoting underlying scalar types. More...
 
struct  promote_scalar_type
 Template metaprogram to calculate a type for converting a convertible type. More...
 
struct  promote_scalar_type< T, Eigen::Matrix< S, 1, Eigen::Dynamic > >
 Template metaprogram to calculate a type for a row vector whose underlying scalar is converted from the second template parameter type to the first. More...
 
struct  promote_scalar_type< T, Eigen::Matrix< S, Eigen::Dynamic, Eigen::Dynamic > >
 Template metaprogram to calculate a type for a matrix whose underlying scalar is converted from the second template parameter type to the first. More...
 
struct  promote_scalar_type< T, Eigen::Matrix< S, Eigen::Dynamic, 1 > >
 Template metaprogram to calculate a type for a vector whose underlying scalar is converted from the second template parameter type to the first. More...
 
struct  promote_scalar_type< T, std::vector< S > >
 Template metaprogram to calculate a type for a container whose underlying scalar is converted from the second template parameter type to the first. More...
 
struct  promoter
 
struct  promoter< Eigen::Matrix< F, R, C >, Eigen::Matrix< T, R, C > >
 
struct  promoter< Eigen::Matrix< T, R, C >, Eigen::Matrix< T, R, C > >
 
struct  promoter< std::vector< F >, std::vector< T > >
 
struct  promoter< std::vector< T >, std::vector< T > >
 
struct  promoter< T, T >
 
class  seq_view
 
class  seq_view< double, std::vector< int > >
 
class  seq_view< T, Eigen::Matrix< S, 1, Eigen::Dynamic > >
 
class  seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, 1 > >
 
class  seq_view< T, Eigen::Matrix< S, Eigen::Dynamic, Eigen::Dynamic > >
 
class  seq_view< T, std::vector< S > >
 
class  seq_view< T, std::vector< std::vector< T > > >
 
class  seq_view< T, std::vector< T > >
 
class  stack_alloc
 An instance of this class provides a memory pool through which blocks of raw memory may be allocated and then collected simultaneously. More...
 
struct  store_type
 
struct  store_type< double >
 
struct  store_type< int >
 
class  stored_gradient_vari
 A var implementation that stores the daughter variable implementation pointers and the partial derivative with respect to the result explicitly in arrays constructed on the auto-diff memory stack. More...
 
class  sum_eigen_v_vari
 Class for representing sums with constructors for Eigen. More...
 
class  sum_v_vari
 Class for sums of variables constructed with standard vectors. More...
 
struct  value_type
 Primary template class for metaprogram to compute the type of values stored in a container. More...
 
struct  value_type< const T >
 Template class for metaprogram to compute the type of values stored in a constant container. More...
 
struct  value_type< Eigen::Matrix< T, R, C > >
 Template metaprogram defining the type of values stored in an Eigen matrix, vector, or row vector. More...
 
struct  value_type< std::vector< T > >
 Template metaprogram class to compute the type of values stored in a standard vector. More...
 
class  var
 Independent (input) and dependent (output) variables for gradients. More...
 
class  vari
 The variable implementation base class. More...
 
class  welford_covar_estimator
 
class  welford_var_estimator
 

Typedefs

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index size_type
 Type for sizes and indexes in an Eigen matrix with double e. More...
 
typedef Eigen::Matrix< fvar< double >, Eigen::Dynamic, Eigen::Dynamic > matrix_fd
 
typedef Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, Eigen::Dynamic > matrix_ffd
 
typedef Eigen::Matrix< fvar< double >, Eigen::Dynamic, 1 > vector_fd
 
typedef Eigen::Matrix< fvar< fvar< double > >, Eigen::Dynamic, 1 > vector_ffd
 
typedef Eigen::Matrix< fvar< double >, 1, Eigen::Dynamic > row_vector_fd
 
typedef Eigen::Matrix< fvar< fvar< double > >, 1, Eigen::Dynamic > row_vector_ffd
 
typedef Eigen::Matrix< fvar< var >, Eigen::Dynamic, Eigen::Dynamic > matrix_fv
 
typedef Eigen::Matrix< fvar< fvar< var > >, Eigen::Dynamic, Eigen::Dynamic > matrix_ffv
 
typedef Eigen::Matrix< fvar< var >, Eigen::Dynamic, 1 > vector_fv
 
typedef Eigen::Matrix< fvar< fvar< var > >, Eigen::Dynamic, 1 > vector_ffv
 
typedef Eigen::Matrix< fvar< var >, 1, Eigen::Dynamic > row_vector_fv
 
typedef Eigen::Matrix< fvar< fvar< var > >, 1, Eigen::Dynamic > row_vector_ffv
 
typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > matrix_d
 Type for matrix of double values. More...
 
typedef Eigen::Matrix< double, Eigen::Dynamic, 1 > vector_d
 Type for (column) vector of double values. More...
 
typedef Eigen::Matrix< double, 1, Eigen::Dynamic > row_vector_d
 Type for (row) vector of double values. More...
 
typedef AutodiffStackStorage< vari, chainable_allocChainableStack
 
typedef Eigen::Matrix< var, Eigen::Dynamic, Eigen::Dynamic > matrix_v
 The type of a matrix holding stan::math::var values. More...
 
typedef Eigen::Matrix< var, Eigen::Dynamic, 1 > vector_v
 The type of a (column) vector holding stan::math::var values. More...
 
typedef Eigen::Matrix< var, 1, Eigen::Dynamic > row_vector_v
 The type of a row vector holding stan::math::var values. More...
 

Functions

template<typename T >
fvar< T > log_sum_exp (const std::vector< fvar< T > > &v)
 
template<typename T >
fvar< T > sum (const std::vector< fvar< T > > &m)
 Return the sum of the entries of the specified standard vector. More...
 
template<typename T >
fvar< T > operator+ (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator+ (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator+ (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > operator/ (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator/ (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > operator/ (const double x1, const fvar< T > &x2)
 
template<typename T >
bool operator== (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator== (const fvar< T > &x, double y)
 
template<typename T >
bool operator== (double x, const fvar< T > &y)
 
template<typename T >
bool operator> (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator> (const fvar< T > &x, double y)
 
template<typename T >
bool operator> (double x, const fvar< T > &y)
 
template<typename T >
bool operator>= (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator>= (const fvar< T > &x, double y)
 
template<typename T >
bool operator>= (double x, const fvar< T > &y)
 
template<typename T >
bool operator< (const fvar< T > &x, double y)
 
template<typename T >
bool operator< (double x, const fvar< T > &y)
 
template<typename T >
bool operator< (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator<= (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator<= (const fvar< T > &x, double y)
 
template<typename T >
bool operator<= (double x, const fvar< T > &y)
 
template<typename T >
fvar< T > operator* (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator* (double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator* (const fvar< T > &x1, double x2)
 
template<typename T >
bool operator!= (const fvar< T > &x, const fvar< T > &y)
 
template<typename T >
bool operator!= (const fvar< T > &x, double y)
 
template<typename T >
bool operator!= (double x, const fvar< T > &y)
 
template<typename T >
fvar< T > operator- (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator- (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > operator- (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > operator- (const fvar< T > &x)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 > columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 > columns_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, 1, C1 > columns_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, 1, C > columns_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, C, C > crossprod (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T , int R, int C>
fvar< T > determinant (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T1 , typename T2 >
stan::return_type< T1, T2 >::type divide (const T1 &v, const T2 &c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > divide (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > divide (const Eigen::Matrix< fvar< T >, R, C > &v, const double c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > divide (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, const fvar< T > &c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > operator/ (const Eigen::Matrix< fvar< T >, R, C > &v, const double c)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > operator/ (const Eigen::Matrix< double, R, C > &v, const fvar< T > &c)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2, size_type &length)
 
template<typename T , int R1, int C1, int R2, int C2>
fvar< T > dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2, size_type &length)
 
template<typename T >
fvar< T > dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2)
 
template<typename T >
fvar< T > dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2)
 
template<typename T >
fvar< T > dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2)
 
template<typename T >
fvar< T > dot_product (const std::vector< fvar< T > > &v1, const std::vector< fvar< T > > &v2, size_type &length)
 
template<typename T >
fvar< T > dot_product (const std::vector< double > &v1, const std::vector< fvar< T > > &v2, size_type &length)
 
template<typename T >
fvar< T > dot_product (const std::vector< fvar< T > > &v1, const std::vector< double > &v2, size_type &length)
 
template<typename T , int R, int C>
fvar< T > dot_self (const Eigen::Matrix< fvar< T >, R, C > &v)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > inverse (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T , int R, int C>
fvar< T > log_determinant (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > log_softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)
 
template<typename T , int R, int C>
fvar< T > log_sum_exp (const Eigen::Matrix< fvar< T >, R, C > &v)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_left (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2, typename T2 >
Eigen::Matrix< fvar< T2 >, R1, C2 > mdivide_left_ldlt (const stan::math::LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< fvar< T2 >, R2, C2 > &b)
 Returns the solution of the system Ax=b given an LDLT_factor of A. More...
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 > mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 > mdivide_left_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 > mdivide_left_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_right (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_right (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C1 > mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_right_tri_low (const Eigen::Matrix< fvar< T >, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > mdivide_right_tri_low (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< fvar< T >, R2, C2 > &b)
 
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 > multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m, const fvar< T > &c)
 
template<typename T , int R2, int C2>
Eigen::Matrix< fvar< T >, R2, C2 > multiply (const Eigen::Matrix< fvar< T >, R2, C2 > &m, const double c)
 
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 > multiply (const Eigen::Matrix< double, R1, C1 > &m, const fvar< T > &c)
 
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 > multiply (const fvar< T > &c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)
 
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 > multiply (const double c, const Eigen::Matrix< fvar< T >, R1, C1 > &m)
 
template<typename T , int R1, int C1>
Eigen::Matrix< fvar< T >, R1, C1 > multiply (const fvar< T > &c, const Eigen::Matrix< double, R1, C1 > &m)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > multiply (const Eigen::Matrix< fvar< T >, R1, C1 > &m1, const Eigen::Matrix< double, R2, C2 > &m2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, C2 > multiply (const Eigen::Matrix< double, R1, C1 > &m1, const Eigen::Matrix< fvar< T >, R2, C2 > &m2)
 
template<typename T , int C1, int R2>
fvar< T > multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)
 
template<typename T , int C1, int R2>
fvar< T > multiply (const Eigen::Matrix< fvar< T >, 1, C1 > &rv, const Eigen::Matrix< double, R2, 1 > &v)
 
template<typename T , int C1, int R2>
fvar< T > multiply (const Eigen::Matrix< double, 1, C1 > &rv, const Eigen::Matrix< fvar< T >, R2, 1 > &v)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R > multiply_lower_tri_self_transpose (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > qr_Q (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)
 
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > qr_R (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &m)
 
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< fvar< T >, CB, CB > quad_form_sym (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
 
template<int RA, int CA, int RB, typename T >
fvar< T > quad_form_sym (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, 1 > &B)
 
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< fvar< T >, CB, CB > quad_form_sym (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
 
template<int RA, int CA, int RB, typename T >
fvar< T > quad_form_sym (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, 1 > &B)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 > rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 > rows_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< fvar< T >, R2, C2 > &v2)
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< fvar< T >, R1, 1 > rows_dot_product (const Eigen::Matrix< fvar< T >, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, 1 > rows_dot_self (const Eigen::Matrix< fvar< T >, R, C > &x)
 
template<typename T >
Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > softmax (const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &alpha)
 
template<typename T >
std::vector< fvar< T > > sort_asc (std::vector< fvar< T > > xs)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > sort_asc (Eigen::Matrix< fvar< T >, R, C > xs)
 
template<typename T >
std::vector< fvar< T > > sort_desc (std::vector< fvar< T > > xs)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > sort_desc (Eigen::Matrix< fvar< T >, R, C > xs)
 
template<typename T , int R, int C>
fvar< T > sum (const Eigen::Matrix< fvar< T >, R, C > &m)
 Return the sum of the entries of the specified matrix. More...
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, R > tcrossprod (const Eigen::Matrix< fvar< T >, R, C > &m)
 
template<typename T >
fvar< T > to_fvar (const T &x)
 
template<typename T >
fvar< T > to_fvar (const fvar< T > &x)
 
template<int R, int C, typename T >
Eigen::Matrix< T, R, C > to_fvar (const Eigen::Matrix< T, R, C > &m)
 
template<int R, int C>
Eigen::Matrix< fvar< double >, R, C > to_fvar (const Eigen::Matrix< double, R, C > &m)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > to_fvar (const Eigen::Matrix< T, R, C > &val, const Eigen::Matrix< T, R, C > &deriv)
 
template<int RD, int CD, int RA, int CA, int RB, int CB, typename T >
fvar< T > trace_gen_quad_form (const Eigen::Matrix< fvar< T >, RD, CD > &D, const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
 
template<int RA, int CA, int RB, int CB, typename T >
fvar< T > trace_quad_form (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
 
template<int RA, int CA, int RB, int CB, typename T >
fvar< T > trace_quad_form (const Eigen::Matrix< fvar< T >, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
 
template<int RA, int CA, int RB, int CB, typename T >
fvar< T > trace_quad_form (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< fvar< T >, RB, CB > &B)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > unit_vector_constrain (const Eigen::Matrix< fvar< T >, R, C > &y)
 
template<typename T , int R, int C>
Eigen::Matrix< fvar< T >, R, C > unit_vector_constrain (const Eigen::Matrix< fvar< T >, R, C > &y, fvar< T > &lp)
 
template<typename T , typename F >
void gradient (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad_fx)
 Calculate the value and the gradient of the specified function at the specified argument. More...
 
template<typename T , typename F >
void jacobian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, Eigen::Matrix< T, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &J)
 
template<typename T >
fvar< T > abs (const fvar< T > &x)
 
template<typename T >
fvar< T > acos (const fvar< T > &x)
 
template<typename T >
fvar< T > acosh (const fvar< T > &x)
 
template<typename T >
fvar< T > asin (const fvar< T > &x)
 
template<typename T >
fvar< T > asinh (const fvar< T > &x)
 
template<typename T >
fvar< T > atan (const fvar< T > &x)
 
template<typename T >
fvar< T > atan2 (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > atan2 (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > atan2 (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > atanh (const fvar< T > &x)
 
template<typename T >
fvar< T > bessel_first_kind (int v, const fvar< T > &z)
 
template<typename T >
fvar< T > bessel_second_kind (int v, const fvar< T > &z)
 
template<typename T >
fvar< T > binary_log_loss (const int y, const fvar< T > &y_hat)
 
template<typename T >
fvar< T > binomial_coefficient_log (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > binomial_coefficient_log (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > binomial_coefficient_log (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > cbrt (const fvar< T > &x)
 
template<typename T >
fvar< T > ceil (const fvar< T > &x)
 
template<typename T >
fvar< T > cos (const fvar< T > &x)
 
template<typename T >
fvar< T > cosh (const fvar< T > &x)
 
template<typename T >
fvar< T > digamma (const fvar< T > &x)
 
template<typename T >
fvar< T > erf (const fvar< T > &x)
 
template<typename T >
fvar< T > erfc (const fvar< T > &x)
 
template<typename T >
fvar< T > exp (const fvar< T > &x)
 
template<typename T >
fvar< T > exp2 (const fvar< T > &x)
 
template<typename T >
fvar< T > expm1 (const fvar< T > &x)
 
template<typename T >
fvar< T > fabs (const fvar< T > &x)
 
template<typename T >
fvar< T > falling_factorial (const fvar< T > &x, const fvar< T > &n)
 
template<typename T >
fvar< T > falling_factorial (const fvar< T > &x, const double n)
 
template<typename T >
fvar< T > falling_factorial (const double x, const fvar< T > &n)
 
template<typename T >
fvar< T > fdim (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fdim (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > fdim (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > floor (const fvar< T > &x)
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
 The fused multiply-add operation (C99). More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const T1 &x1, const fvar< T2 > &x2, const fvar< T3 > &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const fvar< T1 > &x1, const T2 &x2, const fvar< T3 > &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const fvar< T1 > &x1, const fvar< T2 > &x2, const T3 &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const T1 &x1, const T2 &x2, const fvar< T3 > &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const fvar< T1 > &x1, const T2 &x2, const T3 &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T1 , typename T2 , typename T3 >
fvar< typename stan::return_type< T1, T2, T3 >::type > fma (const T1 &x1, const fvar< T2 > &x2, const T3 &x3)
 See all-var input signature for details on the function and derivatives. More...
 
template<typename T >
fvar< T > fmax (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fmax (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fmax (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > fmin (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fmin (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fmin (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > fmod (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > fmod (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > fmod (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > gamma_p (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > gamma_p (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > gamma_p (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > gamma_q (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > gamma_q (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > gamma_q (const double x1, const fvar< T > &x2)
 
template<typename T >
void grad_inc_beta (stan::math::fvar< T > &g1, stan::math::fvar< T > &g2, stan::math::fvar< T > a, stan::math::fvar< T > b, stan::math::fvar< T > z)
 
template<typename T >
fvar< T > hypot (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > hypot (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > hypot (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > inc_beta (const fvar< T > &a, const fvar< T > &b, const fvar< T > &x)
 
template<typename T >
fvar< T > inv (const fvar< T > &x)
 
template<typename T >
fvar< T > inv_cloglog (const fvar< T > &x)
 
template<typename T >
fvar< T > inv_logit (const fvar< T > &x)
 
template<typename T >
fvar< T > inv_Phi (const fvar< T > &p)
 
template<typename T >
fvar< T > inv_sqrt (const fvar< T > &x)
 
template<typename T >
fvar< T > inv_square (const fvar< T > &x)
 
template<typename T >
int is_inf (const fvar< T > &x)
 Returns 1 if the input's value is infinite and 0 otherwise. More...
 
template<typename T >
int is_nan (const fvar< T > &x)
 Returns 1 if the input's value is NaN and 0 otherwise. More...
 
template<typename T >
fvar< T > lbeta (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > lbeta (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > lbeta (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > lgamma (const fvar< T > &x)
 
template<typename T >
fvar< typename stan::return_type< T, int >::type > lmgamma (int x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > log (const fvar< T > &x)
 
template<typename T >
fvar< T > log10 (const fvar< T > &x)
 
template<typename T >
fvar< T > log1m (const fvar< T > &x)
 
template<typename T >
fvar< T > log1m_exp (const fvar< T > &x)
 
template<typename T >
fvar< T > log1m_inv_logit (const fvar< T > &x)
 
template<typename T >
fvar< T > log1p (const fvar< T > &x)
 
template<typename T >
fvar< T > log1p_exp (const fvar< T > &x)
 
template<typename T >
fvar< T > log2 (const fvar< T > &x)
 
template<typename T >
fvar< T > log_diff_exp (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T1 , typename T2 >
fvar< T2 > log_diff_exp (const T1 &x1, const fvar< T2 > &x2)
 
template<typename T1 , typename T2 >
fvar< T1 > log_diff_exp (const fvar< T1 > &x1, const T2 &x2)
 
template<typename T >
fvar< T > log_falling_factorial (const fvar< T > &x, const fvar< T > &n)
 
template<typename T >
fvar< T > log_falling_factorial (const double x, const fvar< T > &n)
 
template<typename T >
fvar< T > log_falling_factorial (const fvar< T > &x, const double n)
 
template<typename T >
fvar< T > log_inv_logit (const fvar< T > &x)
 
template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>
void log_mix_partial_helper (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2, typename boost::math::tools::promote_args< T_theta, T_lambda1, T_lambda2 >::type(&partials_array)[N])
 
template<typename T >
fvar< T > log_mix (const fvar< T > &theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
 Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
 
template<typename T >
fvar< T > log_mix (const fvar< T > &theta, const fvar< T > &lambda1, const double lambda2)
 
template<typename T >
fvar< T > log_mix (const fvar< T > &theta, const double lambda1, const fvar< T > &lambda2)
 
template<typename T >
fvar< T > log_mix (const double theta, const fvar< T > &lambda1, const fvar< T > &lambda2)
 
template<typename T >
fvar< T > log_mix (const fvar< T > &theta, const double lambda1, const double lambda2)
 
template<typename T >
fvar< T > log_mix (const double theta, const fvar< T > &lambda1, const double lambda2)
 
template<typename T >
fvar< T > log_mix (const double theta, const double lambda1, const fvar< T > &lambda2)
 
template<typename T >
fvar< T > log_rising_factorial (const fvar< T > &x, const fvar< T > &n)
 
template<typename T >
fvar< T > log_rising_factorial (const fvar< T > &x, const double n)
 
template<typename T >
fvar< T > log_rising_factorial (const double x, const fvar< T > &n)
 
template<typename T >
fvar< T > log_sum_exp (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > log_sum_exp (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > log_sum_exp (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > logit (const fvar< T > &x)
 
template<typename T >
fvar< T > modified_bessel_first_kind (int v, const fvar< T > &z)
 
template<typename T >
fvar< T > modified_bessel_second_kind (int v, const fvar< T > &z)
 
template<typename T >
fvar< T > multiply_log (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > multiply_log (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > multiply_log (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > owens_t (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > owens_t (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > owens_t (const fvar< T > &x1, const double x2)
 
template<typename T >
fvar< T > Phi (const fvar< T > &x)
 
template<typename T >
fvar< T > pow (const fvar< T > &x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > pow (const double x1, const fvar< T > &x2)
 
template<typename T >
fvar< T > pow (const fvar< T > &x1, const double x2)
 
template<typename T >
double primitive_value (const fvar< T > &v)
 Return the primitive value of the specified forward-mode autodiff variable. More...
 
template<typename T >
fvar< T > rising_factorial (const fvar< T > &x, const fvar< T > &n)
 
template<typename T >
fvar< T > rising_factorial (const fvar< T > &x, const double n)
 
template<typename T >
fvar< T > rising_factorial (const double x, const fvar< T > &n)
 
template<typename T >
fvar< T > round (const fvar< T > &x)
 
template<typename T >
fvar< T > sin (const fvar< T > &x)
 
template<typename T >
fvar< T > sinh (const fvar< T > &x)
 
template<typename T >
fvar< T > sqrt (const fvar< T > &x)
 
template<typename T >
fvar< T > square (const fvar< T > &x)
 
template<typename T >
fvar< T > tan (const fvar< T > &x)
 
template<typename T >
fvar< T > tanh (const fvar< T > &x)
 
template<typename T >
fvar< T > tgamma (const fvar< T > &x)
 
template<typename T >
fvar< T > trunc (const fvar< T > &x)
 
template<typename T >
value_of (const fvar< T > &v)
 Return the value of the specified variable. More...
 
template<typename T >
double value_of_rec (const fvar< T > &v)
 Return the value of the specified variable. More...
 
template<typename T >
bool is_aligned (T *ptr, unsigned int bytes_aligned)
 Return true if the specified pointer is aligned on the number of bytes. More...
 
template<typename T , typename F >
void derivative (const F &f, const T &x, T &fx, T &dfx_dx)
 Return the derivative of the specified univariate function at the specified argument. More...
 
template<typename F >
void finite_diff_grad_hessian (const F &f, const Eigen::Matrix< double,-1, 1 > &x, double &fx, Eigen::Matrix< double,-1,-1 > &hess, std::vector< Eigen::Matrix< double,-1,-1 > > &grad_hess_fx, const double epsilon=1e-04)
 Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference. More...
 
template<typename F >
void grad_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H, std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &grad_H)
 Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument. More...
 
template<typename F >
void grad_tr_mat_times_hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &M, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_tr_MH)
 
template<typename T1 , typename T2 , typename F >
void gradient_dot_vector (const F &f, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &v, T1 &fx, T1 &grad_fx_dot_v)
 
template<typename F >
void hessian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &H)
 Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space. More...
 
template<typename T , typename F >
void hessian (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &grad, Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &H)
 
template<typename F >
void hessian_times_vector (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &v, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &Hv)
 
template<typename T , typename F >
void hessian_times_vector (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, T &fx, Eigen::Matrix< T, Eigen::Dynamic, 1 > &Hv)
 
template<typename T , typename F >
void partial_derivative (const F &f, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int n, T &fx, T &dfx_dxn)
 Return the partial derivative of the specified multiivariate function at the specified argument. More...
 
double dist (const std::vector< double > &x, const std::vector< double > &y)
 
double dot (const std::vector< double > &x, const std::vector< double > &y)
 
double dot_self (const std::vector< double > &x)
 
double log_sum_exp (const std::vector< double > &x)
 Return the log of the sum of the exponentiated values of the specified sequence of values. More...
 
template<typename T >
std::vector< T > rep_array (const T &x, int n)
 
template<typename T >
std::vector< std::vector< T > > rep_array (const T &x, int m, int n)
 
template<typename T >
std::vector< std::vector< std::vector< T > > > rep_array (const T &x, int k, int m, int n)
 
void scaled_add (std::vector< double > &x, const std::vector< double > &y, const double lambda)
 
void sub (std::vector< double > &x, std::vector< double > &y, std::vector< double > &result)
 
template<typename T >
sum (const std::vector< T > &xs)
 Return the sum of the values in the specified standard vector. More...
 
template<typename F , typename T1 , typename T2 >
std::vector< std::vector< typename stan::return_type< T1, T2 >::type > > integrate_ode (const F &f, const std::vector< T1 > y0, const double t0, const std::vector< double > &ts, const std::vector< T2 > &theta, const std::vector< double > &x, const std::vector< int > &x_int, std::ostream *msgs)
 Return the solutions for the specified system of ordinary differential equations given the specified initial state, initial times, times of desired solution, and parameters and data, writing error and warning messages to the specified stream. More...
 
template<typename T_y >
bool check_cholesky_factor (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is a valid Cholesky factor. More...
 
template<typename T_y >
bool check_cholesky_factor_corr (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is a valid Cholesky factor of a correlation matrix. More...
 
template<typename T_y , int R, int C>
bool check_column_index (const char *function, const char *name, const Eigen::Matrix< T_y, R, C > &y, const size_t i)
 Return true if the specified index is a valid column of the matrix. More...
 
template<typename T_y >
bool check_corr_matrix (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is a valid correlation matrix. More...
 
template<typename T_y >
bool check_cov_matrix (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is a valid covariance matrix. More...
 
template<typename T , int R, int C>
bool check_ldlt_factor (const char *function, const char *name, stan::math::LDLT_factor< T, R, C > &A)
 Return true if the argument is a valid stan::math::LDLT_factor. More...
 
template<typename T_y >
bool check_lower_triangular (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is lower triangular. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
bool check_matching_dims (const char *function, const char *name1, const Eigen::Matrix< T1, R1, C1 > &y1, const char *name2, const Eigen::Matrix< T2, R2, C2 > &y2)
 Return true if the two matrices are of the same size. More...
 
template<typename T_y1 , typename T_y2 >
bool check_matching_sizes (const char *function, const char *name1, const T_y1 &y1, const char *name2, const T_y2 &y2)
 Return true if two structures at the same size. More...
 
template<typename T1 , typename T2 >
bool check_multiplicable (const char *function, const char *name1, const T1 &y1, const char *name2, const T2 &y2)
 Return true if the matrices can be multiplied. More...
 
template<typename T_y >
bool check_ordered (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &y)
 Return true if the specified vector is sorted into strictly increasing order. More...
 
template<typename T_y >
bool check_ordered (const char *function, const char *name, const std::vector< T_y > &y)
 Return true if the specified vector is sorted into strictly increasing order. More...
 
template<typename T_y >
bool check_pos_definite (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified square, symmetric matrix is positive definite. More...
 
template<typename Derived >
bool check_pos_definite (const char *function, const char *name, const Eigen::LDLT< Derived > &cholesky)
 Return true if the specified LDLT transform of a matrix is positive definite. More...
 
template<typename Derived >
bool check_pos_definite (const char *function, const char *name, const Eigen::LLT< Derived > &cholesky)
 Return true if the specified LLT transform of a matrix is positive definite. More...
 
template<typename T_y >
bool check_pos_semidefinite (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is positive definite. More...
 
template<typename T_y >
bool check_positive_ordered (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &y)
 Return true if the specified vector contains non-negative values and is sorted into strictly increasing order. More...
 
bool check_range (const char *function, const char *name, const int max, const int index, const int nested_level, const char *error_msg)
 Return true if specified index is within range. More...
 
bool check_range (const char *function, const char *name, const int max, const int index, const char *error_msg)
 Return true if specified index is within range. More...
 
bool check_range (const char *function, const char *name, const int max, const int index)
 Return true if specified index is within range. More...
 
template<typename T_y , int R, int C>
bool check_row_index (const char *function, const char *name, const Eigen::Matrix< T_y, R, C > &y, size_t i)
 Return true if the specified index is a valid row of the matrix. More...
 
template<typename T_prob >
bool check_simplex (const char *function, const char *name, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 Return true if the specified vector is simplex. More...
 
template<typename T_y >
bool check_spsd_matrix (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is a square, symmetric, and positive semi-definite. More...
 
template<typename T_y >
bool check_square (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is square. More...
 
template<typename T >
bool check_std_vector_index (const char *function, const char *name, const std::vector< T > &y, int i)
 Return true if the specified index is valid in std vector. More...
 
template<typename T_y >
bool check_symmetric (const char *function, const char *name, const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return true if the specified matrix is symmetric. More...
 
template<typename T_prob >
bool check_unit_vector (const char *function, const char *name, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 Return true if the specified vector is unit vector. More...
 
template<typename T , int R, int C>
bool check_vector (const char *function, const char *name, const Eigen::Matrix< T, R, C > &x)
 Return true if the matrix is either a row vector or column vector. More...
 
void validate_non_negative_index (const char *var_name, const char *expr, int val)
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > add (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
 Return the sum of the specified matrices. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > add (const Eigen::Matrix< T1, R, C > &m, const T2 &c)
 Return the sum of the specified matrix and specified scalar. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > add (const T1 &c, const Eigen::Matrix< T2, R, C > &m)
 Return the sum of the specified scalar and specified matrix. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename return_type< T1, T2 >::type, Eigen::Dynamic, Eigen::Dynamic > append_col (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &B)
 Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
 
template<typename T1 , typename T2 , int C1, int C2>
Eigen::Matrix< typename return_type< T1, T2 >::type, 1, Eigen::Dynamic > append_col (const Eigen::Matrix< T1, 1, C1 > &A, const Eigen::Matrix< T2, 1, C2 > &B)
 Return the result of concatenaing the first row vector followed by the second row vector side by side, with the result being a row vector. More...
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > append_col (const Eigen::Matrix< T, R1, C1 > &A, const Eigen::Matrix< T, R2, C2 > &B)
 Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix. More...
 
template<typename T , int C1, int C2>
Eigen::Matrix< T, 1, Eigen::Dynamic > append_col (const Eigen::Matrix< T, 1, C1 > &A, const Eigen::Matrix< T, 1, C2 > &B)
 Return the result of concatenaing the first row vector followed by the second row vector side by side, with the result being a row vector. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename return_type< T1, T2 >::type, Eigen::Dynamic, Eigen::Dynamic > append_row (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &B)
 Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
 
template<typename T1 , typename T2 , int R1, int R2>
Eigen::Matrix< typename return_type< T1, T2 >::type, Eigen::Dynamic, 1 > append_row (const Eigen::Matrix< T1, R1, 1 > &A, const Eigen::Matrix< T2, R2, 1 > &B)
 Return the result of stacking the first vector on top of the second vector, with the result being a vector. More...
 
template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > append_row (const Eigen::Matrix< T, R1, C1 > &A, const Eigen::Matrix< T, R2, C2 > &B)
 Return the result of stacking the rows of the first argument matrix on top of the second argument matrix. More...
 
template<typename T , int R1, int R2>
Eigen::Matrix< T, Eigen::Dynamic, 1 > append_row (const Eigen::Matrix< T, R1, 1 > &A, const Eigen::Matrix< T, R2, 1 > &B)
 Return the result of stacking the first vector on top of the second vector, with the result being a vector. More...
 
void print_mat_size (int n, std::ostream &o)
 Helper function to return the matrix size as either "dynamic" or "1". More...
 
template<typename LHS , typename RHS >
void assign (LHS &lhs, const RHS &rhs)
 Copy the right-hand side's value to the left-hand side variable. More...
 
template<typename LHS , typename RHS , int R1, int C1, int R2, int C2>
void assign (Eigen::Matrix< LHS, R1, C1 > &x, const Eigen::Matrix< RHS, R2, C2 > &y)
 Copy the right-hand side's value to the left-hand side variable. More...
 
template<typename LHS , typename RHS , int R, int C>
void assign (Eigen::Matrix< LHS, R, C > &x, const Eigen::Matrix< RHS, R, C > &y)
 Copy the right-hand side's value to the left-hand side variable. More...
 
template<typename LHS , typename RHS , int R, int C>
void assign (Eigen::Block< LHS > x, const Eigen::Matrix< RHS, R, C > &y)
 Copy the right-hand side's value to the left-hand side variable. More...
 
template<typename LHS , typename RHS >
void assign (std::vector< LHS > &x, const std::vector< RHS > &y)
 Copy the right-hand side's value to the left-hand side variable. More...
 
template<typename T >
void autocorrelation (const std::vector< T > &y, std::vector< T > &ac, Eigen::FFT< T > &fft)
 Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
 
template<typename T >
void autocorrelation (const std::vector< T > &y, std::vector< T > &ac)
 Write autocorrelation estimates for every lag for the specified input sequence into the specified result. More...
 
template<typename T >
void autocovariance (const std::vector< T > &y, std::vector< T > &acov, Eigen::FFT< T > &fft)
 Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine. More...
 
template<typename T >
void autocovariance (const std::vector< T > &y, std::vector< T > &acov)
 Write autocovariance estimates for every lag for the specified input sequence into the specified result. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > block (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t nrows, size_t ncols)
 Return a nrows x ncols submatrix starting at (i-1, j-1). More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cholesky_corr_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, int K)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cholesky_corr_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, int K, T &lp)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > cholesky_corr_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cholesky_decompose (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cholesky_factor_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int M, int N)
 Return the Cholesky factor of the specified size read from the specified vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cholesky_factor_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, int M, int N, T &lp)
 Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > cholesky_factor_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return the unconstrained vector of parameters correspdonding to the specified Cholesky factor. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > col (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t j)
 Return the specified column of the specified matrix using start-at-1 indexing. More...
 
template<typename T , int R, int C>
int cols (const Eigen::Matrix< T, R, C > &m)
 Return the number of columns in the specified matrix, vector, or row vector. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, 1, C1 > columns_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 Returns the dot product of the specified vectors. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, 1, C > columns_dot_self (const Eigen::Matrix< T, R, C > &x)
 Returns the dot product of each column of a matrix with itself. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > corr_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type k)
 Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > corr_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type k, T &lp)
 Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > corr_matrix_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cov_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type K)
 Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cov_matrix_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > >::type K, T &lp)
 Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2). More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cov_matrix_constrain_lkj (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t k)
 Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > cov_matrix_constrain_lkj (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t k, T &lp)
 Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > cov_matrix_free (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
 The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2). More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > cov_matrix_free_lkj (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &y)
 Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix. More...
 
matrix_d crossprod (const matrix_d &M)
 Returns the result of pre-multiplying a matrix by its own transpose. More...
 
template<typename T >
const std::vector< int > csr_extract_u (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
 Extract the NZE index for each entry from a sparse matrix. More...
 
template<typename T , int R, int C>
const std::vector< int > csr_extract_u (const Eigen::Matrix< T, R, C > &A)
 Extract the NZE index for each entry from a sparse matrix. More...
 
template<typename T >
const std::vector< int > csr_extract_v (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
 Extract the column indexes for non-zero value from a sparse matrix. More...
 
template<typename T , int R, int C>
const std::vector< int > csr_extract_v (const Eigen::Matrix< T, R, C > &A)
 Extract the column indexes for non-zero values from a dense matrix by converting to sparse and calling the sparse matrix extractor. More...
 
template<typename T >
const Eigen::Matrix< T, Eigen::Dynamic, 1 > csr_extract_w (const Eigen::SparseMatrix< T, Eigen::RowMajor > &A)
 
template<typename T , int R, int C>
const Eigen::Matrix< T, Eigen::Dynamic, 1 > csr_extract_w (const Eigen::Matrix< T, R, C > &A)
 
template<typename T1 , typename T2 >
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, Eigen::Dynamic, 1 > csr_matrix_times_vector (const int &m, const int &n, const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &w, const std::vector< int > &v, const std::vector< int > &u, const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &b)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > csr_to_dense_matrix (const int &m, const int &n, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &w, const std::vector< int > &v, const std::vector< int > &u)
 Construct a dense Eigen matrix from the CSR format components. More...
 
int csr_u_to_z (const std::vector< int > &u, int i)
 Return the z vector computed from the specified u vector at the index for the z vector. More...
 
template<typename T >
std::vector< T > cumulative_sum (const std::vector< T > &x)
 Return the cumulative sum of the specified vector. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > cumulative_sum (const Eigen::Matrix< T, R, C > &m)
 Return the cumulative sum of the specified matrix. More...
 
template<typename T , int R, int C>
determinant (const Eigen::Matrix< T, R, C > &m)
 Returns the determinant of the specified square matrix. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > diag_matrix (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
 Return a square diagonal matrix with the specified vector of coefficients as the diagonal values. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C1 > diag_post_multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R2, C2 > diag_pre_multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > diagonal (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Return a column vector of the diagonal elements of the specified matrix. More...
 
template<typename T >
void dims (const T &x, std::vector< int > &result)
 
template<typename T , int R, int C>
void dims (const Eigen::Matrix< T, R, C > &x, std::vector< int > &result)
 
template<typename T >
void dims (const std::vector< T > &x, std::vector< int > &result)
 
template<typename T >
std::vector< int > dims (const T &x)
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::math::tools::promote_args< T1, T2 >::type distance (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
 Returns the distance between the specified vectors. More...
 
template<int R, int C, typename T >
boost::enable_if_c< boost::is_arithmetic< T >::value, Eigen::Matrix< double, R, C > >::type divide (const Eigen::Matrix< double, R, C > &m, T c)
 Return specified matrix divided by specified scalar. More...
 
template<int R1, int C1, int R2, int C2>
double dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 Returns the dot product of the specified vectors. More...
 
double dot_product (const double *v1, const double *v2, size_t length)
 Returns the dot product of the specified arrays of doubles. More...
 
double dot_product (const std::vector< double > &v1, const std::vector< double > &v2)
 Returns the dot product of the specified arrays of doubles. More...
 
template<int R, int C>
double dot_self (const Eigen::Matrix< double, R, C > &v)
 Returns the dot product of the specified vector with itself. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > eigenvalues_sym (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Return the eigenvalues of the specified symmetric matrix in descending order of magnitude. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > eigenvectors_sym (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > elt_divide (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
 Return the elementwise division of the specified matrices. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > elt_divide (const Eigen::Matrix< T1, R, C > &m, T2 s)
 Return the elementwise division of the specified matrix by the specified scalar. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > elt_divide (T1 s, const Eigen::Matrix< T2, R, C > &m)
 Return the elementwise division of the specified scalar by the specified matrix. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > elt_multiply (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
 Return the elementwise multiplication of the specified matrices. More...
 
template<typename T , int Rows, int Cols>
Eigen::Matrix< T, Rows, Cols > exp (const Eigen::Matrix< T, Rows, Cols > &m)
 Return the element-wise exponentiation of the matrix or vector. More...
 
template<int Rows, int Cols>
Eigen::Matrix< double, Rows, Cols > exp (const Eigen::Matrix< double, Rows, Cols > &m)
 
template<typename T >
bool factor_cov_matrix (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &Sigma, Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, Eigen::Array< T, Eigen::Dynamic, 1 > &sds)
 This function is intended to make starting values, given a covariance matrix Sigma. More...
 
template<typename T >
void factor_U (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &U, Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs)
 This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix. More...
 
template<typename T , typename S >
void fill (T &x, const S &y)
 Fill the specified container with the specified value. More...
 
template<typename T , int R, int C, typename S >
void fill (Eigen::Matrix< T, R, C > &x, const S &y)
 Fill the specified container with the specified value. More...
 
template<typename T , typename S >
void fill (std::vector< T > &x, const S &y)
 Fill the specified container with the specified value. More...
 
template<typename T >
const T & get_base1 (const std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one index. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
const T & get_base1 (const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, const char *error_msg, size_t idx)
 Return a copy of the row of the specified vector at the specified base-one row index. More...
 
template<typename T >
const T & get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, size_t n, const char *error_msg, size_t idx)
 Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
 
template<typename T >
const T & get_base1 (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t m, const char *error_msg, size_t idx)
 Return a reference to the value of the specified column vector at the specified base-one index. More...
 
template<typename T >
const T & get_base1 (const Eigen::Matrix< T, 1, Eigen::Dynamic > &x, size_t n, const char *error_msg, size_t idx)
 Return a reference to the value of the specified row vector at the specified base-one index. More...
 
template<typename T >
T & get_base1_lhs (std::vector< T > &x, size_t i, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one index. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< T > > &x, size_t i1, size_t i2, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< T > > > &x, size_t i1, size_t i2, size_t i3, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< std::vector< T > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
T & get_base1_lhs (std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &x, size_t i1, size_t i2, size_t i3, size_t i4, size_t i5, size_t i6, size_t i7, size_t i8, const char *error_msg, size_t idx)
 Return a reference to the value of the specified vector at the specified base-one indexes. More...
 
template<typename T >
Eigen::Block< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > > get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, const char *error_msg, size_t idx)
 Return a copy of the row of the specified vector at the specified base-one row index. More...
 
template<typename T >
T & get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x, size_t m, size_t n, const char *error_msg, size_t idx)
 Return a reference to the value of the specified matrix at the specified base-one row and column indexes. More...
 
template<typename T >
T & get_base1_lhs (Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, size_t m, const char *error_msg, size_t idx)
 Return a reference to the value of the specified column vector at the specified base-one index. More...
 
template<typename T >
T & get_base1_lhs (Eigen::Matrix< T, 1, Eigen::Dynamic > &x, size_t n, const char *error_msg, size_t idx)
 Return a reference to the value of the specified row vector at the specified base-one index. More...
 
template<typename T_lp , typename T_lp_accum >
boost::math::tools::promote_args< T_lp, T_lp_accum >::type get_lp (const T_lp &lp, const stan::math::accumulator< T_lp_accum > &lp_accum)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > head (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t n)
 Return the specified number of elements as a vector from the front of the specified vector. More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > head (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, size_t n)
 Return the specified number of elements as a row vector from the front of the specified row vector. More...
 
template<typename T >
std::vector< T > head (const std::vector< T > &sv, size_t n)
 Return the specified number of elements as a standard vector from the front of the specified standard vector. More...
 
template<typename T >
void initialize (T &x, const T &v)
 
template<typename T , typename V >
boost::enable_if_c< boost::is_arithmetic< V >::value, void >::type initialize (T &x, V v)
 
template<typename T , int R, int C, typename V >
void initialize (Eigen::Matrix< T, R, C > &x, const V &v)
 
template<typename T , typename V >
void initialize (std::vector< T > &x, const V &v)
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > inverse (const Eigen::Matrix< T, R, C > &m)
 Returns the inverse of the specified matrix. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > inverse_spd (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Returns the inverse of the specified symmetric, pos/neg-definite matrix. More...
 
template<typename T , int Rows, int Cols>
Eigen::Matrix< T, Rows, Cols > log (const Eigen::Matrix< T, Rows, Cols > &m)
 Return the element-wise logarithm of the matrix or vector. More...
 
template<typename T , int R, int C>
log_determinant (const Eigen::Matrix< T, R, C > &m)
 Returns the log absolute determinant of the specified square matrix. More...
 
template<int R, int C, typename T >
log_determinant_ldlt (stan::math::LDLT_factor< T, R, C > &A)
 
template<typename T , int R, int C>
log_determinant_spd (const Eigen::Matrix< T, R, C > &m)
 Returns the log absolute determinant of the specified square matrix. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > log_softmax (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
 Return the natural logarithm of the softmax of the specified vector. More...
 
template<int R, int C>
double log_sum_exp (const Eigen::Matrix< double, R, C > &x)
 Return the log of the sum of the exponentiated values of the specified matrix of values. More...
 
template<typename T >
const Eigen::Array< T, Eigen::Dynamic, 1 > make_nu (const T eta, const size_t K)
 This function calculates the degrees of freedom for the t distribution that corresponds to the shape parameter in the Lewandowski et. More...
 
int max (const std::vector< int > &x)
 Returns the maximum coefficient in the specified column vector. More...
 
template<typename T >
max (const std::vector< T > &x)
 Returns the maximum coefficient in the specified column vector. More...
 
template<typename T , int R, int C>
max (const Eigen::Matrix< T, R, C > &m)
 Returns the maximum coefficient in the specified vector, row vector, or matrix. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_left (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
 Returns the solution of the system Ax=b. More...
 
template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_left_ldlt (const stan::math::LDLT_factor< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
 Returns the solution of the system Ax=b given an LDLT_factor of A. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_left_spd (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
 Returns the solution of the system Ax=b where A is symmetric positive definite. More...
 
template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_left_tri (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
 Returns the solution of the system Ax=b when A is triangular. More...
 
template<int TriView, typename T , int R1, int C1>
Eigen::Matrix< T, R1, C1 > mdivide_left_tri (const Eigen::Matrix< T, R1, C1 > &A)
 Returns the solution of the system Ax=b when A is triangular and b=I. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_left_tri_low (const Eigen::Matrix< T1, R1, C1 > &A, const Eigen::Matrix< T2, R2, C2 > &b)
 
template<typename T , int R1, int C1>
Eigen::Matrix< T, R1, C1 > mdivide_left_tri_low (const Eigen::Matrix< T, R1, C1 > &A)
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_right (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
 Returns the solution of the system Ax=b. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_right_ldlt (const Eigen::Matrix< T1, R1, C1 > &b, const stan::math::LDLT_factor< T2, R2, C2 > &A)
 Returns the solution of the system xA=b given an LDLT_factor of A. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, C2 > mdivide_right_ldlt (const Eigen::Matrix< double, R1, C1 > &b, const stan::math::LDLT_factor< double, R2, C2 > &A)
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_right_spd (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
 Returns the solution of the system Ax=b where A is symmetric positive definite. More...
 
template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_right_tri (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
 Returns the solution of the system Ax=b when A is triangular. More...
 
template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R1, C2 > mdivide_right_tri_low (const Eigen::Matrix< T1, R1, C1 > &b, const Eigen::Matrix< T2, R2, C2 > &A)
 Returns the solution of the system tri(A)x=b when tri(A) is a lower triangular view of the matrix A. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type mean (const std::vector< T > &v)
 Returns the sample mean (i.e., average) of the coefficients in the specified standard vector. More...
 
template<typename T , int R, int C>
boost::math::tools::promote_args< T >::type mean (const Eigen::Matrix< T, R, C > &m)
 Returns the sample mean (i.e., average) of the coefficients in the specified vector, row vector, or matrix. More...
 
int min (const std::vector< int > &x)
 Returns the minimum coefficient in the specified column vector. More...
 
template<typename T >
min (const std::vector< T > &x)
 Returns the minimum coefficient in the specified column vector. More...
 
template<typename T , int R, int C>
min (const Eigen::Matrix< T, R, C > &m)
 Returns the minimum coefficient in the specified matrix, vector, or row vector. More...
 
template<typename T >
minus (const T &x)
 Returns the negation of the specified scalar or matrix. More...
 
template<int R, int C, typename T >
boost::enable_if_c< boost::is_arithmetic< T >::value, Eigen::Matrix< double, R, C > >::type multiply (const Eigen::Matrix< double, R, C > &m, T c)
 Return specified matrix multiplied by specified scalar. More...
 
template<int R, int C, typename T >
boost::enable_if_c< boost::is_arithmetic< T >::value, Eigen::Matrix< double, R, C > >::type multiply (T c, const Eigen::Matrix< double, R, C > &m)
 Return specified scalar multiplied by specified matrix. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, C2 > multiply (const Eigen::Matrix< double, R1, C1 > &m1, const Eigen::Matrix< double, R2, C2 > &m2)
 Return the product of the specified matrices. More...
 
template<int C1, int R2>
double multiply (const Eigen::Matrix< double, 1, C1 > &rv, const Eigen::Matrix< double, R2, 1 > &v)
 Return the scalar product of the specified row vector and specified column vector. More...
 
matrix_d multiply_lower_tri_self_transpose (const matrix_d &L)
 Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose. More...
 
template<typename T >
int num_elements (const T &x)
 Returns 1, the number of elements in a primitive type. More...
 
template<typename T , int R, int C>
int num_elements (const Eigen::Matrix< T, R, C > &m)
 Returns the size of the specified matrix. More...
 
template<typename T >
int num_elements (const std::vector< T > &v)
 Returns the number of elements in the specified vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
 Return an increasing ordered vector derived from the specified free vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &lp)
 Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > ordered_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
 Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > positive_ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
 Return an increasing positive ordered vector derived from the specified free vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > positive_ordered_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x, T &lp)
 Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > positive_ordered_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
 Return the vector of unconstrained scalars that transform to the specified positive ordered vector. More...
 
template<typename T >
prod (const std::vector< T > &v)
 Returns the product of the coefficients of the specified standard vector. More...
 
template<typename T , int R, int C>
prod (const Eigen::Matrix< T, R, C > &v)
 Returns the product of the coefficients of the specified column vector. More...
 
template<typename T1 , typename T2 , typename F >
common_type< T1, T2 >::type promote_common (const F &u)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > qr_Q (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > qr_R (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< T, CB, CB > quad_form (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, CB > &B)
 Compute B^T A B. More...
 
template<int RA, int CA, int RB, typename T >
quad_form (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, 1 > &B)
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, Eigen::Dynamic, Eigen::Dynamic > quad_form_diag (const Eigen::Matrix< T1, Eigen::Dynamic, Eigen::Dynamic > &mat, const Eigen::Matrix< T2, R, C > &vec)
 
template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix< T, CB, CB > quad_form_sym (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, CB > &B)
 
template<int RA, int CA, int RB, typename T >
quad_form_sym (const Eigen::Matrix< T, RA, CA > &A, const Eigen::Matrix< T, RB, 1 > &B)
 
template<typename T >
int rank (const std::vector< T > &v, int s)
 Return the number of components of v less than v[s]. More...
 
template<typename T , int R, int C>
int rank (const Eigen::Matrix< T, R, C > &v, int s)
 Return the number of components of v less than v[s]. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_corr_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const size_t K)
 Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_corr_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const size_t K, T &log_prob)
 Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_corr_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const size_t K)
 Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_corr_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const size_t K, T &log_prob)
 Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_cov_L (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds, T &log_prob)
 This is the function that should be called prior to evaluating the density of any elliptical distribution. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_cov_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds, T &log_prob)
 A generally worse alternative to call prior to evaluating the density of an elliptical distribution. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > read_cov_matrix (const Eigen::Array< T, Eigen::Dynamic, 1 > &CPCs, const Eigen::Array< T, Eigen::Dynamic, 1 > &sds)
 Builds a covariance matrix from CPCs and standard deviations. More...
 
template<typename T >
Eigen::Matrix< typename boost::math::tools::promote_args< T >::type, Eigen::Dynamic, Eigen::Dynamic > rep_matrix (const T &x, int m, int n)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > rep_matrix (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, int n)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > rep_matrix (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, int m)
 
template<typename T >
Eigen::Matrix< typename boost::math::tools::promote_args< T >::type, 1, Eigen::Dynamic > rep_row_vector (const T &x, int m)
 
template<typename T >
Eigen::Matrix< typename boost::math::tools::promote_args< T >::type, Eigen::Dynamic, 1 > rep_vector (const T &x, int n)
 
template<typename T >
void resize (T &x, std::vector< size_t > dims)
 Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices. More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > row (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i)
 Return the specified row of the specified matrix, using start-at-1 indexing. More...
 
template<typename T , int R, int C>
int rows (const Eigen::Matrix< T, R, C > &m)
 Return the number of rows in the specified matrix, vector, or row vector. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< double, R1, 1 > rows_dot_product (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 Returns the dot product of the specified vectors. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, 1 > rows_dot_self (const Eigen::Matrix< T, R, C > &x)
 Returns the dot product of each row of a matrix with itself. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type sd (const std::vector< T > &v)
 Returns the unbiased sample standard deviation of the coefficients in the specified column vector. More...
 
template<typename T , int R, int C>
boost::math::tools::promote_args< T >::type sd (const Eigen::Matrix< T, R, C > &m)
 Returns the unbiased sample standard deviation of the coefficients in the specified vector, row vector, or matrix. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > segment (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t i, size_t n)
 Return the specified number of elements as a vector starting from the specified element - 1 of the specified vector. More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > segment (const Eigen::Matrix< T, 1, Eigen::Dynamic > &v, size_t i, size_t n)
 
template<typename T >
std::vector< T > segment (const std::vector< T > &sv, size_t i, size_t n)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > simplex_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y)
 Return the simplex corresponding to the specified free vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > simplex_constrain (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &y, T &lp)
 Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > simplex_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
 Return an unconstrained vector that when transformed produces the specified simplex. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > singular_values (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Return the vector of the singular values of the specified matrix in decreasing order of magnitude. More...
 
template<typename T >
int size (const std::vector< T > &x)
 Return the size of the specified standard vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > softmax (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v)
 Return the softmax of the specified vector. More...
 
template<typename T >
std::vector< T > sort_asc (std::vector< T > xs)
 Return the specified standard vector in ascending order. More...
 
template<typename T >
std::vector< T > sort_desc (std::vector< T > xs)
 Return the specified standard vector in descending order. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > sort_asc (Eigen::Matrix< T, R, C > xs)
 Return the specified eigen vector in ascending order. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > sort_desc (Eigen::Matrix< T, R, C > xs)
 Return the specified eigen vector in descending order. More...
 
template<typename C >
std::vector< int > sort_indices_asc (const C &xs)
 Return a sorted copy of the argument container in ascending order. More...
 
template<typename C >
std::vector< int > sort_indices_desc (const C &xs)
 Return a sorted copy of the argument container in ascending order. More...
 
template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type squared_distance (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
 Returns the squared distance between the specified vectors. More...
 
template<typename T >
void stan_print (std::ostream *o, const T &x)
 
template<typename T >
void stan_print (std::ostream *o, const std::vector< T > &x)
 
template<typename T >
void stan_print (std::ostream *o, const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
 
template<typename T >
void stan_print (std::ostream *o, const Eigen::Matrix< T, 1, Eigen::Dynamic > &x)
 
template<typename T >
void stan_print (std::ostream *o, const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &x)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > sub_col (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t nrows)
 Return a nrows x 1 subcolumn starting at (i-1, j-1). More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > sub_row (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m, size_t i, size_t j, size_t ncols)
 Return a 1 x nrows subrow starting at (i-1, j-1). More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > subtract (const Eigen::Matrix< T1, R, C > &m1, const Eigen::Matrix< T2, R, C > &m2)
 Return the result of subtracting the second specified matrix from the first specified matrix. More...
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > subtract (const T1 &c, const Eigen::Matrix< T2, R, C > &m)
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args< T1, T2 >::type, R, C > subtract (const Eigen::Matrix< T1, R, C > &m, const T2 &c)
 
template<typename T , int R, int C>
double sum (const Eigen::Matrix< T, R, C > &v)
 Returns the sum of the coefficients of the specified column vector. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > tail (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &v, size_t n)
 Return the specified number of elements as a vector from the back of the specified vector. More...
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > tail (const Eigen::Matrix< T, 1, Eigen::Dynamic > &rv, size_t n)
 Return the specified number of elements as a row vector from the back of the specified row vector. More...
 
template<typename T >
std::vector< T > tail (const std::vector< T > &sv, size_t n)
 
matrix_d tcrossprod (const matrix_d &M)
 Returns the result of post-multiplying a matrix by its own transpose. More...
 
template<typename T , int R, int C>
std::vector< T > to_array_1d (const Eigen::Matrix< T, R, C > &matrix)
 
template<typename T >
std::vector< T > to_array_1d (const std::vector< T > &x)
 
template<typename T >
std::vector< typename scalar_type< T >::type > to_array_1d (const std::vector< std::vector< T > > &x)
 
template<typename T >
std::vector< std::vector< T > > to_array_2d (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &matrix)
 
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > to_matrix (Eigen::Matrix< T, R, C > matrix)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > to_matrix (const std::vector< std::vector< T > > &vec)
 
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > to_matrix (const std::vector< std::vector< int > > &vec)
 
template<typename T , int R, int C>
Eigen::Matrix< T, 1, Eigen::Dynamic > to_row_vector (const Eigen::Matrix< T, R, C > &matrix)
 
template<typename T >
Eigen::Matrix< T, 1, Eigen::Dynamic > to_row_vector (const std::vector< T > &vec)
 
Eigen::Matrix< double, 1, Eigen::Dynamic > to_row_vector (const std::vector< int > &vec)
 
template<typename T , int R, int C>
Eigen::Matrix< T, Eigen::Dynamic, 1 > to_vector (const Eigen::Matrix< T, R, C > &matrix)
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > to_vector (const std::vector< T > &vec)
 
Eigen::Matrix< double, Eigen::Dynamic, 1 > to_vector (const std::vector< int > &vec)
 
template<typename T >
trace (const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &m)
 Returns the trace of the specified matrix. More...
 
template<typename T >
trace (const T &m)
 
template<typename T1 , typename T2 , typename T3 , int R1, int C1, int R2, int C2, int R3, int C3>
boost::enable_if_c<!stan::is_var< T1 >::value &&!stan::is_var< T2 >::value &&!stan::is_var< T3 >::value, typename boost::math::tools::promote_args< T1, T2, T3 >::type >::type trace_gen_inv_quad_form_ldlt (const Eigen::Matrix< T1, R1, C1 > &D, const stan::math::LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
 
template<int RD, int CD, int RA, int CA, int RB, int CB>
double trace_gen_quad_form (const Eigen::Matrix< double, RD, CD > &D, const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
 Compute trace(D B^T A B). More...
 
template<typename T1 , typename T2 , int R2, int C2, int R3, int C3>
boost::enable_if_c<!stan::is_var< T1 >::value &&!stan::is_var< T2 >::value, typename boost::math::tools::promote_args< T1, T2 >::type >::type trace_inv_quad_form_ldlt (const stan::math::LDLT_factor< T1, R2, C2 > &A, const Eigen::Matrix< T2, R3, C3 > &B)
 
template<int RA, int CA, int RB, int CB>
double trace_quad_form (const Eigen::Matrix< double, RA, CA > &A, const Eigen::Matrix< double, RB, CB > &B)
 Compute trace(B^T A B). More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, C, R > transpose (const Eigen::Matrix< T, R, C > &m)
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > unit_vector_constrain (const Eigen::Matrix< T, R, C > &y)
 Return the unit length vector corresponding to the free vector y. More...
 
template<typename T , int R, int C>
Eigen::Matrix< T, R, C > unit_vector_constrain (const Eigen::Matrix< T, R, C > &y, T &lp)
 Return the unit length vector corresponding to the free vector y. More...
 
template<typename T >
Eigen::Matrix< T, Eigen::Dynamic, 1 > unit_vector_free (const Eigen::Matrix< T, Eigen::Dynamic, 1 > &x)
 Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1. More...
 
template<typename T , int R, int C>
Eigen::Matrix< typename child_type< T >::type, R, C > value_of (const Eigen::Matrix< T, R, C > &M)
 Convert a matrix of type T to a matrix of doubles. More...
 
template<typename T , int R, int C>
Eigen::Matrix< double, R, C > value_of_rec (const Eigen::Matrix< T, R, C > &M)
 Convert a matrix of type T to a matrix of doubles. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type variance (const std::vector< T > &v)
 Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector. More...
 
template<typename T , int R, int C>
boost::math::tools::promote_args< T >::type variance (const Eigen::Matrix< T, R, C > &m)
 Returns the sample variance (divide by length - 1) of the coefficients in the specified column vector. More...
 
template<typename F >
void finite_diff_gradient (const F &f, const Eigen::Matrix< double,-1, 1 > &x, double &fx, Eigen::Matrix< double,-1, 1 > &grad_fx, const double epsilon=1e-03)
 Calculate the value and the gradient of the specified function at the specified argument using finite difference. More...
 
template<typename F >
double finite_diff_hess_helper (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, const int lambda, const double epsilon=1e-03)
 
template<typename F >
void finite_diff_hessian (const F &f, const Eigen::Matrix< double,-1, 1 > &x, double &fx, Eigen::Matrix< double,-1, 1 > &grad_fx, Eigen::Matrix< double,-1,-1 > &hess_fx, const double epsilon=1e-03)
 Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference. More...
 
template<bool propto, typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_log (const typename math::index_type< Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > >::type n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<bool propto, typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<bool propto, typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_logit_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
 
template<typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_logit_log (int n, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
 
template<bool propto, typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_logit_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
 
template<typename T_prob >
boost::math::tools::promote_args< T_prob >::type categorical_logit_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &beta)
 
template<class RNG >
int categorical_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &theta, RNG &rng)
 
template<bool propto, typename T_prob , typename T_prior_sample_size >
boost::math::tools::promote_args< T_prob, T_prior_sample_size >::type dirichlet_log (const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta, const Eigen::Matrix< T_prior_sample_size, Eigen::Dynamic, 1 > &alpha)
 The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha. More...
 
template<typename T_prob , typename T_prior_sample_size >
boost::math::tools::promote_args< T_prob, T_prior_sample_size >::type dirichlet_log (const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta, const Eigen::Matrix< T_prior_sample_size, Eigen::Dynamic, 1 > &alpha)
 
template<class RNG >
Eigen::VectorXd dirichlet_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &alpha, RNG &rng)
 Return a draw from a Dirichlet distribution with specified parameters and pseudo-random number generator. More...
 
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type< T_F, T_G, T_V, T_W, T_m0, T_C0 >::type >::type gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
 The log of a Gaussian dynamic linear model (GDLM). More...
 
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type< T_F, T_G, T_V, T_W, T_m0, T_C0 >::type >::type gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
 
template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type< T_F, T_G, T_V, T_W, T_m0, T_C0 >::type >::type gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
 The log of a Gaussian dynamic linear model (GDLM) with uncorrelated observation disturbances. More...
 
template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type< T_F, T_G, T_V, T_W, T_m0, T_C0 >::type >::type gaussian_dlm_obs_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &F, const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &G, const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &V, const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &W, const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &m0, const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &C0)
 
template<bool propto, typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args< T_y, T_dof, T_scale >::type inv_wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
 The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix. More...
 
template<typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args< T_y, T_dof, T_scale >::type inv_wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
 
template<class RNG >
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > inv_wishart_rng (const double nu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
 
template<bool propto, typename T_covar , typename T_shape >
boost::math::tools::promote_args< T_covar, T_shape >::type lkj_corr_cholesky_log (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
 
template<typename T_covar , typename T_shape >
boost::math::tools::promote_args< T_covar, T_shape >::type lkj_corr_cholesky_log (const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const T_shape &eta)
 
template<class RNG >
Eigen::MatrixXd lkj_corr_cholesky_rng (const size_t K, const double eta, RNG &rng)
 
template<typename T_shape >
T_shape do_lkj_constant (const T_shape &eta, const unsigned int &K)
 
template<bool propto, typename T_y , typename T_shape >
boost::math::tools::promote_args< T_y, T_shape >::type lkj_corr_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
 
template<typename T_y , typename T_shape >
boost::math::tools::promote_args< T_y, T_shape >::type lkj_corr_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_shape &eta)
 
template<class RNG >
Eigen::MatrixXd lkj_corr_rng (const size_t K, const double eta, RNG &rng)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args< T_y, T_loc, T_scale, T_shape >::type lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args< T_y, T_loc, T_scale, T_shape >::type lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &sigma, const T_shape &eta)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args< T_y, T_loc, T_scale, T_shape >::type lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args< T_y, T_loc, T_scale, T_shape >::type lkj_cov_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const T_loc &mu, const T_scale &sigma, const T_shape &eta)
 
template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
boost::math::tools::promote_args< T_y, T_Mu, T_Sigma, T_D >::type matrix_normal_prec_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
 The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices. More...
 
template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
boost::math::tools::promote_args< T_y, T_Mu, T_Sigma, T_D >::type matrix_normal_prec_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &Mu, const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &D)
 
template<bool propto, typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args< T_y, T_covar, T_w >::type multi_gp_cholesky_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
 The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma. More...
 
template<typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args< T_y, T_covar, T_w >::type multi_gp_cholesky_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &L, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
 
template<bool propto, typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args< T_y, T_covar, T_w >::type multi_gp_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
 The log of a multivariate Gaussian Process for the given y, Sigma, and w. More...
 
template<typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args< T_y, T_covar, T_w >::type multi_gp_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &y, const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &Sigma, const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &w)
 
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
 The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix. More...
 
template<typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_cholesky_log (const T_y &y, const T_loc &mu, const T_covar &L)
 
template<class RNG >
Eigen::VectorXd multi_normal_cholesky_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
 
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
 
template<typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
 
template<typename T_y , typename T_loc , typename T_covar >
return_type< T_y, T_loc, T_covar >::type multi_normal_prec_log (const T_y &y, const T_loc &mu, const T_covar &Sigma)
 
template<class RNG >
Eigen::VectorXd multi_normal_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
 
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
 Return the log of the multivariate Student t distribution at the specified arguments. More...
 
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type multi_student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &Sigma)
 
template<class RNG >
Eigen::VectorXd multi_student_t_rng (const double nu, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &mu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &s, RNG &rng)
 
template<bool propto, typename T_prob >
boost::math::tools::promote_args< T_prob >::type multinomial_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<typename T_prob >
boost::math::tools::promote_args< T_prob >::type multinomial_log (const std::vector< int > &ns, const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &theta)
 
template<class RNG >
std::vector< int > multinomial_rng (const Eigen::Matrix< double, Eigen::Dynamic, 1 > &theta, const int N, RNG &rng)
 
template<typename T >
log_inv_logit_diff (const T &alpha, const T &beta)
 
template<bool propto, typename T_lambda , typename T_cut >
boost::math::tools::promote_args< T_lambda, T_cut >::type ordered_logistic_log (int y, const T_lambda &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
 Returns the (natural) log probability of the specified integer outcome given the continuous location and specified cutpoints in an ordered logistic model. More...
 
template<typename T_lambda , typename T_cut >
boost::math::tools::promote_args< T_lambda, T_cut >::type ordered_logistic_log (int y, const T_lambda &lambda, const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &c)
 
template<class RNG >
int ordered_logistic_rng (const double eta, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &c, RNG &rng)
 
template<bool propto, typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args< T_y, T_dof, T_scale >::type wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
 The log of the Wishart density for the given W, degrees of freedom, and scale matrix. More...
 
template<typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args< T_y, T_dof, T_scale >::type wishart_log (const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &W, const T_dof &nu, const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &S)
 
template<class RNG >
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > wishart_rng (const double nu, const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &S, RNG &rng)
 
template<typename T_y , typename T_low , typename T_high >
bool check_bounded (const char *function, const char *name, const T_y &y, const T_low &low, const T_high &high)
 Return true if the value is between the low and high values, inclusively. More...
 
template<typename T >
bool check_consistent_size (const char *function, const char *name, const T &x, size_t expected_size)
 Return true if the dimension of x is consistent, which is defined to be expected_size if x is a vector or 1 if x is not a vector. More...
 
template<typename T1 , typename T2 >
bool check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2)
 Return true if the dimension of x1 is consistent with x2. More...
 
template<typename T1 , typename T2 , typename T3 >
bool check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3)
 Return true if the dimension of x1, x2, and x3 are consistent. More...
 
template<typename T1 , typename T2 , typename T3 , typename T4 >
bool check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3, const char *name4, const T4 &x4)
 Return true if the dimension of x1, x2, x3, and x4 are consistent. More...
 
template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >
bool check_consistent_sizes (const char *function, const char *name1, const T1 &x1, const char *name2, const T2 &x2, const char *name3, const T3 &x3, const char *name4, const T4 &x4, const char *name5, const T5 &x5)
 
template<typename T_y , typename T_eq >
bool check_equal (const char *function, const char *name, const T_y &y, const T_eq &eq)
 Return true if y is equal to eq. More...
 
template<typename T_y >
bool check_finite (const char *function, const char *name, const T_y &y)
 Return true if y is finite. More...
 
template<typename T_y , typename T_low >
bool check_greater (const char *function, const char *name, const T_y &y, const T_low &low)
 Return true if y is strictly greater than low. More...
 
template<typename T_y , typename T_low >
bool check_greater_or_equal (const char *function, const char *name, const T_y &y, const T_low &low)
 Return true if y is greater or equal than low. More...
 
template<typename T_y , typename T_high >
bool check_less (const char *function, const char *name, const T_y &y, const T_high &high)
 Return true if y is strictly less than high. More...
 
template<typename T_y , typename T_high >
bool check_less_or_equal (const char *function, const char *name, const T_y &y, const T_high &high)
 Return true if y is less or equal to high. More...
 
template<typename T_y >
bool check_nonnegative (const char *function, const char *name, const T_y &y)
 Return true if y is non-negative. More...
 
template<typename T_y >
bool check_nonzero_size (const char *function, const char *name, const T_y &y)
 Return true if the specified matrix/vector is of non-zero size. More...
 
template<typename T_y >
bool check_not_nan (const char *function, const char *name, const T_y &y)
 Return true if y is not NaN. More...
 
template<typename T_y >
bool check_positive (const char *function, const char *name, const T_y &y)
 Return true if y is positive. More...
 
template<typename T_y >
bool check_positive_finite (const char *function, const char *name, const T_y &y)
 Return true if y is positive and finite. More...
 
bool check_positive_size (const char *function, const char *name, const char *expr, const int size)
 Return true if size is positive. More...
 
template<typename T_size1 , typename T_size2 >
bool check_size_match (const char *function, const char *name_i, T_size1 i, const char *name_j, T_size2 j)
 Return true if the provided sizes match. More...
 
template<typename T_size1 , typename T_size2 >
bool check_size_match (const char *function, const char *expr_i, const char *name_i, T_size1 i, const char *expr_j, const char *name_j, T_size2 j)
 Return true if the provided sizes match. More...
 
template<typename T >
void domain_error (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
 Throw a domain error with a consistently formatted message. More...
 
template<typename T >
void domain_error (const char *function, const char *name, const T &y, const char *msg1)
 Throw a domain error with a consistently formatted message. More...
 
template<typename T >
void domain_error_vec (const char *function, const char *name, const T &y, const size_t i, const char *msg1, const char *msg2)
 Throw a domain error with a consistently formatted message. More...
 
template<typename T >
void domain_error_vec (const char *function, const char *name, const T &y, const size_t i, const char *msg)
 Throw a domain error with a consistently formatted message. More...
 
template<typename T >
void invalid_argument (const char *function, const char *name, const T &y, const char *msg1, const char *msg2)
 Throw an invalid_argument exception with a consistently formatted message. More...
 
template<typename T >
void invalid_argument (const char *function, const char *name, const T &y, const char *msg1)
 Throw an invalid_argument exception with a consistently formatted message. More...
 
template<typename T >
void invalid_argument_vec (const char *function, const char *name, const T &y, const size_t i, const char *msg1, const char *msg2)
 Throw an invalid argument exception with a consistently formatted message. More...
 
template<typename T >
void invalid_argument_vec (const char *function, const char *name, const T &y, const size_t i, const char *msg)
 Throw an invalid argument exception with a consistently formatted message. More...
 
void out_of_range (const char *function, const int max, const int index, const char *msg1="", const char *msg2="")
 Throw an out_of_range exception with a consistently formatted message. More...
 
double abs (double x)
 Return floating-point absolute value. More...
 
template<typename T >
bool as_bool (const T x)
 Return 1 if the argument is unequal to zero and 0 otherwise. More...
 
template<typename T2 >
T2 bessel_first_kind (const int v, const T2 z)
 

\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
template<typename T2 >
T2 bessel_second_kind (const int v, const T2 z)
 

\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
template<typename T >
boost::math::tools::promote_args< T >::type binary_log_loss (const int y, const T y_hat)
 Returns the log loss function for binary classification with specified reference and response values. More...
 
template<typename T_N , typename T_n >
boost::math::tools::promote_args< T_N, T_n >::type binomial_coefficient_log (const T_N N, const T_n n)
 Return the log of the binomial coefficient for the specified arguments. More...
 
double pi ()
 Return the value of pi. More...
 
double e ()
 Return the base of the natural logarithm. More...
 
double sqrt2 ()
 Return the square root of two. More...
 
double log10 ()
 Return natural logarithm of ten. More...
 
double positive_infinity ()
 Return positive infinity. More...
 
double negative_infinity ()
 Return negative infinity. More...
 
double not_a_number ()
 Return (quiet) not-a-number. More...
 
double machine_precision ()
 Returns the difference between 1.0 and the next value representable. More...
 
template<typename T >
corr_constrain (const T x)
 Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive). More...
 
template<typename T >
corr_constrain (const T x, T &lp)
 Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive). More...
 
template<typename T >
corr_free (const T y)
 Return the unconstrained scalar that when transformed to a valid correlation produces the specified value. More...
 
double digamma (double x)
 

\[ \mbox{digamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
int divide (const int x, const int y)
 
template<typename T >
boost::math::tools::promote_args< T >::type exp2 (const T y)
 Return the exponent base 2 of the specified argument (C99). More...
 
template<typename T >
F32 (T a, T b, T c, T d, T e, T z, T precision=1e-6)
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type falling_factorial (const T1 x, const T2 n)
 

\[ \mbox{falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type fdim (T1 a, T2 b)
 The positive difference function (C99). More...
 
double gamma_p (double x, double a)
 

\[ \mbox{gamma\_p}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ P(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

More...
 
double gamma_q (double x, double a)
 

\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

More...
 
template<typename T >
void grad_2F1 (T &gradA, T &gradC, T a, T b, T c, T z, T precision=1e-6)
 
template<typename T >
void grad_F32 (T *g, T a, T b, T c, T d, T e, T z, T precision=1e-6)
 
void grad_inc_beta (double &g1, double &g2, double a, double b, double z)
 
template<typename T >
void grad_reg_inc_beta (T &g1, T &g2, T a, T b, T z, T digammaA, T digammaB, T digammaSum, T betaAB)
 
template<typename T >
grad_reg_inc_gamma (T a, T z, T g, T dig, T precision=1e-6)
 
double ibeta (const double a, const double b, const double x)
 The normalized incomplete beta function of a, b, and x. More...
 
template<typename T >
identity_constrain (T x)
 Returns the result of applying the identity constraint transform to the input. More...
 
template<typename T >
identity_constrain (const T x, T &)
 Returns the result of applying the identity constraint transform to the input and increments the log probability reference with the log absolute Jacobian determinant. More...
 
template<typename T >
identity_free (const T y)
 Returns the result of applying the inverse of the identity constraint transform to the input. More...
 
template<typename T_true , typename T_false >
boost::math::tools::promote_args< T_true, T_false >::type if_else (const bool c, const T_true y_true, const T_false y_false)
 Return the second argument if the first argument is true and otherwise return the second argument. More...
 
double inc_beta (const double &a, const double &b, const double &x)
 
template<typename T >
inc_beta_ddb (T a, T b, T z, T digamma_b, T digamma_ab)
 Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b. More...
 
template<typename T >
inc_beta_dda (T a, T b, T z, T digamma_a, T digamma_ab)
 Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a. More...
 
template<typename T >
inc_beta_ddz (T a, T b, T z)
 Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z. More...
 
template<>
double inc_beta_ddz (double a, double b, double z)
 
template<typename T >
unsigned int int_step (const T y)
 The integer step, or Heaviside, function. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type inv (const T x)
 
template<typename T >
boost::math::tools::promote_args< T >::type inv_cloglog (T x)
 The inverse complementary log-log function. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type inv_logit (const T a)
 Returns the inverse logit function applied to the argument. More...
 
double inv_Phi (double p)
 The inverse of the unit normal cumulative distribution function. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type inv_sqrt (const T x)
 
template<typename T >
boost::math::tools::promote_args< T >::type inv_square (const T x)
 
template<typename Vector >
void inverse_softmax (const Vector &simplex, Vector &y)
 Writes the inverse softmax of the simplex argument into the second argument. More...
 
int is_inf (const double x)
 Returns 1 if the input is infinite and 0 otherwise. More...
 
bool is_nan (double x)
 Returns 1 if the input is NaN and 0 otherwise. More...
 
template<typename T >
bool is_uninitialized (T x)
 Returns true if the specified variable is uninitialized. More...
 
template<typename T , typename TL >
lb_constrain (const T x, const TL lb)
 Return the lower-bounded value for the specified unconstrained input and specified lower bound. More...
 
template<typename T , typename TL >
boost::math::tools::promote_args< T, TL >::type lb_constrain (const T x, const TL lb, T &lp)
 Return the lower-bounded value for the speicifed unconstrained input and specified lower bound, incrementing the specified reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T , typename TL >
boost::math::tools::promote_args< T, TL >::type lb_free (const T y, const TL lb)
 Return the unconstrained value that produces the specified lower-bound constrained value. More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type lbeta (const T1 a, const T2 b)
 Return the log of the beta function applied to the specified arguments. More...
 
double lgamma (double x)
 

\[ \mbox{lgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
template<typename T >
boost::math::tools::promote_args< T >::type lmgamma (const int k, T x)
 Return the natural logarithm of the multivariate gamma function with the speciifed dimensions and argument. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log1m (T x)
 Return the natural logarithm of one minus the specified value. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log1m_exp (const T a)
 Calculates the log of 1 minus the exponential of the specified value without overflow log1m_exp(x) = log(1-exp(x)). More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log1m_inv_logit (const T u)
 Returns the natural logarithm of 1 minus the inverse logit of the specified argument. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log1p (const T x)
 Return the natural logarithm of one plus the specified value. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log1p_exp (const T a)
 Calculates the log of 1 plus the exponential of the specified value without overflow. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log2 (const T a)
 Returns the base 2 logarithm of the argument (C99). More...
 
double log2 ()
 Return natural logarithm of two. More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type log_diff_exp (const T1 x, const T2 y)
 The natural logarithm of the difference of the natural exponentiation of x1 and the natural exponentiation of x2. More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type log_falling_factorial (const T1 x, const T2 n)
 

\[ \mbox{log\_falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

More...
 
template<typename T >
boost::math::tools::promote_args< T >::type log_inv_logit (const T &u)
 Returns the natural logarithm of the inverse logit of the specified argument. More...
 
double log_mix (double theta, double lambda1, double lambda2)
 Return the log mixture density with specified mixing proportion and log densities. More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type log_rising_factorial (const T1 x, const T2 n)
 

\[ \mbox{log\_rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type log_sum_exp (const T2 &a, const T1 &b)
 Calculates the log sum of exponetials without overflow. More...
 
template<typename T1 , typename T2 >
int logical_and (const T1 x1, const T2 x2)
 The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise. More...
 
template<typename T1 , typename T2 >
int logical_eq (const T1 x1, const T2 x2)
 Return 1 if the first argument is equal to the second. More...
 
template<typename T1 , typename T2 >
int logical_gt (const T1 x1, const T2 x2)
 Return 1 if the first argument is strictly greater than the second. More...
 
template<typename T1 , typename T2 >
int logical_gte (const T1 x1, const T2 x2)
 Return 1 if the first argument is greater than or equal to the second. More...
 
template<typename T1 , typename T2 >
int logical_lt (T1 x1, T2 x2)
 Return 1 if the first argument is strictly less than the second. More...
 
template<typename T1 , typename T2 >
int logical_lte (const T1 x1, const T2 x2)
 Return 1 if the first argument is less than or equal to the second. More...
 
template<typename T >
int logical_negation (const T x)
 The logical negation function which returns 1 if the input is equal to zero and 0 otherwise. More...
 
template<typename T1 , typename T2 >
int logical_neq (const T1 x1, const T2 x2)
 Return 1 if the first argument is unequal to the second. More...
 
template<typename T1 , typename T2 >
int logical_or (T1 x1, T2 x2)
 The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type logit (const T a)
 Returns the logit function applied to the argument. More...
 
template<typename T , typename TL , typename TU >
boost::math::tools::promote_args< T, TL, TU >::type lub_constrain (const T x, TL lb, TU ub)
 Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds. More...
 
template<typename T , typename TL , typename TU >
boost::math::tools::promote_args< T, TL, TU >::type lub_constrain (const T x, const TL lb, const TU ub, T &lp)
 Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log probability with the log absolute Jacobian determinant. More...
 
template<typename T , typename TL , typename TU >
boost::math::tools::promote_args< T, TL, TU >::type lub_free (const T y, TL lb, TU ub)
 Return the unconstrained scalar that transforms to the specified lower- and upper-bounded scalar given the specified bounds. More...
 
template<typename T2 >
T2 modified_bessel_first_kind (const int v, const T2 z)
 

\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

More...
 
template<typename T2 >
T2 modified_bessel_second_kind (const int v, const T2 z)
 

\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

More...
 
int modulus (const int x, const int y)
 
template<typename T_a , typename T_b >
boost::math::tools::promote_args< T_a, T_b >::type multiply_log (const T_a a, const T_b b)
 Calculated the value of the first argument times log of the second argument while behaving properly with 0 inputs. More...
 
double owens_t (const double h, const double a)
 The Owen's T function of h and a. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type Phi (const T x)
 The unit normal cumulative distribution function. More...
 
template<typename T >
boost::math::tools::promote_args< T >::type Phi_approx (T x)
 Approximation of the unit normal CDF. More...
 
template<typename T >
positive_constrain (const T x)
 Return the positive value for the specified unconstrained input. More...
 
template<typename T >
positive_constrain (const T x, T &lp)
 Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant. More...
 
template<typename T >
positive_free (const T y)
 Return the unconstrained value corresponding to the specified positive-constrained value. More...
 
template<typename T >
boost::enable_if< boost::is_arithmetic< T >, T >::type primitive_value (T x)
 Return the value of the specified arithmetic argument unmodified with its own declared type. More...
 
template<typename T >
boost::disable_if< boost::is_arithmetic< T >, double >::type primitive_value (const T &x)
 Return the primitive value of the specified argument. More...
 
template<typename T >
prob_constrain (const T x)
 Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar. More...
 
template<typename T >
prob_constrain (const T x, T &lp)
 Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T >
prob_free (const T y)
 Return the free scalar that when transformed to a probability produces the specified scalar. More...
 
template<typename T , typename S >
promote_scalar_type< T, S >::type promote_scalar (const S &x)
 This is the top-level function to call to promote the scalar types of an input of type S to type T. More...
 
template<typename T1 , typename T2 >
boost::math::tools::promote_args< T1, T2 >::type rising_factorial (const T1 x, const T2 n)
 

\[ \mbox{rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

More...
 
template<typename T >
int sign (const T &z)
 
template<typename T >
square (const T x)
 Return the square of the specified argument. More...
 
template<typename T >
int step (const T y)
 The step, or Heaviside, function. More...
 
template<typename T >
trigamma (T x)
 

\[ \mbox{trigamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi_1(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
template<typename T , typename TU >
boost::math::tools::promote_args< T, TU >::type ub_constrain (const T x, const TU ub)
 Return the upper-bounded value for the specified unconstrained scalar and upper bound. More...
 
template<typename T , typename TU >
boost::math::tools::promote_args< T, TU >::type ub_constrain (const T x, const TU ub, T &lp)
 Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform. More...
 
template<typename T , typename TU >
boost::math::tools::promote_args< T, TU >::type ub_free (const T y, const TU ub)
 Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound. More...
 
template<typename T >
double value_of (const T x)
 Return the value of the specified scalar argument converted to a double value. More...
 
template<>
double value_of< double > (const double x)
 Return the specified argument. More...
 
template<typename T >
double value_of_rec (const T x)
 Return the value of the specified scalar argument converted to a double value. More...
 
template<>
double value_of_rec< double > (const double x)
 Return the specified argument. More...
 
template<typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_ccdf_log (const T_n &n, const T_prob &theta)
 
template<typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_cdf (const T_n &n, const T_prob &theta)
 
template<typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_cdf_log (const T_n &n, const T_prob &theta)
 
template<bool propto, typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_log (const T_n &n, const T_prob &theta)
 
template<typename T_y , typename T_prob >
return_type< T_prob >::type bernoulli_log (const T_y &n, const T_prob &theta)
 
template<bool propto, typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_logit_log (const T_n &n, const T_prob &theta)
 
template<typename T_n , typename T_prob >
return_type< T_prob >::type bernoulli_logit_log (const T_n &n, const T_prob &theta)
 
template<class RNG >
int bernoulli_rng (const double theta, RNG &rng)
 
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type< T_size1, T_size2 >::type beta_binomial_ccdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
 
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type< T_size1, T_size2 >::type beta_binomial_cdf (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
 
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type< T_size1, T_size2 >::type beta_binomial_cdf_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
 
template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type< T_size1, T_size2 >::type beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
 
template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type< T_size1, T_size2 >::type beta_binomial_log (const T_n &n, const T_N &N, const T_size1 &alpha, const T_size2 &beta)
 
template<class RNG >
int beta_binomial_rng (const int N, const double alpha, const double beta, RNG &rng)
 
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type< T_y, T_scale_succ, T_scale_fail >::type beta_ccdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
 
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type< T_y, T_scale_succ, T_scale_fail >::type beta_cdf (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
 Calculates the beta cumulative distribution function for the given variate and scale variables. More...
 
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type< T_y, T_scale_succ, T_scale_fail >::type beta_cdf_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
 
template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type< T_y, T_scale_succ, T_scale_fail >::type beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
 The log of the beta density for the specified scalar(s) given the specified sample size(s). More...
 
template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type< T_y, T_scale_succ, T_scale_fail >::type beta_log (const T_y &y, const T_scale_succ &alpha, const T_scale_fail &beta)
 
template<class RNG >
double beta_rng (const double alpha, const double beta, RNG &rng)
 
template<typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_ccdf_log (const T_n &n, const T_N &N, const T_prob &theta)
 
template<typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_cdf (const T_n &n, const T_N &N, const T_prob &theta)
 
template<typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_cdf_log (const T_n &n, const T_N &N, const T_prob &theta)
 
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
 
template<typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_log (const T_n &n, const T_N &N, const T_prob &theta)
 
template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
 
template<typename T_n , typename T_N , typename T_prob >
return_type< T_prob >::type binomial_logit_log (const T_n &n, const T_N &N, const T_prob &alpha)
 
template<class RNG >
int binomial_rng (const int N, const double theta, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type cauchy_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type cauchy_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
 Calculates the cauchy cumulative distribution function for the given variate, location, and scale. More...
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type cauchy_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s). More...
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type cauchy_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double cauchy_rng (const double mu, const double sigma, RNG &rng)
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type chi_square_ccdf_log (const T_y &y, const T_dof &nu)
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type chi_square_cdf (const T_y &y, const T_dof &nu)
 Calculates the chi square cumulative distribution function for the given variate and degrees of freedom. More...
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type chi_square_cdf_log (const T_y &y, const T_dof &nu)
 
template<bool propto, typename T_y , typename T_dof >
return_type< T_y, T_dof >::type chi_square_log (const T_y &y, const T_dof &nu)
 The log of a chi-squared density for y with the specified degrees of freedom parameter. More...
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type chi_square_log (const T_y &y, const T_dof &nu)
 
template<class RNG >
double chi_square_rng (const double nu, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type double_exponential_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type double_exponential_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
 Calculates the double exponential cumulative density function. More...
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type double_exponential_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type double_exponential_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double double_exponential_rng (const double mu, const double sigma, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type< T_y, T_loc, T_scale, T_inv_scale >::type exp_mod_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type< T_y, T_loc, T_scale, T_inv_scale >::type exp_mod_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type< T_y, T_loc, T_scale, T_inv_scale >::type exp_mod_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type< T_y, T_loc, T_scale, T_inv_scale >::type exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type< T_y, T_loc, T_scale, T_inv_scale >::type exp_mod_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_inv_scale &lambda)
 
template<class RNG >
double exp_mod_normal_rng (const double mu, const double sigma, const double lambda, RNG &rng)
 
template<typename T_y , typename T_inv_scale >
return_type< T_y, T_inv_scale >::type exponential_ccdf_log (const T_y &y, const T_inv_scale &beta)
 
template<typename T_y , typename T_inv_scale >
return_type< T_y, T_inv_scale >::type exponential_cdf (const T_y &y, const T_inv_scale &beta)
 Calculates the exponential cumulative distribution function for the given y and beta. More...
 
template<typename T_y , typename T_inv_scale >
return_type< T_y, T_inv_scale >::type exponential_cdf_log (const T_y &y, const T_inv_scale &beta)
 
template<bool propto, typename T_y , typename T_inv_scale >
return_type< T_y, T_inv_scale >::type exponential_log (const T_y &y, const T_inv_scale &beta)
 The log of an exponential density for y with the specified inverse scale parameter. More...
 
template<typename T_y , typename T_inv_scale >
return_type< T_y, T_inv_scale >::type exponential_log (const T_y &y, const T_inv_scale &beta)
 
template<class RNG >
double exponential_rng (const double beta, RNG &rng)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type frechet_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type frechet_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type frechet_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type frechet_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<class RNG >
double frechet_rng (const double alpha, const double sigma, RNG &rng)
 
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type< T_y, T_shape, T_inv_scale >::type gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
 
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type< T_y, T_shape, T_inv_scale >::type gamma_cdf (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
 The cumulative density function for a gamma distribution for y with the specified shape and inverse scale parameters. More...
 
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type< T_y, T_shape, T_inv_scale >::type gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
 
template<bool propto, typename T_y , typename T_shape , typename T_inv_scale >
return_type< T_y, T_shape, T_inv_scale >::type gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
 The log of a gamma density for y with the specified shape and inverse scale parameters. More...
 
template<typename T_y , typename T_shape , typename T_inv_scale >
return_type< T_y, T_shape, T_inv_scale >::type gamma_log (const T_y &y, const T_shape &alpha, const T_inv_scale &beta)
 
template<class RNG >
double gamma_rng (const double alpha, const double beta, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type gumbel_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type gumbel_cdf (const T_y &y, const T_loc &mu, const T_scale &beta)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type gumbel_cdf_log (const T_y &y, const T_loc &mu, const T_scale &beta)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type gumbel_log (const T_y &y, const T_loc &mu, const T_scale &beta)
 
template<class RNG >
double gumbel_rng (const double mu, const double beta, RNG &rng)
 
template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
 
template<typename T_n , typename T_N , typename T_a , typename T_b >
double hypergeometric_log (const T_n &n, const T_N &N, const T_a &a, const T_b &b)
 
template<class RNG >
int hypergeometric_rng (int N, int a, int b, RNG &rng)
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu)
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type inv_chi_square_cdf (const T_y &y, const T_dof &nu)
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type inv_chi_square_cdf_log (const T_y &y, const T_dof &nu)
 
template<bool propto, typename T_y , typename T_dof >
return_type< T_y, T_dof >::type inv_chi_square_log (const T_y &y, const T_dof &nu)
 The log of an inverse chi-squared density for y with the specified degrees of freedom parameter. More...
 
template<typename T_y , typename T_dof >
return_type< T_y, T_dof >::type inv_chi_square_log (const T_y &y, const T_dof &nu)
 
template<class RNG >
double inv_chi_square_rng (const double nu, RNG &rng)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type inv_gamma_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type inv_gamma_cdf (const T_y &y, const T_shape &alpha, const T_scale &beta)
 The CDF of an inverse gamma density for y with the specified shape and scale parameters. More...
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type inv_gamma_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
 
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
 The log of an inverse gamma density for y with the specified shape and scale parameters. More...
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type inv_gamma_log (const T_y &y, const T_shape &alpha, const T_scale &beta)
 
template<class RNG >
double inv_gamma_rng (const double alpha, const double beta, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type logistic_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type logistic_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type logistic_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type logistic_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double logistic_rng (const double mu, const double sigma, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type lognormal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type lognormal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type lognormal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type lognormal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double lognormal_rng (const double mu, const double sigma, RNG &rng)
 
template<typename T_n , typename T_location , typename T_precision >
return_type< T_location, T_precision >::type neg_binomial_2_ccdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
 
template<typename T_n , typename T_location , typename T_precision >
return_type< T_location, T_precision >::type neg_binomial_2_cdf (const T_n &n, const T_location &mu, const T_precision &phi)
 
template<typename T_n , typename T_location , typename T_precision >
return_type< T_location, T_precision >::type neg_binomial_2_cdf_log (const T_n &n, const T_location &mu, const T_precision &phi)
 
template<bool propto, typename T_n , typename T_location , typename T_precision >
return_type< T_location, T_precision >::type neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
 
template<typename T_n , typename T_location , typename T_precision >
return_type< T_location, T_precision >::type neg_binomial_2_log (const T_n &n, const T_location &mu, const T_precision &phi)
 
template<bool propto, typename T_n , typename T_log_location , typename T_precision >
return_type< T_log_location, T_precision >::type neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
 
template<typename T_n , typename T_log_location , typename T_precision >
return_type< T_log_location, T_precision >::type neg_binomial_2_log_log (const T_n &n, const T_log_location &eta, const T_precision &phi)
 
template<class RNG >
int neg_binomial_2_log_rng (const double eta, const double phi, RNG &rng)
 
template<class RNG >
int neg_binomial_2_rng (const double mu, const double phi, RNG &rng)
 
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type< T_shape, T_inv_scale >::type neg_binomial_ccdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
 
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type< T_shape, T_inv_scale >::type neg_binomial_cdf (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
 
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type< T_shape, T_inv_scale >::type neg_binomial_cdf_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
 
template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >
return_type< T_shape, T_inv_scale >::type neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
 
template<typename T_n , typename T_shape , typename T_inv_scale >
return_type< T_shape, T_inv_scale >::type neg_binomial_log (const T_n &n, const T_shape &alpha, const T_inv_scale &beta)
 
template<class RNG >
int neg_binomial_rng (const double alpha, const double beta, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma)
 Calculates the normal cumulative distribution function for the given variate, location, and scale. More...
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s). More...
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double normal_rng (const double mu, const double sigma, RNG &rng)
 
template<typename T_y , typename T_scale , typename T_shape >
return_type< T_y, T_scale, T_shape >::type pareto_ccdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
 
template<typename T_y , typename T_scale , typename T_shape >
return_type< T_y, T_scale, T_shape >::type pareto_cdf (const T_y &y, const T_scale &y_min, const T_shape &alpha)
 
template<typename T_y , typename T_scale , typename T_shape >
return_type< T_y, T_scale, T_shape >::type pareto_cdf_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
 
template<bool propto, typename T_y , typename T_scale , typename T_shape >
return_type< T_y, T_scale, T_shape >::type pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
 
template<typename T_y , typename T_scale , typename T_shape >
return_type< T_y, T_scale, T_shape >::type pareto_log (const T_y &y, const T_scale &y_min, const T_shape &alpha)
 
template<class RNG >
double pareto_rng (const double y_min, const double alpha, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type pareto_type_2_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type pareto_type_2_cdf (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type pareto_type_2_cdf_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type pareto_type_2_log (const T_y &y, const T_loc &mu, const T_scale &lambda, const T_shape &alpha)
 
template<class RNG >
double pareto_type_2_rng (const double mu, const double lambda, const double alpha, RNG &rng)
 
template<typename T_n , typename T_rate >
return_type< T_rate >::type poisson_ccdf_log (const T_n &n, const T_rate &lambda)
 
template<typename T_n , typename T_rate >
return_type< T_rate >::type poisson_cdf (const T_n &n, const T_rate &lambda)
 
template<typename T_n , typename T_rate >
return_type< T_rate >::type poisson_cdf_log (const T_n &n, const T_rate &lambda)
 
template<bool propto, typename T_n , typename T_rate >
return_type< T_rate >::type poisson_log (const T_n &n, const T_rate &lambda)
 
template<typename T_n , typename T_rate >
return_type< T_rate >::type poisson_log (const T_n &n, const T_rate &lambda)
 
template<bool propto, typename T_n , typename T_log_rate >
return_type< T_log_rate >::type poisson_log_log (const T_n &n, const T_log_rate &alpha)
 
template<typename T_n , typename T_log_rate >
return_type< T_log_rate >::type poisson_log_log (const T_n &n, const T_log_rate &alpha)
 
template<class RNG >
int poisson_log_rng (const double alpha, RNG &rng)
 
template<class RNG >
int poisson_rng (const double lambda, RNG &rng)
 
template<typename T_y , typename T_scale >
return_type< T_y, T_scale >::type rayleigh_ccdf_log (const T_y &y, const T_scale &sigma)
 
template<typename T_y , typename T_scale >
return_type< T_y, T_scale >::type rayleigh_cdf (const T_y &y, const T_scale &sigma)
 
template<typename T_y , typename T_scale >
return_type< T_y, T_scale >::type rayleigh_cdf_log (const T_y &y, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_scale >
return_type< T_y, T_scale >::type rayleigh_log (const T_y &y, const T_scale &sigma)
 
template<typename T_y , typename T_scale >
return_type< T_y, T_scale >::type rayleigh_log (const T_y &y, const T_scale &sigma)
 
template<class RNG >
double rayleigh_rng (const double sigma, RNG &rng)
 
template<typename T_y , typename T_dof , typename T_scale >
return_type< T_y, T_dof, T_scale >::type scaled_inv_chi_square_ccdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
 
template<typename T_y , typename T_dof , typename T_scale >
return_type< T_y, T_dof, T_scale >::type scaled_inv_chi_square_cdf (const T_y &y, const T_dof &nu, const T_scale &s)
 The CDF of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
 
template<typename T_y , typename T_dof , typename T_scale >
return_type< T_y, T_dof, T_scale >::type scaled_inv_chi_square_cdf_log (const T_y &y, const T_dof &nu, const T_scale &s)
 
template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type< T_y, T_dof, T_scale >::type scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
 The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter. More...
 
template<typename T_y , typename T_dof , typename T_scale >
return_type< T_y, T_dof, T_scale >::type scaled_inv_chi_square_log (const T_y &y, const T_dof &nu, const T_scale &s)
 
template<class RNG >
double scaled_inv_chi_square_rng (const double nu, const double s, RNG &rng)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type skew_normal_ccdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type skew_normal_cdf (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type skew_normal_cdf_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
 
template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type< T_y, T_loc, T_scale, T_shape >::type skew_normal_log (const T_y &y, const T_loc &mu, const T_scale &sigma, const T_shape &alpha)
 
template<class RNG >
double skew_normal_rng (const double mu, const double sigma, const double alpha, RNG &rng)
 
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type student_t_ccdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type student_t_cdf (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
 
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type student_t_cdf_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
 The log of the Student-t density for the given y, nu, mean, and scale parameter. More...
 
template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type< T_y, T_dof, T_loc, T_scale >::type student_t_log (const T_y &y, const T_dof &nu, const T_loc &mu, const T_scale &sigma)
 
template<class RNG >
double student_t_rng (const double nu, const double mu, const double sigma, RNG &rng)
 
template<typename T_y , typename T_low , typename T_high >
return_type< T_y, T_low, T_high >::type uniform_ccdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
 
template<typename T_y , typename T_low , typename T_high >
return_type< T_y, T_low, T_high >::type uniform_cdf (const T_y &y, const T_low &alpha, const T_high &beta)
 
template<typename T_y , typename T_low , typename T_high >
return_type< T_y, T_low, T_high >::type uniform_cdf_log (const T_y &y, const T_low &alpha, const T_high &beta)
 
template<bool propto, typename T_y , typename T_low , typename T_high >
return_type< T_y, T_low, T_high >::type uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
 The log of a uniform density for the given y, lower, and upper bound. More...
 
template<typename T_y , typename T_low , typename T_high >
return_type< T_y, T_low, T_high >::type uniform_log (const T_y &y, const T_low &alpha, const T_high &beta)
 
template<class RNG >
double uniform_rng (const double alpha, const double beta, RNG &rng)
 
template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
 
template<typename T_y , typename T_loc , typename T_scale >
return_type< T_y, T_loc, T_scale >::type von_mises_log (T_y const &y, T_loc const &mu, T_scale const &kappa)
 
template<class RNG >
double von_mises_rng (const double mu, const double kappa, RNG &rng)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type weibull_ccdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type weibull_cdf (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type weibull_cdf_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<typename T_y , typename T_shape , typename T_scale >
return_type< T_y, T_shape, T_scale >::type weibull_log (const T_y &y, const T_shape &alpha, const T_scale &sigma)
 
template<class RNG >
double weibull_rng (const double alpha, const double sigma, RNG &rng)
 
template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type< T_y, T_alpha, T_tau, T_beta, T_delta >::type wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
 The log of the first passage time density function for a (Wiener) drift diffusion model for the given $y$, boundary separation $\alpha$, nondecision time $\tau$, relative bias $\beta$, and drift rate $\delta$. More...
 
template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type< T_y, T_alpha, T_tau, T_beta, T_delta >::type wiener_log (const T_y &y, const T_alpha &alpha, const T_tau &tau, const T_beta &beta, const T_delta &delta)
 
var log_sum_exp (const std::vector< var > &x)
 Returns the log sum of exponentials. More...
 
var sum (const std::vector< var > &m)
 Returns the sum of the entries of the specified vector. More...
 
void add_initial_values (const std::vector< stan::math::var > &y0, std::vector< std::vector< stan::math::var > > &y)
 Increment the state derived from the coupled system in the with the original initial state. More...
 
static bool empty_nested ()
 Return true if there is no nested autodiff being executed. More...
 
static void grad (vari *vi)
 Compute the gradient for all variables starting from the specified root variable implementation. More...
 
static size_t nested_size ()
 
var operator+ (const var &a, const var &b)
 Addition operator for variables (C++). More...
 
var operator+ (const var &a, const double b)
 Addition operator for variable and scalar (C++). More...
 
var operator+ (const double a, const var &b)
 Addition operator for scalar and variable (C++). More...
 
var operator/ (const var &a, const var &b)
 Division operator for two variables (C++). More...
 
var operator/ (const var &a, const double b)
 Division operator for dividing a variable by a scalar (C++). More...
 
var operator/ (const double a, const var &b)
 Division operator for dividing a scalar by a variable (C++). More...
 
bool operator== (const var &a, const var &b)
 Equality operator comparing two variables' values (C++). More...
 
bool operator== (const var &a, const double b)
 Equality operator comparing a variable's value and a double (C++). More...
 
bool operator== (const double a, const var &b)
 Equality operator comparing a scalar and a variable's value (C++). More...
 
bool operator> (const var &a, const var &b)
 Greater than operator comparing variables' values (C++). More...
 
bool operator> (const var &a, const double b)
 Greater than operator comparing variable's value and double (C++). More...
 
bool operator> (const double a, const var &b)
 Greater than operator comparing a double and a variable's value (C++). More...
 
bool operator>= (const var &a, const var &b)
 Greater than or equal operator comparing two variables' values (C++). More...
 
bool operator>= (const var &a, const double b)
 Greater than or equal operator comparing variable's value and double (C++). More...
 
bool operator>= (const double a, const var &b)
 Greater than or equal operator comparing double and variable's value (C++). More...
 
bool operator< (const var &a, const var &b)
 Less than operator comparing variables' values (C++). More...
 
bool operator< (const var &a, const double b)
 Less than operator comparing variable's value and a double (C++). More...
 
bool operator< (const double a, const var &b)
 Less than operator comparing a double and variable's value (C++). More...
 
bool operator<= (const var &a, const var &b)
 Less than or equal operator comparing two variables' values (C++). More...
 
bool operator<= (const var &a, const double b)
 Less than or equal operator comparing a variable's value and a scalar (C++). More...
 
bool operator<= (const double a, const var &b)
 Less than or equal operator comparing a double and variable's value (C++). More...
 
var operator* (const var &a, const var &b)
 Multiplication operator for two variables (C++). More...
 
var operator* (const var &a, const double b)
 Multiplication operator for a variable and a scalar (C++). More...
 
var operator* (const double a, const var &b)
 Multiplication operator for a scalar and a variable (C++). More...
 
bool operator!= (const var &a, const var &b)
 Inequality operator comparing two variables' values (C++). More...
 
bool operator!= (const var &a, const double b)
 Inequality operator comparing a variable's value and a double (C++). More...
 
bool operator!= (const double a, const var &b)
 Inequality operator comparing a double and a variable's value (C++). More...
 
var operator- (const var &a, const var &b)
 Subtraction operator for variables (C++). More...
 
var operator- (const var &a, const double b)
 Subtraction operator for variable and scalar (C++). More...
 
var operator- (const double a, const var &b)
 Subtraction operator for scalar and variable (C++). More...
 
varoperator-- (var &a)
 Prefix decrement operator for variables (C++). More...
 
var operator-- (var &a, int)
 Postfix decrement operator for variables (C++). More...
 
varoperator++ (var &a)
 Prefix increment operator for variables (C++). More...
 
var operator++ (var &a, int)
 Postfix increment operator for variables (C++). More...
 
var operator- (const var &a)
 Unary negation operator for variables (C++). More...
 
bool operator! (const var &a)
 Prefix logical negation for the value of variables (C++). More...
 
var operator+ (const var &a)
 Unary plus operator for variables (C++). More...
 
var precomputed_gradients (const double value, const std::vector< var > &operands, const std::vector< double > &gradients)
 This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands. More...
 
void print_stack (std::ostream &o)
 Prints the auto-dif variable stack. More...
 
static void recover_memory ()
 Recover memory used for all variables for reuse. More...
 
static void recover_memory_nested ()
 Recover only the memory used for the top nested call. More...
 
static void set_zero_all_adjoints ()
 Reset all adjoint values in the stack to zero. More...
 
static void set_zero_all_adjoints_nested ()
 Reset all adjoint values in the top nested portion of the stack to zero. More...
 
static void start_nested ()
 Record the current position so that recover_memory_nested() can find it. More...
 
static void grad (vari *vi)
 
Eigen::Matrix< var,-1,-1 > cholesky_decompose (const Eigen::Matrix< var,-1,-1 > &A)
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, Eigen::Matrix< var, 1, C1 > >::type columns_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
 
template<int R, int C>
Eigen::Matrix< var, 1, C > columns_dot_self (const Eigen::Matrix< var, R, C > &x)
 Returns the dot product of each column of a matrix with itself. More...
 
matrix_v crossprod (const matrix_v &M)
 Returns the result of pre-multiplying a matrix by its own transpose. More...
 
template<int R, int C>
var determinant (const Eigen::Matrix< var, R, C > &m)
 
double divide (double x, double y)
 Return the division of the first scalar by the second scalar. More...
 
template<typename T1 , typename T2 >
var divide (const T1 &v, const T2 &c)
 
template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< var, R, C > divide (const Eigen::Matrix< T1, R, C > &v, const T2 &c)
 Return the division of the specified column vector by the specified scalar. More...
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, var >::type dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
 Returns the dot product. More...
 
template<typename T1 , typename T2 >
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, var >::type dot_product (const T1 *v1, const T2 *v2, size_t length)
 Returns the dot product. More...
 
template<typename T1 , typename T2 >
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, var >::type dot_product (const std::vector< T1 > &v1, const std::vector< T2 > &v2)
 Returns the dot product. More...
 
template<int R, int C>
var dot_self (const Eigen::Matrix< var, R, C > &v)
 Returns the dot product of a vector with itself. More...
 
void grad (var &v, Eigen::Matrix< var, Eigen::Dynamic, 1 > &x, Eigen::VectorXd &g)
 Propagate chain rule to calculate gradients starting from the specified variable. More...
 
void initialize_variable (var &variable, const var &value)
 Initialize variable to value. More...
 
template<int R, int C>
void initialize_variable (Eigen::Matrix< var, R, C > &matrix, const var &value)
 Initialize every cell in the matrix to the specified value. More...
 
template<typename T >
void initialize_variable (std::vector< T > &variables, const var &value)
 Initialize the variables in the standard vector recursively. More...
 
template<int R, int C>
var log_determinant (const Eigen::Matrix< var, R, C > &m)
 
template<int R, int C>
var log_determinant_ldlt (stan::math::LDLT_factor< var, R, C > &A)
 
template<int R, int C>
var log_determinant_spd (const Eigen::Matrix< var, R, C > &m)
 
Eigen::Matrix< var, Eigen::Dynamic, 1 > log_softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
 Return the softmax of the specified Eigen vector. More...
 
template<int R, int C>
var log_sum_exp (const Eigen::Matrix< var, R, C > &x)
 Returns the log sum of exponentials. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_ldlt (const stan::math::LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 Returns the solution of the system Ax=b given an LDLT_factor of A. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_ldlt (const stan::math::LDLT_factor< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 Returns the solution of the system Ax=b given an LDLT_factor of A. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_ldlt (const stan::math::LDLT_factor< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 Returns the solution of the system Ax=b given an LDLT_factor of A. More...
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_spd (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_spd (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_tri (const Eigen::Matrix< double, R1, C1 > &A, const Eigen::Matrix< var, R2, C2 > &b)
 
template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix< var, R1, C2 > mdivide_left_tri (const Eigen::Matrix< var, R1, C1 > &A, const Eigen::Matrix< double, R2, C2 > &b)
 
template<typename T1 , typename T2 >
boost::enable_if_c< (boost::is_scalar< T1 >::value||boost::is_same< T1, var >::value)&&(boost::is_scalar< T2 >::value||boost::is_same< T2, var >::value), typename boost::math::tools::promote_args< T1, T2 >::type >::type multiply (const T1 &v, const T2 &c)
 Return the product of two scalars. More...
 
template<typename T1 , typename T2 , int R2, int C2>
Eigen::Matrix< var, R2, C2 > multiply (const T1 &c, const Eigen::Matrix< T2, R2, C2 > &m)
 Return the product of scalar and matrix. More...
 
template<typename T1 , int R1, int C1, typename T2 >
Eigen::Matrix< var, R1, C1 > multiply (const Eigen::Matrix< T1, R1, C1 > &m, const T2 &c)
 Return the product of scalar and matrix. More...
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, Eigen::Matrix< var, R1, C2 > >::type multiply (const Eigen::Matrix< T1, R1, C1 > &m1, const Eigen::Matrix< T2, R2, C2 > &m2)
 Return the product of the specified matrices. More...
 
template<typename T1 , int C1, typename T2 , int R2>
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, var >::type multiply (const Eigen::Matrix< T1, 1, C1 > &rv, const Eigen::Matrix< T2, R2, 1 > &v)
 Return the scalar product of the specified row vector and specified column vector. More...
 
matrix_v multiply_lower_tri_self_transpose (const matrix_v &L)
 
template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, Eigen::Matrix< var, CB, CB > >::type quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)
 
template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c< boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, var >::type quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, 1 > &B)
 
template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, Eigen::Matrix< var, CB, CB > >::type quad_form_sym (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)
 
template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c< boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, var >::type quad_form_sym (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, 1 > &B)
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c< boost::is_same< T1, var >::value||boost::is_same< T2, var >::value, Eigen::Matrix< var, R1, 1 > >::type rows_dot_product (const Eigen::Matrix< T1, R1, C1 > &v1, const Eigen::Matrix< T2, R2, C2 > &v2)
 
var sd (const std::vector< var > &v)
 Return the sample standard deviation of the specified standard vector. More...
 
template<int R, int C>
var sd (const Eigen::Matrix< var, R, C > &m)
 
Eigen::Matrix< var, Eigen::Dynamic, 1 > softmax (const Eigen::Matrix< var, Eigen::Dynamic, 1 > &alpha)
 Return the softmax of the specified Eigen vector. More...
 
std::vector< varsort_asc (std::vector< var > xs)
 Return the specified standard vector in ascending order with gradients kept. More...
 
template<int R, int C>
Eigen::Matrix< var, R, C > sort_asc (Eigen::Matrix< var, R, C > xs)
 Return the specified eigen vector in ascending order with gradients kept. More...
 
std::vector< varsort_desc (std::vector< var > xs)
 Return the specified standard vector in descending order with gradients kept. More...
 
template<int R, int C>
Eigen::Matrix< var, R, C > sort_desc (Eigen::Matrix< var, R, C > xs)
 Return the specified eigen vector in descending order with gradients kept. More...
 
template<int R1, int C1, int R2, int C2>
var squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)
 
template<int R1, int C1, int R2, int C2>
var squared_distance (const Eigen::Matrix< var, R1, C1 > &v1, const Eigen::Matrix< double, R2, C2 > &v2)
 
template<int R1, int C1, int R2, int C2>
var squared_distance (const Eigen::Matrix< double, R1, C1 > &v1, const Eigen::Matrix< var, R2, C2 > &v2)
 
void stan_print (std::ostream *o, const var &x)
 
template<int R, int C>
var sum (const Eigen::Matrix< var, R, C > &m)
 Returns the sum of the coefficients of the specified matrix, column vector or row vector. More...
 
matrix_v tcrossprod (const matrix_v &M)
 Returns the result of post-multiplying a matrix by its own transpose. More...
 
var to_var (const double &x)
 Converts argument to an automatic differentiation variable. More...
 
var to_var (const var &x)
 Converts argument to an automatic differentiation variable. More...
 
matrix_v to_var (const stan::math::matrix_d &m)
 Converts argument to an automatic differentiation variable. More...
 
matrix_v to_var (const matrix_v &m)
 Converts argument to an automatic differentiation variable. More...
 
vector_v to_var (const stan::math::vector_d &v)
 Converts argument to an automatic differentiation variable. More...
 
vector_v to_var (const vector_v &v)
 Converts argument to an automatic differentiation variable. More...
 
row_vector_v to_var (const stan::math::row_vector_d &rv)
 Converts argument to an automatic differentiation variable. More...
 
row_vector_v to_var (const row_vector_v &rv)
 Converts argument to an automatic differentiation variable. More...
 
template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c< stan::is_var< T1 >::value||stan::is_var< T2 >::value||stan::is_var< T3 >::value, var >::type trace_gen_inv_quad_form_ldlt (const Eigen::Matrix< T1, R1, C1 > &D, const stan::math::LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
 Compute the trace of an inverse quadratic form. More...
 
template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same< TD, var >::value||boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, var >::type trace_gen_quad_form (const Eigen::Matrix< TD, RD, CD > &D, const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)
 
template<typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c< stan::is_var< T2 >::value||stan::is_var< T3 >::value, var >::type trace_inv_quad_form_ldlt (const stan::math::LDLT_factor< T2, R2, C2 > &A, const Eigen::Matrix< T3, R3, C3 > &B)
 Compute the trace of an inverse quadratic form. More...
 
template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same< TA, var >::value||boost::is_same< TB, var >::value, var >::type trace_quad_form (const Eigen::Matrix< TA, RA, CA > &A, const Eigen::Matrix< TB, RB, CB > &B)
 
template<int R, int C>
Eigen::Matrix< var, R, C > unit_vector_constrain (const Eigen::Matrix< var, R, C > &y)
 Return the unit length vector corresponding to the free vector y. More...
 
template<int R, int C>
Eigen::Matrix< var, R, C > unit_vector_constrain (const Eigen::Matrix< var, R, C > &y, var &lp)
 Return the unit length vector corresponding to the free vector y. More...
 
var variance (const std::vector< var > &v)
 Return the sample variance of the specified standard vector. More...
 
template<int R, int C>
var variance (const Eigen::Matrix< var, R, C > &m)
 
template<typename F >
void gradient (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, double &fx, Eigen::Matrix< double, Eigen::Dynamic, 1 > &grad_fx)
 Calculate the value and the gradient of the specified function at the specified argument. More...
 
template<typename F >
void jacobian (const F &f, const Eigen::Matrix< double, Eigen::Dynamic, 1 > &x, Eigen::Matrix< double, Eigen::Dynamic, 1 > &fx, Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &J)
 
var abs (const var &a)
 Return the absolute value of the variable (std). More...
 
var acos (const var &a)
 Return the principal value of the arc cosine of a variable, in radians (cmath). More...
 
var acosh (const var &a)
 The inverse hyperbolic cosine function for variables (C99). More...
 
int as_bool (const var &v)
 Return 1 if the argument is unequal to zero and 0 otherwise. More...
 
var asin (const var &a)
 Return the principal value of the arc sine, in radians, of the specified variable (cmath). More...
 
var asinh (const var &a)
 The inverse hyperbolic sine function for variables (C99). More...
 
var atan (const var &a)
 Return the principal value of the arc tangent, in radians, of the specified variable (cmath). More...
 
var atan2 (const var &a, const var &b)
 Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath). More...
 
var atan2 (const var &a, const double b)
 Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath). More...
 
var atan2 (const double a, const var &b)
 Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath). More...
 
var atanh (const var &a)
 The inverse hyperbolic tangent function for variables (C99). More...
 
var bessel_first_kind (const int &v, const var &a)
 
var bessel_second_kind (const int &v, const var &a)
 
var binary_log_loss (const int y, const stan::math::var &y_hat)
 The log loss function for variables (stan). More...
 
double calculate_chain (const double &x, const double &val)
 
var cbrt (const var &a)
 Returns the cube root of the specified variable (C99). More...
 
var ceil (const var &a)
 Return the ceiling of the specified variable (cmath). More...
 
var cos (const var &a)
 Return the cosine of a radian-scaled variable (cmath). More...
 
var cosh (const var &a)
 Return the hyperbolic cosine of the specified variable (cmath). More...
 
var digamma (const stan::math::var &a)
 
var erf (const var &a)
 The error function for variables (C99). More...
 
var erfc (const var &a)
 The complementary error function for variables (C99). More...
 
var exp (const var &a)
 Return the exponentiation of the specified variable (cmath). More...
 
var exp2 (const var &a)
 Exponentiation base 2 function for variables (C99). More...
 
var expm1 (const stan::math::var &a)
 The exponentiation of the specified variable minus 1 (C99). More...
 
var fabs (const var &a)
 Return the absolute value of the variable (cmath). More...
 
var falling_factorial (const var &a, const double &b)
 
var falling_factorial (const var &a, const var &b)
 
var falling_factorial (const double &a, const var &b)
 
var fdim (const stan::math::var &a, const stan::math::var &b)
 Return the positive difference between the first variable's the value and the second's (C99). More...
 
var fdim (const double &a, const stan::math::var &b)
 Return the positive difference between the first value and the value of the second variable (C99). More...
 
var fdim (const stan::math::var &a, const double &b)
 Return the positive difference between the first variable's value and the second value (C99). More...
 
var floor (const var &a)
 Return the floor of the specified variable (cmath). More...
 
var fma (const stan::math::var &a, const stan::math::var &b, const stan::math::var &c)
 The fused multiply-add function for three variables (C99). More...
 
var fma (const stan::math::var &a, const stan::math::var &b, const double &c)
 The fused multiply-add function for two variables and a value (C99). More...
 
var fma (const stan::math::var &a, const double &b, const stan::math::var &c)
 The fused multiply-add function for a variable, value, and variable (C99). More...
 
var fma (const stan::math::var &a, const double &b, const double &c)
 The fused multiply-add function for a variable and two values (C99). More...
 
var fma (const double &a, const stan::math::var &b, const double &c)
 The fused multiply-add function for a value, variable, and value (C99). More...
 
var fma (const double &a, const double &b, const stan::math::var &c)
 The fused multiply-add function for two values and a variable, and value (C99). More...
 
var fma (const double &a, const stan::math::var &b, const stan::math::var &c)
 The fused multiply-add function for a value and two variables (C99). More...
 
var fmax (const stan::math::var &a, const stan::math::var &b)
 Returns the maximum of the two variable arguments (C99). More...
 
var fmax (const stan::math::var &a, const double &b)
 Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
 
var fmax (const double &a, const stan::math::var &b)
 Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
 
var fmin (const stan::math::var &a, const stan::math::var &b)
 Returns the minimum of the two variable arguments (C99). More...
 
var fmin (const stan::math::var &a, double b)
 Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99). More...
 
var fmin (double a, const stan::math::var &b)
 Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99). More...
 
var fmod (const var &a, const var &b)
 Return the floating point remainder after dividing the first variable by the second (cmath). More...
 
var fmod (const var &a, const double b)
 Return the floating point remainder after dividing the the first variable by the second scalar (cmath). More...
 
var fmod (const double a, const var &b)
 Return the floating point remainder after dividing the first scalar by the second variable (cmath). More...
 
var gamma_p (const stan::math::var &a, const stan::math::var &b)
 
var gamma_p (const stan::math::var &a, const double &b)
 
var gamma_p (const double &a, const stan::math::var &b)
 
var gamma_q (const stan::math::var &a, const stan::math::var &b)
 
var gamma_q (const stan::math::var &a, const double &b)
 
var gamma_q (const double &a, const stan::math::var &b)
 
void grad_inc_beta (var &g1, var &g2, const var &a, const var &b, const var &z)
 
var hypot (const var &a, const var &b)
 Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
 
var hypot (const var &a, double b)
 Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
 
var hypot (double a, const var &b)
 Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99). More...
 
var ibeta (const var &a, const var &b, const var &x)
 The normalized incomplete beta function of a, b, and x. More...
 
var if_else (bool c, const var &y_true, const var &y_false)
 If the specified condition is true, return the first variable, otherwise return the second variable. More...
 
var if_else (bool c, double y_true, const var &y_false)
 If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable. More...
 
var if_else (bool c, const var &y_true, const double y_false)
 If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar. More...
 
var inc_beta (const stan::math::var &a, const stan::math::var &b, const stan::math::var &c)
 
var inv (const var &a)
 

\[ \mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
var inv_cloglog (const stan::math::var &a)
 Return the inverse complementary log-log function applied specified variable (stan). More...
 
var inv_logit (const stan::math::var &a)
 The inverse logit function for variables (stan). More...
 
var inv_Phi (const stan::math::var &p)
 The inverse of unit normal cumulative density function. More...
 
var inv_sqrt (const var &a)
 

\[ \mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
var inv_square (const var &a)
 

\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

More...
 
int is_inf (const var &v)
 Returns 1 if the input's value is infinite and 0 otherwise. More...
 
bool is_nan (const var &v)
 Returns 1 if the input's value is NaN and 0 otherwise. More...
 
bool is_uninitialized (var x)
 Returns true if the specified variable is uninitialized. More...
 
var lgamma (const stan::math::var &a)
 The log gamma function for variables (C99). More...
 
var lmgamma (int a, const stan::math::var &b)
 
var log (const var &a)
 Return the natural log of the specified variable (cmath). More...
 
var log10 (const var &a)
 Return the base 10 log of the specified variable (cmath). More...
 
var log1m (const stan::math::var &a)
 The log (1 - x) function for variables. More...
 
var log1m_exp (const stan::math::var &a)
 Return the log of 1 minus the exponential of the specified variable. More...
 
var log1p (const stan::math::var &a)
 The log (1 + x) function for variables (C99). More...
 
var log1p_exp (const stan::math::var &a)
 Return the log of 1 plus the exponential of the specified variable. More...
 
var log2 (const stan::math::var &a)
 Returns the base 2 logarithm of the specified variable (C99). More...
 
var log_diff_exp (const stan::math::var &a, const stan::math::var &b)
 Returns the log sum of exponentials. More...
 
var log_diff_exp (const stan::math::var &a, const double &b)
 Returns the log sum of exponentials. More...
 
var log_diff_exp (const double &a, const stan::math::var &b)
 Returns the log sum of exponentials. More...
 
var log_falling_factorial (const var &a, const double &b)
 
var log_falling_factorial (const var &a, const var &b)
 
var log_falling_factorial (const double &a, const var &b)
 
void log_mix_partial_helper (const double &theta_val, const double &lambda1_val, const double &lambda2_val, double &one_m_exp_lam2_m_lam1, double &one_m_t_prod_exp_lam2_m_lam1, double &one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1)
 
template<typename T_theta , typename T_lambda1 , typename T_lambda2 >
return_type< T_theta, T_lambda1, T_lambda2 >::type log_mix (const T_theta &theta, const T_lambda1 &lambda1, const T_lambda2 &lambda2)
 Return the log mixture density with specified mixing proportion and log densities and its derivative at each. More...
 
var log_rising_factorial (const var &a, const double &b)
 
var log_rising_factorial (const var &a, const var &b)
 
var log_rising_factorial (const double &a, const var &b)
 
var log_sum_exp (const stan::math::var &a, const stan::math::var &b)
 Returns the log sum of exponentials. More...
 
var log_sum_exp (const stan::math::var &a, const double &b)
 Returns the log sum of exponentials. More...
 
var log_sum_exp (const double &a, const stan::math::var &b)
 Returns the log sum of exponentials. More...
 
var modified_bessel_first_kind (const int &v, const var &a)
 
var modified_bessel_second_kind (const int &v, const var &a)
 
var multiply_log (const var &a, const var &b)
 Return the value of a*log(b). More...
 
var multiply_log (const var &a, const double b)
 Return the value of a*log(b). More...
 
var multiply_log (const double a, const var &b)
 Return the value of a*log(b). More...
 
var owens_t (const var &h, const var &a)
 The Owen's T function of h and a. More...
 
var owens_t (const var &h, double a)
 The Owen's T function of h and a. More...
 
var owens_t (double h, const var &a)
 The Owen's T function of h and a. More...
 
var Phi (const stan::math::var &a)
 The unit normal cumulative density function for variables (stan). More...
 
var Phi_approx (const stan::math::var &a)
 Approximation of the unit normal CDF for variables (stan). More...
 
var pow (const var &base, const var &exponent)
 Return the base raised to the power of the exponent (cmath). More...
 
var pow (const var &base, const double exponent)
 Return the base variable raised to the power of the exponent scalar (cmath). More...
 
var pow (const double base, const var &exponent)
 Return the base scalar raised to the power of the exponent variable (cmath). More...
 
double primitive_value (const var &v)
 Return the primitive double value for the specified auto-diff variable. More...
 
var rising_factorial (const var &a, const double &b)
 
var rising_factorial (const var &a, const var &b)
 
var rising_factorial (const double &a, const var &b)
 
var round (const var &a)
 Returns the rounded form of the specified variable (C99). More...
 
var sin (const var &a)
 Return the sine of a radian-scaled variable (cmath). More...
 
var sinh (const var &a)
 Return the hyperbolic sine of the specified variable (cmath). More...
 
var sqrt (const var &a)
 Return the square root of the specified variable (cmath). More...
 
var square (const var &x)
 Return the square of the input variable. More...
 
var step (const stan::math::var &a)
 Return the step, or heaviside, function applied to the specified variable (stan). More...
 
var tan (const var &a)
 Return the tangent of a radian-scaled variable (cmath). More...
 
var tanh (const var &a)
 Return the hyperbolic tangent of the specified variable (cmath). More...
 
var tgamma (const stan::math::var &a)
 Return the Gamma function applied to the specified variable (C99). More...
 
var trunc (const var &a)
 Returns the truncatation of the specified variable (C99). More...
 
double value_of (const var &v)
 Return the value of the specified variable. More...
 
double value_of_rec (const var &v)
 Return the value of the specified variable. More...
 

Variables

const double CONSTRAINT_TOLERANCE = 1E-8
 The tolerance for checking arithmetic bounds In rank and in simplexes. More...
 
const double E = boost::math::constants::e<double>()
 The base of the natural logarithm, $ e $. More...
 
const double SQRT_2 = std::sqrt(2.0)
 The value of the square root of 2, $ \sqrt{2} $. More...
 
const double INV_SQRT_2 = 1.0 / SQRT_2
 The value of 1 over the square root of 2, $ 1 / \sqrt{2} $. More...
 
const double LOG_2 = std::log(2.0)
 The natural logarithm of 2, $ \log 2 $. More...
 
const double LOG_10 = std::log(10.0)
 The natural logarithm of 10, $ \log 10 $. More...
 
const double INFTY = std::numeric_limits<double>::infinity()
 Positive infinity. More...
 
const double NEGATIVE_INFTY = - std::numeric_limits<double>::infinity()
 Negative infinity. More...
 
const double NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()
 (Quiet) not-a-number value. More...
 
const double EPSILON = std::numeric_limits<double>::epsilon()
 Smallest positive value. More...
 
const double NEGATIVE_EPSILON = - std::numeric_limits<double>::epsilon()
 Largest negative value (i.e., smallest absolute value). More...
 
const double POISSON_MAX_RATE = std::pow(2.0, 30)
 Largest rate parameter allowed in Poisson RNG. More...
 
const double LOG_PI_OVER_FOUR = std::log(boost::math::constants::pi<double>()) / 4.0
 Log pi divided by 4 $ \log \pi / 4 $. More...
 
const double SQRT_PI = std::sqrt(boost::math::constants::pi<double>())
 
const double SQRT_2_TIMES_SQRT_PI = SQRT_2 * SQRT_PI
 
const double TWO_OVER_SQRT_PI = 2.0 / SQRT_PI
 
const double NEG_TWO_OVER_SQRT_PI = -TWO_OVER_SQRT_PI
 
const double INV_SQRT_TWO_PI = 1.0 / std::sqrt(2.0 * boost::math::constants::pi<double>())
 
const double LOG_PI = std::log(boost::math::constants::pi<double>())
 
const double LOG_SQRT_PI = std::log(SQRT_PI)
 
const double LOG_ZERO = std::log(0.0)
 
const double LOG_TWO = std::log(2.0)
 
const double LOG_HALF = std::log(0.5)
 
const double NEG_LOG_TWO = - LOG_TWO
 
const double NEG_LOG_SQRT_TWO_PI = - std::log(std::sqrt(2.0 * boost::math::constants::pi<double>()))
 
const double NEG_LOG_PI = - LOG_PI
 
const double NEG_LOG_SQRT_PI = -std::log(std::sqrt(boost::math::constants::pi<double>()))
 
const double NEG_LOG_TWO_OVER_TWO = - LOG_TWO / 2.0
 
const double LOG_TWO_PI = LOG_TWO + LOG_PI
 
const double NEG_LOG_TWO_PI = - LOG_TWO_PI
 
const std::string MAJOR_VERSION = STAN_STRING(STAN_MATH_MAJOR)
 Major version number for Stan math library. More...
 
const std::string MINOR_VERSION = STAN_STRING(STAN_MATH_MINOR)
 Minor version number for Stan math library. More...
 
const std::string PATCH_VERSION = STAN_STRING(STAN_MATH_PATCH)
 Patch version for Stan math library. More...
 

Detailed Description

Matrices and templated mathematical functions.

Templated probability distributions. All paramaterizations are based on Bayesian Data Analysis. Function gradients via reverse-mode automatic differentiation.

Typedef Documentation

Definition at line 10 of file chainablestack.hpp.

typedef Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_d

Type for matrix of double values.

Definition at line 23 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_fd

Definition at line 17 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_ffd

Definition at line 21 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_ffv

Definition at line 18 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_fv

Definition at line 14 of file typedefs.hpp.

typedef Eigen::Matrix<var, Eigen::Dynamic, Eigen::Dynamic> stan::math::matrix_v

The type of a matrix holding stan::math::var values.

Definition at line 21 of file typedefs.hpp.

typedef Eigen::Matrix<double, 1, Eigen::Dynamic> stan::math::row_vector_d

Type for (row) vector of double values.

Definition at line 37 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<double>, 1, Eigen::Dynamic> stan::math::row_vector_fd

Definition at line 33 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >, 1, Eigen::Dynamic> stan::math::row_vector_ffd

Definition at line 37 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >, 1, Eigen::Dynamic> stan::math::row_vector_ffv

Definition at line 34 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>, 1, Eigen::Dynamic> stan::math::row_vector_fv

Definition at line 30 of file typedefs.hpp.

typedef Eigen::Matrix<var, 1, Eigen::Dynamic> stan::math::row_vector_v

The type of a row vector holding stan::math::var values.

Definition at line 37 of file typedefs.hpp.

typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic >::Index stan::math::size_type

Type for sizes and indexes in an Eigen matrix with double e.

Definition at line 13 of file typedefs.hpp.

typedef Eigen::Matrix<double, Eigen::Dynamic, 1> stan::math::vector_d

Type for (column) vector of double values.

Definition at line 30 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<double>, Eigen::Dynamic, 1> stan::math::vector_fd

Definition at line 25 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<double> >, Eigen::Dynamic, 1> stan::math::vector_ffd

Definition at line 29 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<fvar<var> >, Eigen::Dynamic, 1> stan::math::vector_ffv

Definition at line 26 of file typedefs.hpp.

typedef Eigen::Matrix<fvar<var>, Eigen::Dynamic, 1> stan::math::vector_fv

Definition at line 22 of file typedefs.hpp.

typedef Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::vector_v

The type of a (column) vector holding stan::math::var values.

Definition at line 29 of file typedefs.hpp.

Function Documentation

template<typename T >
fvar<T> stan::math::abs ( const fvar< T > &  x)
inline

Definition at line 15 of file abs.hpp.

double stan::math::abs ( double  x)

Return floating-point absolute value.

Delegates to fabs(double) rather than std::abs(int).

Parameters
xscalar
Returns
absolute value of scalar

Definition at line 19 of file abs.hpp.

var stan::math::abs ( const var a)
inline

Return the absolute value of the variable (std).

Delegates to fabs() (see for doc).

\[ \mbox{abs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{abs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable input.
Returns
Absolute value of variable.

Definition at line 35 of file abs.hpp.

template<typename T >
fvar<T> stan::math::acos ( const fvar< T > &  x)
inline

Definition at line 14 of file acos.hpp.

var stan::math::acos ( const var a)
inline

Return the principal value of the arc cosine of a variable, in radians (cmath).

The derivative is defined by

$\frac{d}{dx} \arccos x = \frac{-1}{\sqrt{1 - x^2}}$.

\[ \mbox{acos}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arccos(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acos}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arccos(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arccos(x)}{\partial x} = -\frac{1}{\sqrt{1-x^2}} \]

Parameters
aVariable in range [-1, 1].
Returns
Arc cosine of variable, in radians.

Definition at line 59 of file acos.hpp.

template<typename T >
fvar<T> stan::math::acosh ( const fvar< T > &  x)
inline

Definition at line 14 of file acosh.hpp.

var stan::math::acosh ( const var a)
inline

The inverse hyperbolic cosine function for variables (C99).

For non-variable function, see acosh().

The derivative is defined by

$\frac{d}{dx} \mbox{acosh}(x) = \frac{x}{x^2 - 1}$.

\[ \mbox{acosh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \cosh^{-1}(x) & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{acosh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 1 \\ \frac{\partial\, \cosh^{-1}(x)}{\partial x} & \mbox{if } x \geq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \cosh^{-1}(x)=\ln\left(x+\sqrt{x^2-1}\right) \]

\[ \frac{\partial \, \cosh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2-1}} \]

Parameters
aThe variable.
Returns
Inverse hyperbolic cosine of the variable.

Definition at line 68 of file acosh.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::add ( const Eigen::Matrix< T1, R, C > &  m1,
const Eigen::Matrix< T2, R, C > &  m2 
)
inline

Return the sum of the specified matrices.

The two matrices must have the same dimensions.

Template Parameters
T1Scalar type of first matrix.
T2Scalar type of second matrix.
RRow type of matrices.
CColumn type of matrices.
Parameters
m1First matrix.
m2Second matrix.
Returns
Sum of the matrices.
Exceptions
std::invalid_argumentif m1 and m2 do not have the same dimensions.

Definition at line 27 of file add.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::add ( const Eigen::Matrix< T1, R, C > &  m,
const T2 &  c 
)
inline

Return the sum of the specified matrix and specified scalar.

Template Parameters
T1Scalar type of matrix.
T2Type of scalar.
Parameters
mMatrix.
cScalar.
Returns
The matrix plus the scalar.

Definition at line 52 of file add.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::add ( const T1 &  c,
const Eigen::Matrix< T2, R, C > &  m 
)
inline

Return the sum of the specified scalar and specified matrix.

Template Parameters
T1Type of scalar.
T2Scalar type of matrix.
Parameters
cScalar.
mMatrix.
Returns
The scalar plus the matrix.

Definition at line 74 of file add.hpp.

void stan::math::add_initial_values ( const std::vector< stan::math::var > &  y0,
std::vector< std::vector< stan::math::var > > &  y 
)

Increment the state derived from the coupled system in the with the original initial state.

This is necessary because the coupled system subtracts out the initial state in its representation when the initial state is unknown.

Parameters
[in]y0original initial values to add back into the coupled system.
[in,out]ystate of the coupled system on input, incremented with initial values on output.

Definition at line 37 of file coupled_ode_system.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename return_type<T1, T2>::type, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  B 
)
inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

The inputs can be (matrix, matrix), (matrix, vector), (vector, matrix), or (vector, vector) and the output is always a matrix.

Template Parameters
T1Scalar type of first matrix.
T2Scalar type of second matrix.
R1Row specification of first matrix.
C1Column specification of first matrix.
R2Row specification of second matrix.
C2Column specification of second matrix.
Parameters
AFirst matrix.
BSecond matrix.
Returns
Result of appending the first matrix followed by the second matrix side by side.

Definition at line 39 of file append_col.hpp.

template<typename T1 , typename T2 , int C1, int C2>
Eigen::Matrix<typename return_type<T1, T2>::type, 1, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T1, 1, C1 > &  A,
const Eigen::Matrix< T2, 1, C2 > &  B 
)
inline

Return the result of concatenaing the first row vector followed by the second row vector side by side, with the result being a row vector.

This function applies to (row_vector, row_vector) and returns a row_vector.

Template Parameters
T1Scalar type of first row vector.
T2Scalar type of second row vector.
C1Column specification of first row vector.
C2Column specification of second row vector.
Parameters
AFirst vector.
BSecond vector
Returns
Result of appending the second row vector to the right of the first row vector.

Definition at line 85 of file append_col.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T, R1, C1 > &  A,
const Eigen::Matrix< T, R2, C2 > &  B 
)
inline

Return the result of appending the second argument matrix after the first argument matrix, that is, putting them side by side, with the first matrix followed by the second matrix.

This is an overloaded template function for the case when both matrices have the same type.

The inputs can be (matrix, matrix), (matrix, vector), (vector, matrix), or (vector, vector), and the output is always a matrix.

Template Parameters
TScalar type of both matrices.
R1Row specification of first matrix.
C1Column specification of first matrix.
R2Row specification of second matrix.
C2Column specification of second matrix.
Parameters
AFirst matrix.
BSecond matrix.
Returns
Result of appending the first matrix followed by the second matrix side by side.

Definition at line 128 of file append_col.hpp.

template<typename T , int C1, int C2>
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::append_col ( const Eigen::Matrix< T, 1, C1 > &  A,
const Eigen::Matrix< T, 1, C2 > &  B 
)
inline

Return the result of concatenaing the first row vector followed by the second row vector side by side, with the result being a row vector.

This function applies to (row_vector, row_vector) and returns a row_vector.

Template Parameters
TScalar type of both vectors.
C1Column specification of first row vector.
C2Column specification of second row vector.
Parameters
AFirst vector.
BSecond vector
Returns
Result of appending the second row vector to the right of the first row vector.

Definition at line 160 of file append_col.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename return_type<T1, T2>::type, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_row ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  B 
)
inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

The inputs can be (matrix, matrix), (matrix, row_vector), (row_vector, matrix), or (row_vector, row_vector), and the output is always a matrix.

Template Parameters
T1Scalar type of first matrix.
T2Scalar type of second matrix.
R1Row specification of first matrix.
C1Column specification of first matrix.
R2Row specification of second matrix.
C2Column specification of second matrix.
Parameters
AFirst matrix.
BSecond matrix.
Returns
Result of stacking first matrix on top of second.

Definition at line 37 of file append_row.hpp.

template<typename T1 , typename T2 , int R1, int R2>
Eigen::Matrix<typename return_type<T1, T2>::type, Eigen::Dynamic, 1> stan::math::append_row ( const Eigen::Matrix< T1, R1, 1 > &  A,
const Eigen::Matrix< T2, R2, 1 > &  B 
)
inline

Return the result of stacking the first vector on top of the second vector, with the result being a vector.

This function applies to (vector, vector) and returns a vector.

Template Parameters
T1Scalar type of first vector.
T2Scalar type of second vector.
R1Row specification of first vector.
R2Row specification of second vector.
Parameters
AFirst vector.
BSecond vector
Returns
Result of stacking first vector on top of the second vector.

Definition at line 80 of file append_row.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::append_row ( const Eigen::Matrix< T, R1, C1 > &  A,
const Eigen::Matrix< T, R2, C2 > &  B 
)
inline

Return the result of stacking the rows of the first argument matrix on top of the second argument matrix.

This is an overload for the case when the scalar types of the two input matrix are the same.

The inputs can be (matrix, matrix), (matrix, row_vector), (row_vector, matrix), or (row_vector, row_vector), and the output is always a matrix.

Template Parameters
TScalar type of both matrices.
R1Row specification of first matrix.
C1Column specification of first matrix.
R2Row specification of second matrix.
C2Column specification of second matrix.
Parameters
AFirst matrix.
BSecond matrix.
Returns
Result of stacking first matrix on top of second.

Definition at line 121 of file append_row.hpp.

template<typename T , int R1, int R2>
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::append_row ( const Eigen::Matrix< T, R1, 1 > &  A,
const Eigen::Matrix< T, R2, 1 > &  B 
)
inline

Return the result of stacking the first vector on top of the second vector, with the result being a vector.

This is an overloaded template function for the case where both inputs have the same scalar type.

This function applies to (vector, vector) and returns a vector.

Template Parameters
TScalar type of both vectors.
R1Row specification of first vector.
R2Row specification of second vector.
Parameters
AFirst vector.
BSecond vector
Returns
Result of stacking first vector on top of the second vector.

Definition at line 155 of file append_row.hpp.

template<typename T >
bool stan::math::as_bool ( const T  x)
inline

Return 1 if the argument is unequal to zero and 0 otherwise.

Parameters
xValue.
Returns
1 if argument is equal to zero (or NaN) and 0 otherwise.

Definition at line 14 of file as_bool.hpp.

int stan::math::as_bool ( const var v)
inline

Return 1 if the argument is unequal to zero and 0 otherwise.

Parameters
vValue.
Returns
1 if argument is equal to zero (or NaN) and 0 otherwise.

Definition at line 15 of file as_bool.hpp.

template<typename T >
fvar<T> stan::math::asin ( const fvar< T > &  x)
inline

Definition at line 12 of file asin.hpp.

var stan::math::asin ( const var a)
inline

Return the principal value of the arc sine, in radians, of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$.

\[ \mbox{asin}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \arcsin(x) & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asin}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \arcsin(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x < -1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arcsin(x)}{\partial x} = \frac{1}{\sqrt{1-x^2}} \]

Parameters
aVariable in range [-1, 1].
Returns
Arc sine of variable, in radians.

Definition at line 58 of file asin.hpp.

template<typename T >
fvar<T> stan::math::asinh ( const fvar< T > &  x)
inline

Definition at line 13 of file asinh.hpp.

var stan::math::asinh ( const var a)
inline

The inverse hyperbolic sine function for variables (C99).

For non-variable function, see asinh().

The derivative is defined by

$\frac{d}{dx} \mbox{asinh}(x) = \frac{x}{x^2 + 1}$.

\[ \mbox{asinh}(x) = \begin{cases} \sinh^{-1}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} \frac{\partial\, \sinh^{-1}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right) \]

\[ \frac{\partial \, \sinh^{-1}(x)}{\partial x} = \frac{1}{\sqrt{x^2+1}} \]

Parameters
aThe variable.
Returns
Inverse hyperbolic sine of the variable.

Definition at line 67 of file asinh.hpp.

template<typename LHS , typename RHS >
void stan::math::assign ( LHS &  lhs,
const RHS &  rhs 
)
inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will match arguments where the right-hand side is assignable to the left and they are not both std::vector or Eigen::Matrix types.

Template Parameters
LHSType of left-hand side.
RHSType of right-hand side.
Parameters
lhsLeft-hand side.
rhsRight-hand side.

Definition at line 51 of file assign.hpp.

template<typename LHS , typename RHS , int R1, int C1, int R2, int C2>
void stan::math::assign ( Eigen::Matrix< LHS, R1, C1 > &  x,
const Eigen::Matrix< RHS, R2, C2 > &  y 
)
inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types, but whose shapes are not compatible. The shapes are specified in the row and column template parameters.

Template Parameters
LHSType of left-hand side matrix elements.
RHSType of right-hand side matrix elements.
R1Row shape of left-hand side matrix.
C1Column shape of left-hand side matrix.
R2Row shape of right-hand side matrix.
C2Column shape of right-hand side matrix.
Parameters
xLeft-hand side matrix.
yRight-hand side matrix.
Exceptions
std::invalid_argument

Definition at line 77 of file assign.hpp.

template<typename LHS , typename RHS , int R, int C>
void stan::math::assign ( Eigen::Matrix< LHS, R, C > &  x,
const Eigen::Matrix< RHS, R, C > &  y 
)
inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types and whose shapes match. The shapes are specified in the row and column template parameters.

Template Parameters
LHSType of left-hand side matrix elements.
RHSType of right-hand side matrix elements.
RRow shape of both matrices.
CColumn shape of both mtarices.
Parameters
xLeft-hand side matrix.
yRight-hand side matrix.
Exceptions
std::invalid_argumentif sizes do not match.

Definition at line 113 of file assign.hpp.

template<typename LHS , typename RHS , int R, int C>
void stan::math::assign ( Eigen::Block< LHS >  x,
const Eigen::Matrix< RHS, R, C > &  y 
)
inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both Eigen::Matrix types and whose shapes match. The shape of the right-hand side matrix is specified in the row and column shape template parameters.

Template Parameters
LHSType of matrix block elements.
RHSType of right-hand side matrix elements.
RRow shape for right-hand side matrix.
CColumn shape for right-hand side matrix.
Parameters
xLeft-hand side block view of matrix.
yRight-hand side matrix.
Exceptions
std::invalid_argumentif sizes do not match.

Definition at line 142 of file assign.hpp.

template<typename LHS , typename RHS >
void stan::math::assign ( std::vector< LHS > &  x,
const std::vector< RHS > &  y 
)
inline

Copy the right-hand side's value to the left-hand side variable.

The assign() function is overloaded. This instance will be called for arguments that are both std::vector, and will call assign() element-by element.

For example, a std::vector<int> can be assigned to a std::vector<double> using this function.

Template Parameters
LHSType of left-hand side vector elements.
RHSType of right-hand side vector elements.
Parameters
xLeft-hand side vector.
yRight-hand side vector.
Exceptions
std::invalid_argumentif sizes do not match.

Definition at line 177 of file assign.hpp.

template<typename T >
fvar<T> stan::math::atan ( const fvar< T > &  x)
inline

Definition at line 12 of file atan.hpp.

var stan::math::atan ( const var a)
inline

Return the principal value of the arc tangent, in radians, of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$.

\[ \mbox{atan}(x) = \begin{cases} \arctan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan}(x)}{\partial x} = \begin{cases} \frac{\partial\, \arctan(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial \, \arctan(x)}{\partial x} = \frac{1}{x^2+1} \]

Parameters
aVariable in range [-1, 1].
Returns
Arc tangent of variable, in radians.

Definition at line 55 of file atan.hpp.

template<typename T >
fvar<T> stan::math::atan2 ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 12 of file atan2.hpp.

template<typename T >
fvar<T> stan::math::atan2 ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 21 of file atan2.hpp.

template<typename T >
fvar<T> stan::math::atan2 ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 29 of file atan2.hpp.

var stan::math::atan2 ( const var a,
const var b 
)
inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second (cmath).

The partial derivatives are defined by

$ \frac{\partial}{\partial x} \arctan \frac{x}{y} = \frac{y}{x^2 + y^2}$, and

$ \frac{\partial}{\partial y} \arctan \frac{x}{y} = \frac{-x}{x^2 + y^2}$.

Parameters
aNumerator variable.
bDenominator variable.
Returns
The arc tangent of the fraction, in radians.

Definition at line 62 of file atan2.hpp.

var stan::math::atan2 ( const var a,
const double  b 
)
inline

Return the principal value of the arc tangent, in radians, of the first variable divided by the second scalar (cmath).

The derivative with respect to the variable is

$ \frac{d}{d x} \arctan \frac{x}{c} = \frac{c}{x^2 + c^2}$.

Parameters
aNumerator variable.
bDenominator scalar.
Returns
The arc tangent of the fraction, in radians.

Definition at line 78 of file atan2.hpp.

var stan::math::atan2 ( const double  a,
const var b 
)
inline

Return the principal value of the arc tangent, in radians, of the first scalar divided by the second variable (cmath).

The derivative with respect to the variable is

$ \frac{\partial}{\partial y} \arctan \frac{c}{y} = \frac{-c}{c^2 + y^2}$.

\[ \mbox{atan2}(x, y) = \begin{cases} \arctan\left(\frac{x}{y}\right) & \mbox{if } -\infty\leq x \leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial x} = \begin{cases} \frac{y}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atan2}(x, y)}{\partial y} = \begin{cases} -\frac{x}{x^2+y^2} & \mbox{if } -\infty\leq x\leq \infty, -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aNumerator scalar.
bDenominator variable.
Returns
The arc tangent of the fraction, in radians.

Definition at line 119 of file atan2.hpp.

template<typename T >
fvar<T> stan::math::atanh ( const fvar< T > &  x)
inline

Definition at line 13 of file atanh.hpp.

var stan::math::atanh ( const var a)
inline

The inverse hyperbolic tangent function for variables (C99).

For non-variable function, see atanh().

The derivative is defined by

$\frac{d}{dx} \mbox{atanh}(x) = \frac{1}{1 - x^2}$.

\[ \mbox{atanh}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \tanh^{-1}(x) & \mbox{if } -1\leq x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{atanh}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{\partial\, \tanh^{-1}(x)}{\partial x} & \mbox{if } -1\leq x\leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \tanh^{-1}(x)=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \]

\[ \frac{\partial \, \tanh^{-1}(x)}{\partial x} = \frac{1}{1-x^2} \]

Parameters
aThe variable.
Returns
Inverse hyperbolic tangent of the variable.

Definition at line 70 of file atanh.hpp.

template<typename T >
void stan::math::autocorrelation ( const std::vector< T > &  y,
std::vector< T > &  ac,
Eigen::FFT< T > &  fft 
)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;
Template Parameters
TScalar type.
Parameters
yInput sequence.
acAutocorrelations.
fftFFT engine instance.

Definition at line 54 of file autocorrelation.hpp.

template<typename T >
void stan::math::autocorrelation ( const std::vector< T > &  y,
std::vector< T > &  ac 
)

Write autocorrelation estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocorrelation function

Template Parameters
TScalar type.
Parameters
yInput sequence.
acAutocorrelations.

Definition at line 123 of file autocorrelation.hpp.

template<typename T >
void stan::math::autocovariance ( const std::vector< T > &  y,
std::vector< T > &  acov,
Eigen::FFT< T > &  fft 
)

Write autocovariance estimates for every lag for the specified input sequence into the specified result using the specified FFT engine.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

An FFT engine can be created for reuse for type double with:

    Eigen::FFT<double> fft;
Template Parameters
TScalar type.
Parameters
yInput sequence.
acovAutocovariance.
fftFFT engine instance.

Definition at line 34 of file autocovariance.hpp.

template<typename T >
void stan::math::autocovariance ( const std::vector< T > &  y,
std::vector< T > &  acov 
)

Write autocovariance estimates for every lag for the specified input sequence into the specified result.

The return vector be resized to the same length as the input sequence with lags given by array index.

The implementation involves a fast Fourier transform, followed by a normalization, followed by an inverse transform.

This method is just a light wrapper around the three-argument autocovariance function

Template Parameters
TScalar type.
Parameters
yInput sequence.
acovAutocovariances.

Definition at line 62 of file autocovariance.hpp.

template<typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_ccdf_log ( const T_n &  n,
const T_prob &  theta 
)

Definition at line 24 of file bernoulli_ccdf_log.hpp.

template<typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_cdf ( const T_n &  n,
const T_prob &  theta 
)

Definition at line 24 of file bernoulli_cdf.hpp.

template<typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_cdf_log ( const T_n &  n,
const T_prob &  theta 
)

Definition at line 24 of file bernoulli_cdf_log.hpp.

template<bool propto, typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_log ( const T_n &  n,
const T_prob &  theta 
)

Definition at line 26 of file bernoulli_log.hpp.

template<typename T_y , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_log ( const T_y &  n,
const T_prob &  theta 
)
inline

Definition at line 120 of file bernoulli_log.hpp.

template<bool propto, typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_logit_log ( const T_n &  n,
const T_prob &  theta 
)

Definition at line 26 of file bernoulli_logit_log.hpp.

template<typename T_n , typename T_prob >
return_type<T_prob>::type stan::math::bernoulli_logit_log ( const T_n &  n,
const T_prob &  theta 
)
inline

Definition at line 104 of file bernoulli_logit_log.hpp.

template<class RNG >
int stan::math::bernoulli_rng ( const double  theta,
RNG &  rng 
)
inline

Definition at line 23 of file bernoulli_rng.hpp.

template<typename T >
fvar<T> stan::math::bessel_first_kind ( int  v,
const fvar< T > &  z 
)
inline

Definition at line 15 of file bessel_first_kind.hpp.

var stan::math::bessel_first_kind ( const int &  v,
const var a 
)
inline

Definition at line 27 of file bessel_first_kind.hpp.

template<typename T2 >
T2 stan::math::bessel_first_kind ( const int  v,
const T2  z 
)
inline

\[ \mbox{bessel\_first\_kind}(v, x) = \begin{cases} J_v(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_first\_kind}(v, x)}{\partial x} = \begin{cases} \frac{\partial\, J_v(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ J_v(x)=\left(\frac{1}{2}x\right)^v \sum_{k=0}^\infty \frac{\left(-\frac{1}{4}x^2\right)^k}{k!\, \Gamma(v+k+1)} \]

\[ \frac{\partial \, J_v(x)}{\partial x} = \frac{v}{x}J_v(x)-J_{v+1}(x) \]

Definition at line 40 of file bessel_first_kind.hpp.

template<typename T >
fvar<T> stan::math::bessel_second_kind ( int  v,
const fvar< T > &  z 
)
inline

Definition at line 15 of file bessel_second_kind.hpp.

var stan::math::bessel_second_kind ( const int &  v,
const var a 
)
inline

Definition at line 27 of file bessel_second_kind.hpp.

template<typename T2 >
T2 stan::math::bessel_second_kind ( const int  v,
const T2  z 
)
inline

\[ \mbox{bessel\_second\_kind}(v, x) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ Y_v(x) & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{bessel\_second\_kind}(v, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0 \\ \frac{\partial\, Y_v(x)}{\partial x} & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)} \]

\[ \frac{\partial \, Y_v(x)}{\partial x} = \frac{v}{x}Y_v(x)-Y_{v+1}(x) \]

Definition at line 40 of file bessel_second_kind.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type<T_size1, T_size2>::type stan::math::beta_binomial_ccdf_log ( const T_n &  n,
const T_N &  N,
const T_size1 &  alpha,
const T_size2 &  beta 
)

Definition at line 28 of file beta_binomial_ccdf_log.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type<T_size1, T_size2>::type stan::math::beta_binomial_cdf ( const T_n &  n,
const T_N &  N,
const T_size1 &  alpha,
const T_size2 &  beta 
)

Definition at line 29 of file beta_binomial_cdf.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type<T_size1, T_size2>::type stan::math::beta_binomial_cdf_log ( const T_n &  n,
const T_N &  N,
const T_size1 &  alpha,
const T_size2 &  beta 
)

Definition at line 28 of file beta_binomial_cdf_log.hpp.

template<bool propto, typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type<T_size1, T_size2>::type stan::math::beta_binomial_log ( const T_n &  n,
const T_N &  N,
const T_size1 &  alpha,
const T_size2 &  beta 
)

Definition at line 30 of file beta_binomial_log.hpp.

template<typename T_n , typename T_N , typename T_size1 , typename T_size2 >
return_type<T_size1, T_size2>::type stan::math::beta_binomial_log ( const T_n &  n,
const T_N &  N,
const T_size1 &  alpha,
const T_size2 &  beta 
)

Definition at line 175 of file beta_binomial_log.hpp.

template<class RNG >
int stan::math::beta_binomial_rng ( const int  N,
const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 26 of file beta_binomial_rng.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type<T_y, T_scale_succ, T_scale_fail>::type stan::math::beta_ccdf_log ( const T_y &  y,
const T_scale_succ &  alpha,
const T_scale_fail &  beta 
)

Definition at line 34 of file beta_ccdf_log.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type<T_y, T_scale_succ, T_scale_fail>::type stan::math::beta_cdf ( const T_y &  y,
const T_scale_succ &  alpha,
const T_scale_fail &  beta 
)

Calculates the beta cumulative distribution function for the given variate and scale variables.

Parameters
yA scalar variate.
alphaPrior sample size.
betaPrior sample size.
Returns
The beta cdf evaluated at the specified arguments.
Template Parameters
T_yType of y.
T_scale_succType of alpha.
T_scale_failType of beta.

Definition at line 52 of file beta_cdf.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type<T_y, T_scale_succ, T_scale_fail>::type stan::math::beta_cdf_log ( const T_y &  y,
const T_scale_succ &  alpha,
const T_scale_fail &  beta 
)

Definition at line 33 of file beta_cdf_log.hpp.

template<bool propto, typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type<T_y, T_scale_succ, T_scale_fail>::type stan::math::beta_log ( const T_y &  y,
const T_scale_succ &  alpha,
const T_scale_fail &  beta 
)

The log of the beta density for the specified scalar(s) given the specified sample size(s).

y, alpha, or beta can each either be scalar or a vector. Any vector inputs must be the same length.

The result log probability is defined to be the sum of the log probabilities for each observation/alpha/beta triple.

Prior sample sizes, alpha and beta, must be greater than 0.

Parameters
y(Sequence of) scalar(s).
alpha(Sequence of) prior sample size(s).
beta(Sequence of) prior sample size(s).
Returns
The log of the product of densities.
Template Parameters
T_yType of scalar outcome.
T_scale_succType of prior scale for successes.
T_scale_failType of prior scale for failures.

Definition at line 52 of file beta_log.hpp.

template<typename T_y , typename T_scale_succ , typename T_scale_fail >
return_type<T_y, T_scale_succ, T_scale_fail>::type stan::math::beta_log ( const T_y &  y,
const T_scale_succ &  alpha,
const T_scale_fail &  beta 
)
inline

Definition at line 209 of file beta_log.hpp.

template<class RNG >
double stan::math::beta_rng ( const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 30 of file beta_rng.hpp.

template<typename T >
fvar<T> stan::math::binary_log_loss ( const int  y,
const fvar< T > &  y_hat 
)
inline

Definition at line 15 of file binary_log_loss.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::binary_log_loss ( const int  y,
const T  y_hat 
)
inline

Returns the log loss function for binary classification with specified reference and response values.

The log loss function for prediction $\hat{y} \in [0, 1]$ given outcome $y \in \{ 0, 1 \}$ is

$\mbox{logloss}(1, \hat{y}) = -\log \hat{y} $, and

$\mbox{logloss}(0, \hat{y}) = -\log (1 - \hat{y}) $.

Parameters
yReference value in { 0 , 1 }.
y_hatResponse value in [0, 1].
Returns
Log loss for response given reference value.

Definition at line 26 of file binary_log_loss.hpp.

var stan::math::binary_log_loss ( const int  y,
const stan::math::var y_hat 
)
inline

The log loss function for variables (stan).

See stan::math::binary_log_loss() for the double-based version.

The derivative with respect to the variable $\hat{y}$ is

$\frac{d}{d\hat{y}} \mbox{logloss}(1, \hat{y}) = - \frac{1}{\hat{y}}$, and

$\frac{d}{d\hat{y}} \mbox{logloss}(0, \hat{y}) = \frac{1}{1 - \hat{y}}$.

\[ \mbox{binary\_log\_loss}(y, \hat{y}) = \begin{cases} y \log \hat{y} + (1 - y) \log (1 - \hat{y}) & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binary\_log\_loss}(y, \hat{y})}{\partial \hat{y}} = \begin{cases} \frac{y}{\hat{y}}-\frac{1-y}{1-\hat{y}} & \mbox{if } 0\leq \hat{y}\leq 1, y\in\{ 0, 1 \}\\[6pt] \textrm{NaN} & \mbox{if } \hat{y} = \textrm{NaN} \end{cases} \]

Parameters
yReference value.
y_hatResponse variable.
Returns
Log loss of response versus reference value.

Definition at line 68 of file binary_log_loss.hpp.

template<typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_ccdf_log ( const T_n &  n,
const T_N &  N,
const T_prob &  theta 
)

Definition at line 31 of file binomial_ccdf_log.hpp.

template<typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_cdf ( const T_n &  n,
const T_N &  N,
const T_prob &  theta 
)

Definition at line 32 of file binomial_cdf.hpp.

template<typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_cdf_log ( const T_n &  n,
const T_N &  N,
const T_prob &  theta 
)

Definition at line 31 of file binomial_cdf_log.hpp.

template<typename T >
fvar<T> stan::math::binomial_coefficient_log ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 16 of file binomial_coefficient_log.hpp.

template<typename T >
fvar<T> stan::math::binomial_coefficient_log ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 46 of file binomial_coefficient_log.hpp.

template<typename T_N , typename T_n >
boost::math::tools::promote_args<T_N, T_n>::type stan::math::binomial_coefficient_log ( const T_N  N,
const T_n  n 
)
inline

Return the log of the binomial coefficient for the specified arguments.

The binomial coefficient, ${N \choose n}$, read "N choose n", is defined for $0 \leq n \leq N$ by

${N \choose n} = \frac{N!}{n! (N-n)!}$.

This function uses Gamma functions to define the log and generalize the arguments to continuous N and n.

$ \log {N \choose n} = \log \ \Gamma(N+1) - \log \Gamma(n+1) - \log \Gamma(N-n+1)$.

\[ \mbox{binomial\_coefficient\_log}(x, y) = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ \ln\Gamma(x+1) & \mbox{if } 0\leq y \leq x \\ \quad -\ln\Gamma(y+1)& \\ \quad -\ln\Gamma(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ \Psi(x+1) & \mbox{if } 0\leq y \leq x \\ \quad -\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{binomial\_coefficient\_log}(x, y)}{\partial y} = \begin{cases} \textrm{error} & \mbox{if } y > x \textrm{ or } y < 0\\ -\Psi(y+1) & \mbox{if } 0\leq y \leq x \\ \quad +\Psi(x-y+1)& \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
Ntotal number of objects.
nnumber of objects chosen.
Returns
log (N choose n).

Definition at line 63 of file binomial_coefficient_log.hpp.

template<typename T >
fvar<T> stan::math::binomial_coefficient_log ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 70 of file binomial_coefficient_log.hpp.

template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_log ( const T_n &  n,
const T_N &  N,
const T_prob &  theta 
)

Definition at line 36 of file binomial_log.hpp.

template<typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_log ( const T_n &  n,
const T_N &  N,
const T_prob &  theta 
)
inline

Definition at line 127 of file binomial_log.hpp.

template<bool propto, typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_logit_log ( const T_n &  n,
const T_N &  N,
const T_prob &  alpha 
)

Definition at line 37 of file binomial_logit_log.hpp.

template<typename T_n , typename T_N , typename T_prob >
return_type<T_prob>::type stan::math::binomial_logit_log ( const T_n &  n,
const T_N &  N,
const T_prob &  alpha 
)
inline

Definition at line 131 of file binomial_logit_log.hpp.

template<class RNG >
int stan::math::binomial_rng ( const int  N,
const double  theta,
RNG &  rng 
)
inline

Definition at line 31 of file binomial_rng.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::block ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m,
size_t  i,
size_t  j,
size_t  nrows,
size_t  ncols 
)
inline

Return a nrows x ncols submatrix starting at (i-1, j-1).

Parameters
mMatrix
iStarting row
jStarting column
nrowsNumber of rows in block
ncolsNumber of columns in block

Definition at line 23 of file block.hpp.

double stan::math::calculate_chain ( const double &  x,
const double &  val 
)
inline

Definition at line 8 of file calculate_chain.hpp.

template<bool propto, typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_log ( int  n,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Definition at line 25 of file categorical_log.hpp.

template<typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_log ( const typename math::index_type< Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > >::type  n,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)
inline

Definition at line 56 of file categorical_log.hpp.

template<bool propto, typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Definition at line 68 of file categorical_log.hpp.

template<typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)
inline

Definition at line 116 of file categorical_log.hpp.

template<bool propto, typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_logit_log ( int  n,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta 
)

Definition at line 22 of file categorical_logit_log.hpp.

template<typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_logit_log ( int  n,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta 
)
inline

Definition at line 45 of file categorical_logit_log.hpp.

template<bool propto, typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_logit_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta 
)

Definition at line 54 of file categorical_logit_log.hpp.

template<typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::categorical_logit_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  beta 
)
inline

Definition at line 89 of file categorical_logit_log.hpp.

template<class RNG >
int stan::math::categorical_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  theta,
RNG &  rng 
)
inline

Definition at line 20 of file categorical_rng.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::cauchy_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file cauchy_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::cauchy_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Calculates the cauchy cumulative distribution function for the given variate, location, and scale.

$\frac{1}{\pi}\arctan\left(\frac{y-\mu}{\sigma}\right) + \frac{1}{2}$

Parameters
yA scalar variate.
muThe location parameter.
sigmaThe scale parameter.
Returns

Definition at line 36 of file cauchy_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::cauchy_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file cauchy_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::cauchy_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

The log of the Cauchy density for the specified scalar(s) given the specified location parameter(s) and scale parameter(s).

y, mu, or sigma can each either be scalar a vector. Any vector inputs must be the same length.

The result log probability is defined to be the sum of the log probabilities for each observation/mu/sigma triple.

Parameters
y(Sequence of) scalar(s).
mu(Sequence of) location(s).
sigma(Sequence of) scale(s).
Returns
The log of the product of densities.
Template Parameters
T_yType of scalar outcome.
T_locType of location.
T_scaleType of scale.

Definition at line 43 of file cauchy_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::cauchy_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)
inline

Definition at line 145 of file cauchy_log.hpp.

template<class RNG >
double stan::math::cauchy_rng ( const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 23 of file cauchy_rng.hpp.

template<typename T >
fvar<T> stan::math::cbrt ( const fvar< T > &  x)
inline

Definition at line 14 of file cbrt.hpp.

var stan::math::cbrt ( const var a)
inline

Returns the cube root of the specified variable (C99).

See cbrt() for the double-based version.

The derivative is

$\frac{d}{dx} x^{1/3} = \frac{1}{3 x^{2/3}}$.

\[ \mbox{cbrt}(x) = \begin{cases} \sqrt[3]{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cbrt}(x)}{\partial x} = \begin{cases} \frac{1}{3x^{2/3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aSpecified variable.
Returns
Cube root of the variable.

Definition at line 56 of file cbrt.hpp.

template<typename T >
fvar<T> stan::math::ceil ( const fvar< T > &  x)
inline

Definition at line 11 of file ceil.hpp.

var stan::math::ceil ( const var a)
inline

Return the ceiling of the specified variable (cmath).

The derivative of the ceiling function is defined and zero everywhere but at integers, and we set them to zero for convenience,

$\frac{d}{dx} {\lceil x \rceil} = 0$.

The ceiling function rounds up. For double values, this is the smallest integral value that is not less than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{ceil}(x) = \begin{cases} \lceil x\rceil & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{ceil}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aInput variable.
Returns
Ceiling of the variable.

Definition at line 60 of file ceil.hpp.

template<typename T_y , typename T_low , typename T_high >
bool stan::math::check_bounded ( const char *  function,
const char *  name,
const T_y &  y,
const T_low &  low,
const T_high &  high 
)
inline

Return true if the value is between the low and high values, inclusively.

Template Parameters
T_yType of value
T_lowType of low value
T_highType of high value
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yValue to check
lowLow bound
highHigh bound
Returns
true if the value is between low and high, inclusively.
Exceptions
<code>std::domain_error</code>otherwise. This also throws if any of the arguments are NaN.

Definition at line 95 of file check_bounded.hpp.

template<typename T_y >
bool stan::math::check_cholesky_factor ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is a valid Cholesky factor.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Template Parameters
T_yType of elements of Cholesky factor
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is a valid Cholesky factor
Exceptions
<code>std::domain_error</code>if y is not a valid Choleksy factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 35 of file check_cholesky_factor.hpp.

template<typename T_y >
bool stan::math::check_cholesky_factor_corr ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)

Return true if the specified matrix is a valid Cholesky factor of a correlation matrix.

A Cholesky factor is a lower triangular matrix whose diagonal elements are all positive. Note that Cholesky factors need not be square, but require at least as many rows M as columns N (i.e., M >= N).

Tolerance is specified by math::CONSTRAINT_TOLERANCE.

Template Parameters
T_yType of elements of Cholesky factor
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is a valid Cholesky factor of a correlation matrix
Exceptions
<code>std::domain_error</code>if y is not a valid Choleksy factor, if number of rows is less than the number of columns, if there are 0 columns, or if any element in matrix is NaN

Definition at line 39 of file check_cholesky_factor_corr.hpp.

template<typename T_y , int R, int C>
bool stan::math::check_column_index ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, R, C > &  y,
const size_t  i 
)
inline

Return true if the specified index is a valid column of the matrix.

By default, this is a 1-indexed check (as opposed to 0-indexed). Behavior can be changed by setting stan::error_index::value. This function will throw an std::out_of_range exception if the index is out of bounds.

Template Parameters
T_yType of scalar.
RNumber of rows of the matrix
CNumber of columns of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix
iIndex to check
Returns
true if the index is a valid column index of the matrix.
Exceptions
std::out_of_rangeif index is an invalid column index

Definition at line 37 of file check_column_index.hpp.

template<typename T >
bool stan::math::check_consistent_size ( const char *  function,
const char *  name,
const T &  x,
size_t  expected_size 
)
inline

Return true if the dimension of x is consistent, which is defined to be expected_size if x is a vector or 1 if x is not a vector.

Template Parameters
TType of value
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
xVariable to check for consistent size
expected_sizeExpected size if x is a vector
Returns
true if x is scalar or if x is vector-like and has size of expected_size
Exceptions
<code>invalid_argument</code>if the size is inconsistent

Definition at line 29 of file check_consistent_size.hpp.

template<typename T1 , typename T2 >
bool stan::math::check_consistent_sizes ( const char *  function,
const char *  name1,
const T1 &  x1,
const char *  name2,
const T2 &  x2 
)
inline

Return true if the dimension of x1 is consistent with x2.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters
T1Type of x1
T2Type of x2
Parameters
functionFunction name (for error messages)
name1Variable name (for error messages)
x1Variable to check for consistent size
name2Variable name (for error messages)
x2Variable to check for consistent size
Returns
true if x1 and x2 have consistent sizes
Exceptions
<code>invalid_argument</code>if sizes are inconsistent

Definition at line 31 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 >
bool stan::math::check_consistent_sizes ( const char *  function,
const char *  name1,
const T1 &  x1,
const char *  name2,
const T2 &  x2,
const char *  name3,
const T3 &  x3 
)
inline

Return true if the dimension of x1, x2, and x3 are consistent.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters
T1Type of x1
T2Type of x2
T3Type of x3
Parameters
functionFunction name (for error messages)
name1Variable name (for error messages)
x1Variable to check for consistent size
name2Variable name (for error messages)
x2Variable to check for consistent size
name3Variable name (for error messages)
x3Variable to check for consistent size
Returns
true if x1, x2, and x3 have consistent sizes
Exceptions
<code>invalid_argument</code>if sizes are inconsistent

Definition at line 66 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 >
bool stan::math::check_consistent_sizes ( const char *  function,
const char *  name1,
const T1 &  x1,
const char *  name2,
const T2 &  x2,
const char *  name3,
const T3 &  x3,
const char *  name4,
const T4 &  x4 
)
inline

Return true if the dimension of x1, x2, x3, and x4 are consistent.

Consistent size is defined as having the same size if vector-like or being a scalar.

Template Parameters
T1Type of x1
T2Type of x2
T3Type of x3
T4Type of x4
Parameters
functionFunction name (for error messages)
name1Variable name (for error messages)
x1Variable to check for consistent size
name2Variable name (for error messages)
x2Variable to check for consistent size
name3Variable name (for error messages)
x3Variable to check for consistent size
name4Variable name (for error messages)
x4Variable to check for consistent size
Returns
true if x1, x2, x3, and x4 have consistent sizes
Exceptions
<code>invalid_argument</code>if sizes are inconsistent

Definition at line 107 of file check_consistent_sizes.hpp.

template<typename T1 , typename T2 , typename T3 , typename T4 , typename T5 >
bool stan::math::check_consistent_sizes ( const char *  function,
const char *  name1,
const T1 &  x1,
const char *  name2,
const T2 &  x2,
const char *  name3,
const T3 &  x3,
const char *  name4,
const T4 &  x4,
const char *  name5,
const T5 &  x5 
)
inline

Definition at line 128 of file check_consistent_sizes.hpp.

template<typename T_y >
bool stan::math::check_corr_matrix ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is a valid correlation matrix.

A valid correlation matrix is symmetric, has a unit diagonal (all 1 values), and has all values between -1 and 1 (inclusive).

This function throws exceptions if the variable is not a valid correlation matrix.

Template Parameters
T_yType of scalar
Parameters
functionName of the function this was called from
nameName of the variable
yMatrix to test
Returns
true if the specified matrix is a valid correlation matrix
Exceptions
<code>std::invalid_argument</code>if the matrix is not square or if the matrix is 0x0
<code>std::domain_error</code>if the matrix is non-symmetric, diagonals not near 1, not positive definite, or any of the elements nan.

Definition at line 45 of file check_corr_matrix.hpp.

template<typename T_y >
bool stan::math::check_cov_matrix ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is a valid covariance matrix.

A valid covariance matrix is a square, symmetric matrix that is positive definite.

Template Parameters
TType of scalar.
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is a valid covariance matrix
Exceptions
<code>std::invalid_argument</code>if the matrix is not square or if the matrix is 0x0
<code>std::domain_error</code>if the matrix is not symmetric, if the matrix is not positive definite, or if any element of the matrix is nan

Definition at line 31 of file check_cov_matrix.hpp.

template<typename T_y , typename T_eq >
bool stan::math::check_equal ( const char *  function,
const char *  name,
const T_y &  y,
const T_eq &  eq 
)
inline

Return true if y is equal to eq.

This function is vectorized over both y and eq. If both y and eq are scalar or vector-like, then each element is compared in order. If one of y or eq are vector and the other is scalar, then the scalar is broadcast to the size of the vector.

Template Parameters
T_yType of variable
T_eqType of comparison
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check equality
eqExpected value for y
Returns
true if y is equal to eq
Exceptions
<code>std::domain_error</code>if y is unequal to eq or if any element of y or eq is NaN.

Definition at line 90 of file check_equal.hpp.

template<typename T_y >
bool stan::math::check_finite ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if y is finite.

This function is vectorized and will check each element of y.

Template Parameters
T_yType of y
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
Returns
true if y is finite.
Exceptions
<code>domain_error</code>if y is infinity, -infinity, or NaN.

Definition at line 62 of file check_finite.hpp.

template<typename T_y , typename T_low >
bool stan::math::check_greater ( const char *  function,
const char *  name,
const T_y &  y,
const T_low &  low 
)
inline

Return true if y is strictly greater than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters
T_yType of y
T_lowType of lower bound
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
lowLower bound
Returns
true if y is strictly greater than low.
Exceptions
<code>domain_error</code>if y is not greater than low or if any element of y or low is NaN.

Definition at line 84 of file check_greater.hpp.

template<typename T_y , typename T_low >
bool stan::math::check_greater_or_equal ( const char *  function,
const char *  name,
const T_y &  y,
const T_low &  low 
)
inline

Return true if y is greater or equal than low.

This function is vectorized and will check each element of y against each element of low.

Template Parameters
T_yType of y
T_lowType of lower bound
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
lowLower bound
Returns
true if y is greater or equal than low.
Exceptions
<code>domain_error</code>if y is not greater or equal to low or if any element of y or low is NaN.

Definition at line 84 of file check_greater_or_equal.hpp.

template<typename T , int R, int C>
bool stan::math::check_ldlt_factor ( const char *  function,
const char *  name,
stan::math::LDLT_factor< T, R, C > &  A 
)
inline

Return true if the argument is a valid stan::math::LDLT_factor.

LDLT_factor can be constructed in an invalid state, so it must be checked. A invalid LDLT_factor is constructed from a non positive definite matrix.

Template Parameters
TType of scalar
RRows of the matrix
CColumns of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
Astan::math::LDLT_factor to check for validity.
Returns
true if the matrix is positive definite.
throws std::domain_error the LDLT_factor was created improperly (A.success() == false)

Definition at line 34 of file check_ldlt_factor.hpp.

template<typename T_y , typename T_high >
bool stan::math::check_less ( const char *  function,
const char *  name,
const T_y &  y,
const T_high &  high 
)
inline

Return true if y is strictly less than high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters
T_yType of y
T_highType of upper bound
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
highUpper bound
Returns
true if y is strictly less than low.
Exceptions
<code>domain_error</code>if y is not less than low or if any element of y or high is NaN.

Definition at line 81 of file check_less.hpp.

template<typename T_y , typename T_high >
bool stan::math::check_less_or_equal ( const char *  function,
const char *  name,
const T_y &  y,
const T_high &  high 
)
inline

Return true if y is less or equal to high.

This function is vectorized and will check each element of y against each element of high.

Template Parameters
T_yType of y
T_highType of upper bound
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
highUpper bound
Returns
true if y is less than or equal to low.
Exceptions
<code>std::domain_error</code>if y is not less than or equal to low or if any element of y or high is NaN.

Definition at line 81 of file check_less_or_equal.hpp.

template<typename T_y >
bool stan::math::check_lower_triangular ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is lower triangular.

A matrix x is not lower triangular if there is a non-zero entry x[m, n] with m < n. This function only inspects the upper triangular portion of the matrix, not including the diagonal.

Template Parameters
TType of scalar of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is lower triangular.
Exceptions
<code>std::domain_error</code>if the matrix is not lower triangular or if any element in the upper triangular portion is NaN

Definition at line 34 of file check_lower_triangular.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
bool stan::math::check_matching_dims ( const char *  function,
const char *  name1,
const Eigen::Matrix< T1, R1, C1 > &  y1,
const char *  name2,
const Eigen::Matrix< T2, R2, C2 > &  y2 
)
inline

Return true if the two matrices are of the same size.

This function checks not only the runtime sizes, but the static sizes as well. For example, a 4x1 matrix is not the same as a vector with 4 elements.

Template Parameters
T1Scalar type of the first matrix
T2Scalar type of the second matrix
R1Rows specified at compile time of the first matrix
C1Columns specified at compile time of the first matrix
R2Rows specified at compile time of the second matrix
C2Columns specified at compile time of the second matrix
Parameters
functionFunction name (for error messages)
name1Variable name for the first matrix (for error messages)
y1First matrix
name2Variable name for the second matrix (for error messages)
y2Second matrix
Returns
true if the dimensions of the two matrices match
Exceptions
<code>std::invalid_argument</code>if the dimensions of the matrices do not match

Definition at line 37 of file check_matching_dims.hpp.

template<typename T_y1 , typename T_y2 >
bool stan::math::check_matching_sizes ( const char *  function,
const char *  name1,
const T_y1 &  y1,
const char *  name2,
const T_y2 &  y2 
)
inline

Return true if two structures at the same size.

This function only checks the runtime sizes for variables that implement a size() method.

Template Parameters
T_y1Type of the first variable
T_y2Type of the second variable
Parameters
functionFunction name (for error messages)
name1First variable name (for error messages)
y1First variable
name2Second variable name (for error messages)
y2Second variable
Returns
true if the sizes match
Exceptions
<code>std::invalid_argument</code>if the sizes do not match

Definition at line 29 of file check_matching_sizes.hpp.

template<typename T1 , typename T2 >
bool stan::math::check_multiplicable ( const char *  function,
const char *  name1,
const T1 &  y1,
const char *  name2,
const T2 &  y2 
)
inline

Return true if the matrices can be multiplied.

This checks the runtime sizes to determine whether the two matrices are multiplicable. This allows Eigen matrices, vectors, and row vectors to be checked.

Template Parameters
T1Type of first matrix
T2Type of second matrix
Parameters
functionFunction name (for error messages)
name1Variable name for the first matrix (for error messages)
y1First matrix
name2Variable name for the second matrix (for error messages)
y2Second matrix
Returns
true if the two matrices are multiplicable
Exceptions
<code>std::invalid_argument</code>if the matrices are not multiplicable or if either matrix is size 0 for either rows or columns

Definition at line 33 of file check_multiplicable.hpp.

template<typename T_y >
bool stan::math::check_nonnegative ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if y is non-negative.

This function is vectorized and will check each element of y.

Template Parameters
T_yType of y
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
Returns
true if y is greater than or equal to 0.
Exceptions
<code>domain_error</code>if y is negative or if any element of y is NaN.

Definition at line 66 of file check_nonnegative.hpp.

template<typename T_y >
bool stan::math::check_nonzero_size ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if the specified matrix/vector is of non-zero size.

Throws a std:invalid_argument otherwise. The message will indicate that the variable name "has size 0".

Template Parameters
T_yType of container
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yContainer to test. This will accept matrices and vectors
Returns
true if the the specified matrix/vector is of non-zero size
Exceptions
<code>std::invalid_argument</code>if the specified matrix/vector has zero size

Definition at line 31 of file check_nonzero_size.hpp.

template<typename T_y >
bool stan::math::check_not_nan ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if y is not NaN.

This function is vectorized and will check each element of y. If any element is NaN, this function will throw an exception.

Template Parameters
T_yType of y
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
Returns
true if y is not NaN.
Exceptions
<code>domain_error</code>if any element of y is NaN.

Definition at line 63 of file check_not_nan.hpp.

template<typename T_y >
bool stan::math::check_ordered ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &  y 
)

Return true if the specified vector is sorted into strictly increasing order.

Template Parameters
T_yType of scalar
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVector to test
Returns
true if the vector is ordered
Exceptions
<code>std::domain_error</code>if the vector elements are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 31 of file check_ordered.hpp.

template<typename T_y >
bool stan::math::check_ordered ( const char *  function,
const char *  name,
const std::vector< T_y > &  y 
)

Return true if the specified vector is sorted into strictly increasing order.

Template Parameters
T_yType of scalar
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
ystd::vector to test
Returns
true if the vector is ordered
Exceptions
<code>std::domain_error</code>if the vector elements are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 78 of file check_ordered.hpp.

template<typename T_y >
bool stan::math::check_pos_definite ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified square, symmetric matrix is positive definite.

Template Parameters
T_yType of scalar of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is positive definite
Exceptions
<code>std::invalid_argument</code>if the matrix is not square or if the matrix has 0 size.
<code>std::domain_error</code>if the matrix is not symmetric, if it is not positive definite, or if any element is NaN.

Definition at line 37 of file check_pos_definite.hpp.

template<typename Derived >
bool stan::math::check_pos_definite ( const char *  function,
const char *  name,
const Eigen::LDLT< Derived > &  cholesky 
)
inline

Return true if the specified LDLT transform of a matrix is positive definite.

Template Parameters
DerivedDerived type of the Eigen::LDLT transform.
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
choleskyEigen::LDLT to test, whose progenitor must not have any NaN elements
Returns
true if the matrix is positive definite
Exceptions
<code>std::domain_error</code>if the matrix is not positive definite.

Definition at line 77 of file check_pos_definite.hpp.

template<typename Derived >
bool stan::math::check_pos_definite ( const char *  function,
const char *  name,
const Eigen::LLT< Derived > &  cholesky 
)
inline

Return true if the specified LLT transform of a matrix is positive definite.

Template Parameters
DerivedDerived type of the Eigen::LLT transform.
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
choleskyEigen::LLT to test, whose progenitor must not have any NaN elements
Returns
true if the matrix is positive definite
Exceptions
<code>std::domain_error</code>if the diagonal of the L matrix is not positive.

Definition at line 103 of file check_pos_definite.hpp.

template<typename T_y >
bool stan::math::check_pos_semidefinite ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is positive definite.

Template Parameters
T_yscalar type of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is positive semi-definite.
Exceptions
<code>std::invalid_argument</code>if the matrix is not square or if the matrix has 0 size.
<code>std::domain_error</code>if the matrix is not symmetric, or if it is not positive semi-definite, or if any element of the matrix is NaN.

Definition at line 35 of file check_pos_semidefinite.hpp.

template<typename T_y >
bool stan::math::check_positive ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if y is positive.

This function is vectorized and will check each element of y.

Template Parameters
T_yType of y
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
Returns
true if y is greater than 0.
Exceptions
<code>domain_error</code>if y is negative or zero or if any element of y is NaN.

Definition at line 68 of file check_positive.hpp.

template<typename T_y >
bool stan::math::check_positive_finite ( const char *  function,
const char *  name,
const T_y &  y 
)
inline

Return true if y is positive and finite.

This function is vectorized and will check each element of y.

Template Parameters
T_yType of y
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVariable to check
Returns
true if every element of y is greater than 0 and y is not infinite.
Exceptions
<code>domain_error</code>if any element of y is not positive or if any element of y is NaN.

Definition at line 28 of file check_positive_finite.hpp.

template<typename T_y >
bool stan::math::check_positive_ordered ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, 1 > &  y 
)

Return true if the specified vector contains non-negative values and is sorted into strictly increasing order.

Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yVector to test
Returns
true if the vector is positive, ordered
Exceptions
<code>std::domain_error</code>if the vector contains non-positive values, if the values are not ordered, if there are duplicated values, or if any element is NaN.

Definition at line 32 of file check_positive_ordered.hpp.

bool stan::math::check_positive_size ( const char *  function,
const char *  name,
const char *  expr,
const int  size 
)
inline

Return true if size is positive.

Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
exprExpression for the dimension size (for error messages)
sizeSize value to check
Returns
true if size is greater than 0.
Exceptions
<code>std::invalid_argument</code>if size is zero or negative.

Definition at line 23 of file check_positive_size.hpp.

bool stan::math::check_range ( const char *  function,
const char *  name,
const int  max,
const int  index,
const int  nested_level,
const char *  error_msg 
)
inline

Return true if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
maxMaximum size of the variable
indexIndex to check
nested_levelNested level (for error messages)
error_msgAdditional error message (for error messages)
Returns
true if the index is within range
Exceptions
<code>std::out_of_range</code>if the index is not in range

Definition at line 29 of file check_range.hpp.

bool stan::math::check_range ( const char *  function,
const char *  name,
const int  max,
const int  index,
const char *  error_msg 
)
inline

Return true if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
maxMaximum size of the variable
indexIndex to check
error_msgAdditional error message (for error messages)
Returns
true if the index is within range
Exceptions
<code>std::out_of_range</code>if the index is not in range

Definition at line 62 of file check_range.hpp.

bool stan::math::check_range ( const char *  function,
const char *  name,
const int  max,
const int  index 
)
inline

Return true if specified index is within range.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
maxMaximum size of the variable
indexIndex to check
Returns
true if the index is within range
Exceptions
<code>std::out_of_range</code>if the index is not in range

Definition at line 89 of file check_range.hpp.

template<typename T_y , int R, int C>
bool stan::math::check_row_index ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, R, C > &  y,
size_t  i 
)
inline

Return true if the specified index is a valid row of the matrix.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Template Parameters
TScalar type
RCompile time rows
CCompile time columns
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
iis index
Returns
true if the index is a valid row index in the matrix
Exceptions
<code>std::out_of_range</code>if the index is out of range.

Definition at line 32 of file check_row_index.hpp.

template<typename T_prob >
bool stan::math::check_simplex ( const char *  function,
const char *  name,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Return true if the specified vector is simplex.

To be a simplex, all values must be greater than or equal to 0 and the values must sum to 1.

A valid simplex is one where the sum of hte elements is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters
T_probScalar type of the vector
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
thetaVector to test.
Returns
true if the vector is a simplex
Exceptions
<code>std::invalid_argument</code>if theta is a 0-vector.
<code>std::domain_error</code>if the vector is not a simplex or if any element is NaN.

Definition at line 41 of file check_simplex.hpp.

template<typename T_size1 , typename T_size2 >
bool stan::math::check_size_match ( const char *  function,
const char *  name_i,
T_size1  i,
const char *  name_j,
T_size2  j 
)
inline

Return true if the provided sizes match.

Template Parameters
T_size1Type of size 1
T_size2Type of size 2
Parameters
functionFunction name (for error messages)
name_iVariable name 1 (for error messages)
iSize 1
name_jVariable name 2 (for error messages)
jSize 2
Returns
true if the sizes match
Exceptions
<code>std::invalid_argument</code>if the sizes do not match

Definition at line 30 of file check_size_match.hpp.

template<typename T_size1 , typename T_size2 >
bool stan::math::check_size_match ( const char *  function,
const char *  expr_i,
const char *  name_i,
T_size1  i,
const char *  expr_j,
const char *  name_j,
T_size2  j 
)
inline

Return true if the provided sizes match.

Template Parameters
T_size1Type of size 1
T_size2Type of size 2
Parameters
functionFunction name (for error messages)
expr_iExpression for variable name 1 (for error messages)
name_iVariable name 1 (for error messages)
iSize 1
expr_jExpression for variable name 2 (for error messages)
name_jVariable name 2 (for error messages)
jSize 2
Returns
true if the sizes match
Exceptions
<code>std::invalid_argument</code>if the sizes do not match

Definition at line 67 of file check_size_match.hpp.

template<typename T_y >
bool stan::math::check_spsd_matrix ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is a square, symmetric, and positive semi-definite.

Template Parameters
TScalar type of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is a square, symmetric, and positive semi-definite.
Exceptions
<code>std::invalid_argument</code>if the matrix is not square or if the matrix is 0x0
<code>std::domain_error</code>if the matrix is not symmetric or if the matrix is not positive semi-definite

Definition at line 31 of file check_spsd_matrix.hpp.

template<typename T_y >
bool stan::math::check_square ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is square.

This check allows 0x0 matrices.

Template Parameters
TType of scalar.
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is a square matrix.
Exceptions
<code>std::invalid_argument</code>if the matrix is not square

Definition at line 28 of file check_square.hpp.

template<typename T >
bool stan::math::check_std_vector_index ( const char *  function,
const char *  name,
const std::vector< T > &  y,
int  i 
)
inline

Return true if the specified index is valid in std vector.

This check is 1-indexed by default. This behavior can be changed by setting stan::error_index::value.

Template Parameters
TScalar type
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
ystd::vector to test
iIndex
Returns
true if the index is a valid in std vector.
Exceptions
<code>std::out_of_range</code>if the index is out of range.

Definition at line 30 of file check_std_vector_index.hpp.

template<typename T_y >
bool stan::math::check_symmetric ( const char *  function,
const char *  name,
const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y 
)
inline

Return true if the specified matrix is symmetric.

The error message is either 0 or 1 indexed, specified by stan::error_index::value.

Template Parameters
T_yType of scalar.
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
yMatrix to test
Returns
true if the matrix is symmetric
Exceptions
<code>std::invalid_argument</code>if the matrix is not square.
<code>std::domain_error</code>if any element not on the main diagonal is NaN

Definition at line 37 of file check_symmetric.hpp.

template<typename T_prob >
bool stan::math::check_unit_vector ( const char *  function,
const char *  name,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Return true if the specified vector is unit vector.

A valid unit vector is one where the square of the elements summed is equal to 1. This function tests that the sum is within the tolerance specified by CONSTRAINT_TOLERANCE. This function only accepts Eigen vectors, statically typed vectors, not general matrices with 1 column.

Template Parameters
T_probScalar type of the vector
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
thetaVector to test.
Returns
true if the vector is a unit vector.
Exceptions
<code>std::invalid_argument</code>if theta is a 0-vector.
<code>std::domain_error</code>if the vector is not a unit vector or if any element is NaN.

Definition at line 36 of file check_unit_vector.hpp.

template<typename T , int R, int C>
bool stan::math::check_vector ( const char *  function,
const char *  name,
const Eigen::Matrix< T, R, C > &  x 
)
inline

Return true if the matrix is either a row vector or column vector.

This function checks the runtime size of the matrix to check whether it is a row or column vector.

Template Parameters
TScalar type of the matrix
RCompile time rows of the matrix
CCompile time columns of the matrix
Parameters
functionFunction name (for error messages)
nameVariable name (for error messages)
xMatrix
Returns
true if x either has 1 columns or 1 rows
Exceptions
<code>std::invalid_argument</code>if x is not a row or column vector.

Definition at line 34 of file check_vector.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::chi_square_ccdf_log ( const T_y &  y,
const T_dof &  nu 
)

Definition at line 28 of file chi_square_ccdf_log.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::chi_square_cdf ( const T_y &  y,
const T_dof &  nu 
)

Calculates the chi square cumulative distribution function for the given variate and degrees of freedom.

y A scalar variate. nu Degrees of freedom.

Returns
The cdf of the chi square distribution

Definition at line 37 of file chi_square_cdf.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::chi_square_cdf_log ( const T_y &  y,
const T_dof &  nu 
)

Definition at line 28 of file chi_square_cdf_log.hpp.

template<bool propto, typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::chi_square_log ( const T_y &  y,
const T_dof &  nu 
)

The log of a chi-squared density for y with the specified degrees of freedom parameter.

The degrees of freedom prarameter must be greater than 0. y must be greater than or equal to 0.

\begin{eqnarray*} y &\sim& \chi^2_\nu \\ \log (p (y \, |\, \nu)) &=& \log \left( \frac{2^{-\nu / 2}}{\Gamma (\nu / 2)} y^{\nu / 2 - 1} \exp^{- y / 2} \right) \\ &=& - \frac{\nu}{2} \log(2) - \log (\Gamma (\nu / 2)) + (\frac{\nu}{2} - 1) \log(y) - \frac{y}{2} \\ & & \mathrm{ where } \; y \ge 0 \end{eqnarray*}

Parameters
yA scalar variable.
nuDegrees of freedom.
Exceptions
std::domain_errorif nu is not greater than or equal to 0
std::domain_errorif y is not greater than or equal to 0.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.

Definition at line 47 of file chi_square_log.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::chi_square_log ( const T_y &  y,
const T_dof &  nu 
)
inline

Definition at line 144 of file chi_square_log.hpp.

template<class RNG >
double stan::math::chi_square_rng ( const double  nu,
RNG &  rng 
)
inline

Definition at line 25 of file chi_square_rng.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y,
int  K 
)

Definition at line 20 of file cholesky_corr_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_corr_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y,
int  K,
T &  lp 
)

Definition at line 58 of file cholesky_corr_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cholesky_corr_free ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x)

Definition at line 18 of file cholesky_corr_free.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_decompose ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Return the lower-triangular Cholesky factor (i.e., matrix square root) of the specified square, symmetric matrix.

The return value $L$ will be a lower-traingular matrix such that the original matrix $A$ is given by

$A = L \times L^T$.

Parameters
mSymmetrix matrix.
Returns
Square root of matrix.
Exceptions
std::domain_errorif m is not a symmetric matrix or if m is not positive definite (if m has more than 0 elements)

Definition at line 25 of file cholesky_decompose.hpp.

Eigen::Matrix<var, -1, -1> stan::math::cholesky_decompose ( const Eigen::Matrix< var,-1,-1 > &  A)

Definition at line 131 of file cholesky_decompose.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
int  M,
int  N 
)

Return the Cholesky factor of the specified size read from the specified vector.

A total of (N choose 2) + N + (M - N) * N elements are required to read an M by N Cholesky factor.

Template Parameters
TType of scalars in matrix
Parameters
xVector of unconstrained values
MNumber of rows
NNumber of columns
Returns
Cholesky factor

Definition at line 29 of file cholesky_factor_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cholesky_factor_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
int  M,
int  N,
T &  lp 
)

Return the Cholesky factor of the specified size read from the specified vector and increment the specified log probability reference with the log Jacobian adjustment of the transform.

A total of (N choose 2) + N + N * (M - N) free parameters are required to read an M by N Cholesky factor.

Template Parameters
TType of scalars in matrix
Parameters
xVector of unconstrained values
MNumber of rows
NNumber of columns
lpLog probability that is incremented with the log Jacobian
Returns
Cholesky factor

Definition at line 73 of file cholesky_factor_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cholesky_factor_free ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  y)

Return the unconstrained vector of parameters correspdonding to the specified Cholesky factor.

A Cholesky factor must be lower triangular and have positive diagonal elements.

Parameters
yCholesky factor.
Returns
Unconstrained parameters for Cholesky factor.
Exceptions
std::domain_errorIf the matrix is not a Cholesky factor.

Definition at line 24 of file cholesky_factor_free.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::col ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m,
size_t  j 
)
inline

Return the specified column of the specified matrix using start-at-1 indexing.

This is equivalent to calling m.col(i - 1) and assigning the resulting template expression to a column vector.

Parameters
mMatrix.
jColumn index (count from 1).
Returns
Specified column of the matrix.

Definition at line 24 of file col.hpp.

template<typename T , int R, int C>
int stan::math::cols ( const Eigen::Matrix< T, R, C > &  m)
inline

Return the number of columns in the specified matrix, vector, or row vector.

Template Parameters
TType of matrix entries.
RRow type of matrix.
CColumn type of matrix.
Parameters
[in]mInput matrix, vector, or row vector.
Returns
Number of columns.

Definition at line 20 of file cols.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 18 of file columns_dot_product.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<double, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Returns the dot product of the specified vectors.

Parameters
v1First vector.
v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorIf the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 22 of file columns_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c<boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, Eigen::Matrix<var, 1, C1> >::type stan::math::columns_dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1,
const Eigen::Matrix< T2, R2, C2 > &  v2 
)
inline

Definition at line 25 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Definition at line 35 of file columns_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, 1, C1> stan::math::columns_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 52 of file columns_dot_product.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  x)
inline

Definition at line 15 of file columns_dot_self.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< T, R, C > &  x)
inline

Returns the dot product of each column of a matrix with itself.

Parameters
xMatrix.
Template Parameters
Tscalar type

Definition at line 16 of file columns_dot_self.hpp.

template<int R, int C>
Eigen::Matrix<var, 1, C> stan::math::columns_dot_self ( const Eigen::Matrix< var, R, C > &  x)
inline

Returns the dot product of each column of a matrix with itself.

Parameters
xMatrix.
Template Parameters
Tscalar type

Definition at line 22 of file columns_dot_self.hpp.

template<typename T >
T stan::math::corr_constrain ( const T  x)
inline

Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive).

The transform used is the hyperbolic tangent function,

$f(x) = \tanh x = \frac{\exp(2x) - 1}{\exp(2x) + 1}$.

Parameters
xScalar input.
Returns
Result of transforming the input to fall between -1 and 1.
Template Parameters
TType of scalar.

Definition at line 25 of file corr_constrain.hpp.

template<typename T >
T stan::math::corr_constrain ( const T  x,
T &  lp 
)
inline

Return the result of transforming the specified scalar to have a valid correlation value between -1 and 1 (inclusive).

The transform used is as specified for corr_constrain(T). The log absolute Jacobian determinant is

$\log | \frac{d}{dx} \tanh x | = \log (1 - \tanh^2 x)$.

Template Parameters
TType of scalar.

Definition at line 43 of file corr_constrain.hpp.

template<typename T >
T stan::math::corr_free ( const T  y)
inline

Return the unconstrained scalar that when transformed to a valid correlation produces the specified value.

This function inverts the transform defined for corr_constrain(T), which is the inverse hyperbolic tangent,

$ f^{-1}(y) = \mbox{atanh}\, y = \frac{1}{2} \log \frac{y + 1}{y - 1}$.

Parameters
yCorrelation scalar input.
Returns
Free scalar that transforms to the specified input.
Template Parameters
TType of scalar.

Definition at line 29 of file corr_free.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type  k 
)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length ${k \choose 2} = \frac{k(k-1)}{2}$. The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform based on partial correlations is as specified in

  • Lewandowski, Daniel, Dorota Kurowicka, and Harry Joe. 2009. Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis 100:1989–-2001.

The free vector entries are first constrained to be valid correlation values using corr_constrain(T).

Parameters
xVector of unconstrained partial correlations.
kDimensionality of returned correlation matrix.
Template Parameters
TType of scalar.
Exceptions
std::invalid_argumentif x is not a valid correlation matrix.

Definition at line 40 of file corr_matrix_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::corr_matrix_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type  k,
T &  lp 
)

Return the correlation matrix of the specified dimensionality derived from the specified vector of unconstrained values.

The input vector must be of length ${k \choose 2} = \frac{k(k-1)}{2}$. The values in the input vector represent unconstrained (partial) correlations among the dimensions.

The transform is as specified for corr_matrix_constrain(Matrix, size_t); the paper it cites also defines the Jacobians for correlation inputs, which are composed with the correlation constrained Jacobians defined in corr_constrain(T, double) for this function.

Parameters
xVector of unconstrained partial correlations.
kDimensionality of returned correlation matrix.
lpLog probability reference to increment.
Template Parameters
TType of scalar.

Definition at line 78 of file corr_matrix_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::corr_matrix_free ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  y)

Return the vector of unconstrained partial correlations that define the specified correlation matrix when transformed.

The constraining transform is defined as for corr_matrix_constrain(Matrix, size_t). The inverse transform in this function is simpler in that it only needs to compute the $k \choose 2$ partial correlations and then free those.

Parameters
yThe correlation matrix to free.
Returns
Vector of unconstrained values that produce the specified correlation matrix when transformed.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif the correlation matrix has no elements or is not a square matrix.
std::runtime_errorif the correlation matrix cannot be factorized by factor_cov_matrix() or if the sds returned by factor_cov_matrix() on log scale are unconstrained.

Definition at line 38 of file corr_matrix_free.hpp.

template<typename T >
fvar<T> stan::math::cos ( const fvar< T > &  x)
inline

Definition at line 13 of file cos.hpp.

var stan::math::cos ( const var a)
inline

Return the cosine of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \cos x = - \sin x$.

\[ \mbox{cos}(x) = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cos}(x)}{\partial x} = \begin{cases} -\sin(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable for radians of angle.
Returns
Cosine of variable.

Definition at line 49 of file cos.hpp.

template<typename T >
fvar<T> stan::math::cosh ( const fvar< T > &  x)
inline

Definition at line 13 of file cosh.hpp.

var stan::math::cosh ( const var a)
inline

Return the hyperbolic cosine of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \cosh x = \sinh x$.

\[ \mbox{cosh}(x) = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{cosh}(x)}{\partial x} = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable.
Returns
Hyperbolic cosine of variable.

Definition at line 50 of file cosh.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, 1 > >::type  K 
)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Parameters
xThe vector to convert to a covariance matrix.
KThe number of rows and columns of the resulting covariance matrix.
Exceptions
std::domain_errorif (x.size() != K + (K choose 2)).

Definition at line 30 of file cov_matrix_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
typename math::index_type< Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > >::type  K,
T &  lp 
)

Return the symmetric, positive-definite matrix of dimensions K by K resulting from transforming the specified finite vector of size K plus (K choose 2).

See cov_matrix_free() for the inverse transform.

Parameters
xThe vector to convert to a covariance matrix.
KThe dimensions of the resulting covariance matrix.
lpReference
Exceptions
std::domain_errorif (x.size() != K + (K choose 2)).

Definition at line 70 of file cov_matrix_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
size_t  k 
)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values.

The input vector must be of length $k \choose 2 + k$. The first $k \choose 2$ values in the input represent unconstrained (partial) correlations and the last $k$ are unconstrained standard deviations of the dimensions.

The transform scales the correlation matrix transform defined in corr_matrix_constrain(Matrix, size_t) with the constrained deviations.

Parameters
xInput vector of unconstrained partial correlations and standard deviations.
kDimensionality of returned covariance matrix.
Returns
Covariance matrix derived from the unconstrained partial correlations and deviations.
Template Parameters
TType of scalar.

Definition at line 34 of file cov_matrix_constrain_lkj.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::cov_matrix_constrain_lkj ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
size_t  k,
T &  lp 
)

Return the covariance matrix of the specified dimensionality derived from constraining the specified vector of unconstrained values and increment the specified log probability reference with the log absolute Jacobian determinant.

The transform is defined as for cov_matrix_constrain(Matrix, size_t).

The log absolute Jacobian determinant is derived by composing the log absolute Jacobian determinant for the underlying correlation matrix as defined in cov_matrix_constrain(Matrix, size_t, T&) with the Jacobian of the transfrom of the correlation matrix into a covariance matrix by scaling by standard deviations.

Parameters
xInput vector of unconstrained partial correlations and standard deviations.
kDimensionality of returned covariance matrix.
lpLog probability reference to increment.
Returns
Covariance matrix derived from the unconstrained partial correlations and deviations.
Template Parameters
TType of scalar.

Definition at line 73 of file cov_matrix_constrain_lkj.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cov_matrix_free ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  y)

The covariance matrix derived from the symmetric view of the lower-triangular view of the K by K specified matrix is freed to return a vector of size K + (K choose 2).

This is the inverse of the cov_matrix_constrain() function so that for any finite vector x of size K

  • (K choose 2),

x == cov_matrix_free(cov_matrix_constrain(x, K)).

In order for this round-trip to work (and really for this function to work), the symmetric view of its lower-triangular view must be positive definite.

Parameters
yMatrix of dimensions K by K such that he symmetric view of the lower-triangular view is positive definite.
Returns
Vector of size K plus (K choose 2) in (-inf, inf) that produces
Exceptions
std::domain_errorif y is not square, has zero dimensionality, or has a non-positive diagonal element.

Definition at line 37 of file cov_matrix_free.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::cov_matrix_free_lkj ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  y)

Return the vector of unconstrained partial correlations and deviations that transform to the specified covariance matrix.

The constraining transform is defined as for cov_matrix_constrain(Matrix, size_t). The inverse first factors out the deviations, then applies the freeing transfrom of corr_matrix_free(Matrix&).

Parameters
yCovariance matrix to free.
Returns
Vector of unconstrained values that transforms to the specified covariance matrix.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif the correlation matrix has no elements or is not a square matrix.
std::runtime_errorif the correlation matrix cannot be factorized by factor_cov_matrix()

Definition at line 32 of file cov_matrix_free_lkj.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, C, C> stan::math::crossprod ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 17 of file crossprod.hpp.

matrix_d stan::math::crossprod ( const matrix_d M)
inline

Returns the result of pre-multiplying a matrix by its own transpose.

Parameters
MMatrix to multiply.
Returns
Transpose of M times M

Definition at line 17 of file crossprod.hpp.

matrix_v stan::math::crossprod ( const matrix_v M)
inline

Returns the result of pre-multiplying a matrix by its own transpose.

Parameters
MMatrix to multiply.
Returns
Transpose of M times M

Definition at line 17 of file crossprod.hpp.

template<typename T1 , typename T2 >
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, Eigen::Dynamic, 1> stan::math::csr_matrix_times_vector ( const int &  m,
const int &  n,
const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &  w,
const std::vector< int > &  v,
const std::vector< int > &  u,
const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &  b 
)
inline

Definition at line 79 of file csr_matrix_times_vector.hpp.

template<typename T >
std::vector<T> stan::math::cumulative_sum ( const std::vector< T > &  x)
inline

Return the cumulative sum of the specified vector.

The cumulative sum of a vector of values

1  is the
2 
3 @code x[0], x[1] + x[2], ..., x[1] + , ..., + x[x.size()-1]
Template Parameters
TScalar type of vector.
Parameters
xVector of values.
Returns
Cumulative sum of values.

Definition at line 23 of file cumulative_sum.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::cumulative_sum ( const Eigen::Matrix< T, R, C > &  m)
inline

Return the cumulative sum of the specified matrix.

The cumulative sum is of the same type as the input and has values defined by

x(0), x(1) + x(2), ..., x(1) + , ..., + x(x.size()-1)
Template Parameters
TScalar type of matrix.
RRow type of matrix.
CColumn type of matrix.
Parameters
mMatrix of values.
Returns
Cumulative sum of values.

Definition at line 49 of file cumulative_sum.hpp.

template<typename T , typename F >
void stan::math::derivative ( const F &  f,
const T &  x,
T &  fx,
T &  dfx_dx 
)

Return the derivative of the specified univariate function at the specified argument.

Template Parameters
TArgument type
FFunction type
Parameters
[in]fFunction
[in]xArgument
[out]fxValue of function applied to argument
[out]dfx_dxValue of derivative

Definition at line 26 of file derivative.hpp.

template<typename T , int R, int C>
T stan::math::determinant ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the determinant of the specified square matrix.

Parameters
mSpecified matrix.
Returns
Determinant of the matrix.
Exceptions
std::domain_errorif matrix is not square.

Definition at line 18 of file determinant.hpp.

template<typename T , int R, int C>
fvar<T> stan::math::determinant ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 21 of file determinant.hpp.

template<int R, int C>
var stan::math::determinant ( const Eigen::Matrix< var, R, C > &  m)
inline

Definition at line 66 of file determinant.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::diag_matrix ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v)
inline

Return a square diagonal matrix with the specified vector of coefficients as the diagonal values.

Parameters
[in]vSpecified vector.
Returns
Diagonal matrix with vector as diagonal values.

Definition at line 18 of file diag_matrix.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C1> stan::math::diag_post_multiply ( const Eigen::Matrix< T1, R1, C1 > &  m1,
const Eigen::Matrix< T2, R2, C2 > &  m2 
)

Definition at line 14 of file diag_post_multiply.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R2, C2> stan::math::diag_pre_multiply ( const Eigen::Matrix< T1, R1, C1 > &  m1,
const Eigen::Matrix< T2, R2, C2 > &  m2 
)

Definition at line 14 of file diag_pre_multiply.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::diagonal ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)
inline

Return a column vector of the diagonal elements of the specified matrix.

The matrix is not required to be square.

Parameters
mSpecified matrix.
Returns
Diagonal of the matrix.

Definition at line 18 of file diagonal.hpp.

template<typename T >
fvar<T> stan::math::digamma ( const fvar< T > &  x)
inline

Definition at line 16 of file digamma.hpp.

var stan::math::digamma ( const stan::math::var a)
inline

Definition at line 24 of file digamma.hpp.

double stan::math::digamma ( double  x)

\[ \mbox{digamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{digamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi(x)=\frac{\Gamma'(x)}{\Gamma(x)} \]

\[ \frac{\partial \, \Psi(x)}{\partial x} = \frac{\Gamma''(x)\Gamma(x)-(\Gamma'(x))^2}{\Gamma^2(x)} \]

Definition at line 39 of file digamma.hpp.

template<typename T >
void stan::math::dims ( const T &  x,
std::vector< int > &  result 
)
inline

Definition at line 13 of file dims.hpp.

template<typename T , int R, int C>
void stan::math::dims ( const Eigen::Matrix< T, R, C > &  x,
std::vector< int > &  result 
)
inline

Definition at line 18 of file dims.hpp.

template<typename T >
void stan::math::dims ( const std::vector< T > &  x,
std::vector< int > &  result 
)
inline

Definition at line 25 of file dims.hpp.

template<typename T >
std::vector<int> stan::math::dims ( const T &  x)
inline

Definition at line 34 of file dims.hpp.

template<bool propto, typename T_prob , typename T_prior_sample_size >
boost::math::tools::promote_args<T_prob, T_prior_sample_size>::type stan::math::dirichlet_log ( const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta,
const Eigen::Matrix< T_prior_sample_size, Eigen::Dynamic, 1 > &  alpha 
)

The log of the Dirichlet density for the given theta and a vector of prior sample sizes, alpha.

Each element of alpha must be greater than 0. Each element of theta must be greater than or 0. Theta sums to 1.

\begin{eqnarray*} \theta &\sim& \mbox{\sf{Dirichlet}} (\alpha_1, \ldots, \alpha_k) \\ \log (p (\theta \, |\, \alpha_1, \ldots, \alpha_k) ) &=& \log \left( \frac{\Gamma(\alpha_1 + \cdots + \alpha_k)}{\Gamma(\alpha_1) \cdots \Gamma(\alpha_k)} \theta_1^{\alpha_1 - 1} \cdots \theta_k^{\alpha_k - 1} \right) \\ &=& \log (\Gamma(\alpha_1 + \cdots + \alpha_k)) - \log(\Gamma(\alpha_1)) - \cdots - \log(\Gamma(\alpha_k)) + (\alpha_1 - 1) \log (\theta_1) + \cdots + (\alpha_k - 1) \log (\theta_k) \end{eqnarray*}

Parameters
thetaA scalar vector.
alphaPrior sample sizes.
Returns
The log of the Dirichlet density.
Exceptions
std::domain_errorif any element of alpha is less than or equal to 0.
std::domain_errorif any element of theta is less than 0.
std::domain_errorif the sum of theta is not 1.
Template Parameters
T_probType of scalar.
T_prior_sample_sizeType of prior sample sizes.

Definition at line 46 of file dirichlet_log.hpp.

template<typename T_prob , typename T_prior_sample_size >
boost::math::tools::promote_args<T_prob, T_prior_sample_size>::type stan::math::dirichlet_log ( const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta,
const Eigen::Matrix< T_prior_sample_size, Eigen::Dynamic, 1 > &  alpha 
)
inline

Definition at line 79 of file dirichlet_log.hpp.

template<class RNG >
Eigen::VectorXd stan::math::dirichlet_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  alpha,
RNG &  rng 
)
inline

Return a draw from a Dirichlet distribution with specified parameters and pseudo-random number generator.

For prior counts greater than zero, the usual algorithm that draws gamma variates and normalizes is used.

For prior counts less than zero (i.e., parameters with value less than one), a log-scale version of the following algorithm is used to deal with underflow:

G. Marsaglia and W. Tsang. A simple method for generating gamma variables. ACM Transactions on Mathematical Software. 26(3):363–372, 2000.

Template Parameters
RNGType of pseudo-random number generator.
Parameters
alphaPrior count (plus 1) parameter for Dirichlet.
rngPseudo-random number generator.

Definition at line 46 of file dirichlet_rng.hpp.

double stan::math::dist ( const std::vector< double > &  x,
const std::vector< double > &  y 
)
inline

Definition at line 11 of file dist.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::math::tools::promote_args<T1, T2>::type stan::math::distance ( const Eigen::Matrix< T1, R1, C1 > &  v1,
const Eigen::Matrix< T2, R2, C2 > &  v2 
)
inline

Returns the distance between the specified vectors.

Parameters
v1First vector.
v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorIf the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 25 of file distance.hpp.

int stan::math::divide ( const int  x,
const int  y 
)
inline

Definition at line 10 of file divide.hpp.

template<typename T1 , typename T2 >
stan::return_type<T1, T2>::type stan::math::divide ( const T1 &  v,
const T2 &  c 
)
inline

Definition at line 17 of file divide.hpp.

double stan::math::divide ( double  x,
double  y 
)
inline

Return the division of the first scalar by the second scalar.

Parameters
[in]xSpecified vector.
[in]ySpecified scalar.
Returns
Vector divided by the scalar.

Definition at line 22 of file divide.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< fvar< T >, R, C > &  v,
const fvar< T > &  c 
)
inline

Definition at line 23 of file divide.hpp.

template<int R, int C, typename T >
boost::enable_if_c<boost::is_arithmetic<T>::value, Eigen::Matrix<double, R, C> >::type stan::math::divide ( const Eigen::Matrix< double, R, C > &  m,
c 
)
inline

Return specified matrix divided by specified scalar.

Template Parameters
RRow type for matrix.
CColumn type for matrix.
Parameters
mMatrix.
cScalar.
Returns
Matrix divided by scalar.

Definition at line 23 of file divide.hpp.

template<typename T1 , typename T2 >
var stan::math::divide ( const T1 &  v,
const T2 &  c 
)
inline

Definition at line 27 of file divide.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< fvar< T >, R, C > &  v,
const double  c 
)
inline

Definition at line 34 of file divide.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<var, R, C> stan::math::divide ( const Eigen::Matrix< T1, R, C > &  v,
const T2 &  c 
)
inline

Return the division of the specified column vector by the specified scalar.

Parameters
[in]vSpecified vector.
[in]cSpecified scalar.
Returns
Vector divided by the scalar.

Definition at line 39 of file divide.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::divide ( const Eigen::Matrix< double, R, C > &  v,
const fvar< T > &  c 
)
inline

Definition at line 46 of file divide.hpp.

template<typename T_shape >
T_shape stan::math::do_lkj_constant ( const T_shape &  eta,
const unsigned int &  K 
)

Definition at line 53 of file lkj_corr_log.hpp.

template<typename T >
void stan::math::domain_error ( const char *  function,
const char *  name,
const T &  y,
const char *  msg1,
const char *  msg2 
)
inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
msg1Message to print before the variable
msg2Message to print after the variable
Exceptions
std::domain_error

Definition at line 32 of file domain_error.hpp.

template<typename T >
void stan::math::domain_error ( const char *  function,
const char *  name,
const T &  y,
const char *  msg1 
)
inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name> <msg1><y>"

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
msg1Message to print before the variable
Exceptions
std::domain_error

Definition at line 67 of file domain_error.hpp.

template<typename T >
void stan::math::domain_error_vec ( const char *  function,
const char *  name,
const T &  y,
const size_t  i,
const char *  msg1,
const char *  msg2 
)
inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
iIndex
msg1Message to print before the variable
msg2Message to print after the variable
Exceptions
std::domain_error

Definition at line 38 of file domain_error_vec.hpp.

template<typename T >
void stan::math::domain_error_vec ( const char *  function,
const char *  name,
const T &  y,
const size_t  i,
const char *  msg 
)
inline

Throw a domain error with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing domain errors. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
iIndex
msgMessage to print before the variable
Exceptions
std::domain_error

Definition at line 73 of file domain_error_vec.hpp.

double stan::math::dot ( const std::vector< double > &  x,
const std::vector< double > &  y 
)
inline

Definition at line 11 of file dot.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 20 of file dot_product.hpp.

template<int R1, int C1, int R2, int C2>
double stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Returns the dot product of the specified vectors.

Parameters
v1First vector.
v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorIf the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 22 of file dot_product.hpp.

double stan::math::dot_product ( const double *  v1,
const double *  v2,
size_t  length 
)
inline

Returns the dot product of the specified arrays of doubles.

Parameters
v1First array.
v2Second array.
lengthLength of both arrays.

Definition at line 37 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Definition at line 37 of file dot_product.hpp.

double stan::math::dot_product ( const std::vector< double > &  v1,
const std::vector< double > &  v2 
)
inline

Returns the dot product of the specified arrays of doubles.

Parameters
v1First array.
v2Second array.
Exceptions
std::domain_errorif the vectors are not the same size.

Definition at line 50 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 54 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2,
size_type length 
)
inline

Definition at line 71 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2,
size_type length 
)
inline

Definition at line 86 of file dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
fvar<T> stan::math::dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2,
size_type length 
)
inline

Definition at line 101 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1,
const std::vector< fvar< T > > &  v2 
)
inline

Definition at line 116 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< double > &  v1,
const std::vector< fvar< T > > &  v2 
)
inline

Definition at line 130 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1,
const std::vector< double > &  v2 
)
inline

Definition at line 144 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1,
const std::vector< fvar< T > > &  v2,
size_type length 
)
inline

Definition at line 158 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< double > &  v1,
const std::vector< fvar< T > > &  v2,
size_type length 
)
inline

Definition at line 170 of file dot_product.hpp.

template<typename T >
fvar<T> stan::math::dot_product ( const std::vector< fvar< T > > &  v1,
const std::vector< double > &  v2,
size_type length 
)
inline

Definition at line 182 of file dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c<boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, var>::type stan::math::dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1,
const Eigen::Matrix< T2, R2, C2 > &  v2 
)
inline

Returns the dot product.

Parameters
[in]v1First column vector.
[in]v2Second column vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorif length of v1 is not equal to length of v2.

Definition at line 212 of file dot_product.hpp.

template<typename T1 , typename T2 >
boost::enable_if_c<boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, var>::type stan::math::dot_product ( const T1 *  v1,
const T2 *  v2,
size_t  length 
)
inline

Returns the dot product.

Parameters
[in]v1First array.
[in]v2Second array.
[in]lengthLength of both arrays.
Returns
Dot product of the arrays.

Definition at line 233 of file dot_product.hpp.

template<typename T1 , typename T2 >
boost::enable_if_c<boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, var>::type stan::math::dot_product ( const std::vector< T1 > &  v1,
const std::vector< T2 > &  v2 
)
inline

Returns the dot product.

Parameters
[in]v1First vector.
[in]v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorif sizes of v1 and v2 do not match.

Definition at line 249 of file dot_product.hpp.

double stan::math::dot_self ( const std::vector< double > &  x)
inline

Definition at line 11 of file dot_self.hpp.

template<typename T , int R, int C>
fvar<T> stan::math::dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  v)
inline

Definition at line 16 of file dot_self.hpp.

template<int R, int C>
double stan::math::dot_self ( const Eigen::Matrix< double, R, C > &  v)
inline

Returns the dot product of the specified vector with itself.

Parameters
vVector.
Template Parameters
Rnumber of rows or Eigen::Dynamic for dynamic
Cnumber of rows or Eigen::Dyanmic for dynamic
Exceptions
std::domain_errorIf v is not vector dimensioned.

Definition at line 18 of file dot_self.hpp.

template<int R, int C>
var stan::math::dot_self ( const Eigen::Matrix< var, R, C > &  v)
inline

Returns the dot product of a vector with itself.

Parameters
[in]vVector.
Returns
Dot product of the vector with itself.
Template Parameters
Rnumber of rows or Eigen::Dynamic for dynamic; one of R or C must be 1
Cnumber of rows or Eigen::Dyanmic for dynamic; one of R or C must be 1

Definition at line 80 of file dot_self.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::double_exponential_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file double_exponential_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::double_exponential_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Calculates the double exponential cumulative density function.

$ f(y|\mu, \sigma) = \begin{cases} \ \frac{1}{2} \exp\left(\frac{y-\mu}{\sigma}\right), \mbox{if } y < \mu \\ 1 - \frac{1}{2} \exp\left(-\frac{y-\mu}{\sigma}\right), \mbox{if } y \ge \mu \ \end{cases}$

Parameters
yA scalar variate.
muThe location parameter.
sigmaThe scale parameter.
Returns
The cumulative density function.

Definition at line 38 of file double_exponential_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::double_exponential_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file double_exponential_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::double_exponential_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 29 of file double_exponential_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::double_exponential_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 128 of file double_exponential_log.hpp.

template<class RNG >
double stan::math::double_exponential_rng ( const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 24 of file double_exponential_rng.hpp.

double stan::math::e ( )
inline

Return the base of the natural logarithm.

Returns
Base of natural logarithm.

Definition at line 95 of file constants.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::eigenvalues_sym ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Return the eigenvalues of the specified symmetric matrix in descending order of magnitude.

This function is more efficient than the general eigenvalues function for symmetric matrices.

See eigen_decompose() for more information.

Parameters
mSpecified matrix.
Returns
Eigenvalues of matrix.

Definition at line 22 of file eigenvalues_sym.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::eigenvectors_sym ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Definition at line 13 of file eigenvectors_sym.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::elt_divide ( const Eigen::Matrix< T1, R, C > &  m1,
const Eigen::Matrix< T2, R, C > &  m2 
)

Return the elementwise division of the specified matrices.

Template Parameters
T1Type of scalars in first matrix.
T2Type of scalars in second matrix.
RRow type of both matrices.
CColumn type of both matrices.
Parameters
m1First matrix
m2Second matrix
Returns
Elementwise division of matrices.

Definition at line 24 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::elt_divide ( const Eigen::Matrix< T1, R, C > &  m,
T2  s 
)

Return the elementwise division of the specified matrix by the specified scalar.

Template Parameters
T1Type of scalars in the matrix.
T2Type of the scalar.
RRow type of the matrix.
CColumn type of the matrix.
Parameters
mmatrix
sscalar
Returns
Elementwise division of a scalar by matrix.

Definition at line 50 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::elt_divide ( T1  s,
const Eigen::Matrix< T2, R, C > &  m 
)

Return the elementwise division of the specified scalar by the specified matrix.

Template Parameters
T1Type of the scalar.
T2Type of scalars in the matrix.
RRow type of the matrix.
CColumn type of the matrix.
Parameters
sscalar
mmatrix
Returns
Elementwise division of a scalar by matrix.

Definition at line 68 of file elt_divide.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::elt_multiply ( const Eigen::Matrix< T1, R, C > &  m1,
const Eigen::Matrix< T2, R, C > &  m2 
)

Return the elementwise multiplication of the specified matrices.

Template Parameters
T1Type of scalars in first matrix.
T2Type of scalars in second matrix.
RRow type of both matrices.
CColumn type of both matrices.
Parameters
m1First matrix
m2Second matrix
Returns
Elementwise product of matrices.

Definition at line 25 of file elt_multiply.hpp.

static bool stan::math::empty_nested ( )
inlinestatic

Return true if there is no nested autodiff being executed.

Definition at line 12 of file empty_nested.hpp.

template<typename T >
fvar<T> stan::math::erf ( const fvar< T > &  x)
inline

Definition at line 14 of file erf.hpp.

var stan::math::erf ( const var a)
inline

The error function for variables (C99).

For non-variable function, see erf() from cmath.

The derivative is

$\frac{d}{dx} \mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \exp(-x^2)$.

\[ \mbox{erf}(x) = \begin{cases} \operatorname{erf}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erf}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erf}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erf}(x)}{\partial x} = \frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters
aThe variable.
Returns
Error function applied to the variable.

Definition at line 68 of file erf.hpp.

template<typename T >
fvar<T> stan::math::erfc ( const fvar< T > &  x)
inline

Definition at line 14 of file erfc.hpp.

var stan::math::erfc ( const var a)
inline

The complementary error function for variables (C99).

For non-variable function, see erfc() from <cmath>.

The derivative is

$\frac{d}{dx} \mbox{erfc}(x) = - \frac{2}{\sqrt{\pi}} \exp(-x^2)$.

\[ \mbox{erfc}(x) = \begin{cases} \operatorname{erfc}(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{erfc}(x)}{\partial x} = \begin{cases} \frac{\partial\, \operatorname{erfc}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \operatorname{erfc}(x)=\frac{2}{\sqrt{\pi}}\int_x^\infty e^{-t^2}dt \]

\[ \frac{\partial \, \operatorname{erfc}(x)}{\partial x} = -\frac{2}{\sqrt{\pi}} e^{-x^2} \]

Parameters
aThe variable.
Returns
Complementary error function applied to the variable.

Definition at line 68 of file erfc.hpp.

template<typename T >
fvar<T> stan::math::exp ( const fvar< T > &  x)
inline

Definition at line 10 of file exp.hpp.

template<typename T , int Rows, int Cols>
Eigen::Matrix<T, Rows, Cols> stan::math::exp ( const Eigen::Matrix< T, Rows, Cols > &  m)
inline

Return the element-wise exponentiation of the matrix or vector.

Parameters
mThe matrix or vector.
Returns
ret(i, j) = exp(m(i, j))

Definition at line 19 of file exp.hpp.

template<int Rows, int Cols>
Eigen::Matrix<double, Rows, Cols> stan::math::exp ( const Eigen::Matrix< double, Rows, Cols > &  m)
inline

Definition at line 28 of file exp.hpp.

var stan::math::exp ( const var a)
inline

Return the exponentiation of the specified variable (cmath).

\[ \mbox{exp}(x) = \begin{cases} e^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable to exponentiate.
Returns
Exponentiated variable.

Definition at line 44 of file exp.hpp.

template<typename T >
fvar<T> stan::math::exp2 ( const fvar< T > &  x)
inline

Definition at line 14 of file exp2.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::exp2 ( const T  y)
inline

Return the exponent base 2 of the specified argument (C99).

The exponent base 2 function is defined by

exp2(y) = pow(2.0, y).

Parameters
yValue.
Template Parameters
TType of scalar.
Returns
Exponent base 2 of value.

Definition at line 23 of file exp2.hpp.

var stan::math::exp2 ( const var a)
inline

Exponentiation base 2 function for variables (C99).

For non-variable function, see boost::math::exp2().

The derivatie is

$\frac{d}{dx} 2^x = (\log 2) 2^x$.

\[ \mbox{exp2}(x) = \begin{cases} 2^x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{exp2}(x)}{\partial x} = \begin{cases} 2^x\ln2 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aThe variable.
Returns
Two to the power of the specified variable.

Definition at line 52 of file exp2.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type<T_y, T_loc, T_scale, T_inv_scale>::type stan::math::exp_mod_normal_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_inv_scale &  lambda 
)

Definition at line 24 of file exp_mod_normal_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type<T_y, T_loc, T_scale, T_inv_scale>::type stan::math::exp_mod_normal_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_inv_scale &  lambda 
)

Definition at line 24 of file exp_mod_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type<T_y, T_loc, T_scale, T_inv_scale>::type stan::math::exp_mod_normal_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_inv_scale &  lambda 
)

Definition at line 24 of file exp_mod_normal_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type<T_y, T_loc, T_scale, T_inv_scale>::type stan::math::exp_mod_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_inv_scale &  lambda 
)

Definition at line 25 of file exp_mod_normal_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_inv_scale >
return_type<T_y, T_loc, T_scale, T_inv_scale>::type stan::math::exp_mod_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_inv_scale &  lambda 
)
inline

Definition at line 139 of file exp_mod_normal_log.hpp.

template<class RNG >
double stan::math::exp_mod_normal_rng ( const double  mu,
const double  sigma,
const double  lambda,
RNG &  rng 
)
inline

Definition at line 29 of file exp_mod_normal_rng.hpp.

template<typename T >
fvar<T> stan::math::expm1 ( const fvar< T > &  x)
inline

Definition at line 12 of file expm1.hpp.

var stan::math::expm1 ( const stan::math::var a)
inline

The exponentiation of the specified variable minus 1 (C99).

The derivative is given by

$\frac{d}{dx} \exp(a) - 1 = \exp(a)$.

\[ \mbox{expm1}(x) = \begin{cases} e^x-1 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{expm1}(x)}{\partial x} = \begin{cases} e^x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aThe variable.
Returns
Two to the power of the specified variable.

Definition at line 57 of file expm1.hpp.

template<typename T_y , typename T_inv_scale >
return_type<T_y, T_inv_scale>::type stan::math::exponential_ccdf_log ( const T_y &  y,
const T_inv_scale &  beta 
)

Definition at line 27 of file exponential_ccdf_log.hpp.

template<typename T_y , typename T_inv_scale >
return_type<T_y, T_inv_scale>::type stan::math::exponential_cdf ( const T_y &  y,
const T_inv_scale &  beta 
)

Calculates the exponential cumulative distribution function for the given y and beta.

Inverse scale parameter must be greater than 0. y must be greater than or equal to 0.

Parameters
yA scalar variable.
betaInverse scale parameter.
Template Parameters
T_yType of scalar.
T_inv_scaleType of inverse scale.

Definition at line 40 of file exponential_cdf.hpp.

template<typename T_y , typename T_inv_scale >
return_type<T_y, T_inv_scale>::type stan::math::exponential_cdf_log ( const T_y &  y,
const T_inv_scale &  beta 
)

Definition at line 28 of file exponential_cdf_log.hpp.

template<bool propto, typename T_y , typename T_inv_scale >
return_type<T_y, T_inv_scale>::type stan::math::exponential_log ( const T_y &  y,
const T_inv_scale &  beta 
)

The log of an exponential density for y with the specified inverse scale parameter.

Inverse scale parameter must be greater than 0. y must be greater than or equal to 0.

\begin{eqnarray*} y &\sim& \mbox{\sf{Expon}}(\beta) \\ \log (p (y \, |\, \beta) ) &=& \log \left( \beta \exp^{-\beta y} \right) \\ &=& \log (\beta) - \beta y \\ & & \mathrm{where} \; y > 0 \end{eqnarray*}

Parameters
yA scalar variable.
betaInverse scale parameter.
Exceptions
std::domain_errorif beta is not greater than 0.
std::domain_errorif y is not greater than or equal to 0.
Template Parameters
T_yType of scalar.
T_inv_scaleType of inverse scale.

Definition at line 54 of file exponential_log.hpp.

template<typename T_y , typename T_inv_scale >
return_type<T_y, T_inv_scale>::type stan::math::exponential_log ( const T_y &  y,
const T_inv_scale &  beta 
)
inline

Definition at line 111 of file exponential_log.hpp.

template<class RNG >
double stan::math::exponential_rng ( const double  beta,
RNG &  rng 
)
inline

Definition at line 27 of file exponential_rng.hpp.

template<typename T >
T stan::math::F32 ( a,
b,
c,
d,
e,
z,
precision = 1e-6 
)

Definition at line 11 of file F32.hpp.

template<typename T >
fvar<T> stan::math::fabs ( const fvar< T > &  x)
inline

Definition at line 14 of file fabs.hpp.

var stan::math::fabs ( const var a)
inline

Return the absolute value of the variable (cmath).

Choosing an arbitrary value at the non-differentiable point 0,

$\frac{d}{dx}|x| = \mbox{sgn}(x)$.

where $\mbox{sgn}(x)$ is the signum function, taking values -1 if $x < 0$, 0 if $x == 0$, and 1 if $x == 1$.

The function abs() provides the same behavior, with abs() defined in stdlib.h and fabs() defined in cmath. The derivative is 0 if the input is 0.

Returns std::numeric_limits<double>::quiet_NaN() for NaN inputs.

\[ \mbox{fabs}(x) = \begin{cases} |x| & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fabs}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } x < 0 \\ 0 & \mbox{if } x = 0 \\ 1 & \mbox{if } x > 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aInput variable.
Returns
Absolute value of variable.

Definition at line 50 of file fabs.hpp.

template<typename T >
bool stan::math::factor_cov_matrix ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  Sigma,
Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
Eigen::Array< T, Eigen::Dynamic, 1 > &  sds 
)

This function is intended to make starting values, given a covariance matrix Sigma.

The transformations are hard coded as log for standard deviations and Fisher transformations (atanh()) of CPCs

Parameters
[in]Sigmacovariance matrix
[out]CPCsfill this unbounded (does not resize)
[out]sdsfill this unbounded (does not resize)
Returns
false if any of the diagonals of Sigma are 0

Definition at line 27 of file factor_cov_matrix.hpp.

template<typename T >
void stan::math::factor_U ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  U,
Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs 
)

This function is intended to make starting values, given a unit upper-triangular matrix U such that U'DU is a correlation matrix.

Parameters
USigma matrix
CPCsfill this unbounded

Definition at line 29 of file factor_U.hpp.

template<typename T >
fvar<T> stan::math::falling_factorial ( const fvar< T > &  x,
const fvar< T > &  n 
)
inline

Definition at line 15 of file falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::falling_factorial ( const fvar< T > &  x,
const double  n 
)
inline

Definition at line 26 of file falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::falling_factorial ( const double  x,
const fvar< T > &  n 
)
inline

Definition at line 37 of file falling_factorial.hpp.

var stan::math::falling_factorial ( const var a,
const double &  b 
)
inline

Definition at line 52 of file falling_factorial.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::falling_factorial ( const T1  x,
const T2  n 
)
inline

\[ \mbox{falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, (x)_n}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ (x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)} \]

\[ \frac{\partial \, (x)_n}{\partial x} = (x)_n\Psi(x+1) \]

\[ \frac{\partial \, (x)_n}{\partial n} = -(x)_n\Psi(n+1) \]

Definition at line 54 of file falling_factorial.hpp.

var stan::math::falling_factorial ( const var a,
const var b 
)
inline

Definition at line 57 of file falling_factorial.hpp.

var stan::math::falling_factorial ( const double &  a,
const var b 
)
inline

Definition at line 62 of file falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::fdim ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 11 of file fdim.hpp.

template<typename T >
fvar<T> stan::math::fdim ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 22 of file fdim.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::fdim ( T1  a,
T2  b 
)
inline

The positive difference function (C99).

The function is defined by

fdim(a, b) = (a > b) ? (a - b) : 0.0.

Parameters
aFirst value.
bSecond value.
Returns
Returns min(a - b, 0.0).

Definition at line 26 of file fdim.hpp.

template<typename T >
fvar<T> stan::math::fdim ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 32 of file fdim.hpp.

var stan::math::fdim ( const stan::math::var a,
const stan::math::var b 
)
inline

Return the positive difference between the first variable's the value and the second's (C99).

See stan::math::fdim() for the double-based version.

The partial derivative with respect to the first argument is

$\frac{\partial}{\partial x} \mbox{fdim}(x, y) = 0.0$ if $x < y$, and

$\frac{\partial}{\partial x} \mbox{fdim}(x, y) = 1.0$ if $x \geq y$.

With respect to the second argument, the partial is

$\frac{\partial}{\partial y} \mbox{fdim}(x, y) = 0.0$ if $x < y$, and

$\frac{\partial}{\partial y} \mbox{fdim}(x, y) = -\lfloor\frac{x}{y}\rfloor$ if $x \geq y$.

\[ \mbox{fdim}(x, y) = \begin{cases} 0 & \mbox{if } x < y\\ x-y & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial x} = \begin{cases} 0 & \mbox{if } x < y \\ 1 & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fdim}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x < y \\ -\lfloor\frac{x}{y}\rfloor & \mbox{if } x \geq y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
The positive difference between the first and second variable.

Definition at line 110 of file fdim.hpp.

var stan::math::fdim ( const double &  a,
const stan::math::var b 
)
inline

Return the positive difference between the first value and the value of the second variable (C99).

See fdim(var, var) for definitions of values and derivatives.

The derivative with respect to the variable is

$\frac{d}{d y} \mbox{fdim}(c, y) = 0.0$ if $c < y$, and

$\frac{d}{d y} \mbox{fdim}(c, y) = -\lfloor\frac{c}{y}\rfloor$ if $c \geq y$.

Parameters
aFirst value.
bSecond variable.
Returns
The positive difference between the first and second arguments.

Definition at line 135 of file fdim.hpp.

var stan::math::fdim ( const stan::math::var a,
const double &  b 
)
inline

Return the positive difference between the first variable's value and the second value (C99).

See fdim(var, var) for definitions of values and derivatives.

The derivative with respect to the variable is

$\frac{d}{d x} \mbox{fdim}(x, c) = 0.0$ if $x < c$, and

$\frac{d}{d x} \mbox{fdim}(x, c) = 1.0$ if $x \geq yc$.

Parameters
aFirst value.
bSecond variable.
Returns
The positive difference between the first and second arguments.

Definition at line 158 of file fdim.hpp.

template<typename T , typename S >
void stan::math::fill ( T &  x,
const S &  y 
)

Fill the specified container with the specified value.

This base case simply assigns the value to the container.

Template Parameters
TType of reference container.
SType of value.
Parameters
xContainer.
yValue.

Definition at line 22 of file fill.hpp.

template<typename T , int R, int C, typename S >
void stan::math::fill ( Eigen::Matrix< T, R, C > &  x,
const S &  y 
)

Fill the specified container with the specified value.

The specified matrix is filled by element.

Template Parameters
TType of scalar for matrix container.
RRow type of matrix.
CColumn type of matrix.
SType of value.
Parameters
xContainer.
yValue.

Definition at line 39 of file fill.hpp.

template<typename T , typename S >
void stan::math::fill ( std::vector< T > &  x,
const S &  y 
)

Fill the specified container with the specified value.

Each container in the specified standard vector is filled recursively by calling fill.

Template Parameters
TType of container in vector.
SType of value.
Parameters
xContainer.
yValue.

Definition at line 55 of file fill.hpp.

template<typename F >
void stan::math::finite_diff_grad_hessian ( const F &  f,
const Eigen::Matrix< double,-1, 1 > &  x,
double &  fx,
Eigen::Matrix< double,-1,-1 > &  hess,
std::vector< Eigen::Matrix< double,-1,-1 > > &  grad_hess_fx,
const double  epsilon = 1e-04 
)

Calculate the value and the gradient of the hessian of the specified function at the specified argument using second-order autodiff and first-order finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, Eigen::Dynamic, 1>&)

Reference:

De Levie: An improved numerical approximation for the first derivative, page 3

4 calls to the function, f.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]hessHessian matrix
[out]grad_hess_fxgradient of Hessian of function at argument
[in]epsilonperturbation size

Definition at line 43 of file finite_diff_grad_hessian.hpp.

template<typename F >
void stan::math::finite_diff_gradient ( const F &  f,
const Eigen::Matrix< double,-1, 1 > &  x,
double &  fx,
Eigen::Matrix< double,-1, 1 > &  grad_fx,
const double  epsilon = 1e-03 
)

Calculate the value and the gradient of the specified function at the specified argument using finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, Eigen::Dynamic, 1>&)

Error should be on order of epsilon ^ 6. The reference for this algorithm is:

De Levie: An improved numerical approximation for the first derivative, page 3

This function involves 6 calls to f.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]grad_fxGradient of function at argument
[in]epsilonperturbation size

Definition at line 39 of file finite_diff_gradient.hpp.

template<typename F >
double stan::math::finite_diff_hess_helper ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
const int  lambda,
const double  epsilon = 1e-03 
)

Definition at line 13 of file finite_diff_hessian.hpp.

template<typename F >
void stan::math::finite_diff_hessian ( const F &  f,
const Eigen::Matrix< double,-1, 1 > &  x,
double &  fx,
Eigen::Matrix< double,-1, 1 > &  grad_fx,
Eigen::Matrix< double,-1,-1 > &  hess_fx,
const double  epsilon = 1e-03 
)

Calculate the value and the Hessian of the specified function at the specified argument using second-order finite difference.

The functor must implement

double operator()(const Eigen::Matrix<double, Eigen::Dynamic, 1>&)

Error should be on order of epsilon ^ 4, with 4 calls to the function f.

Reference: Eberly: Derivative Approximation by Finite Differences Page 6

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]grad_fxGradient of function at argument
[out]hess_fxHessian of function at argument
[in]epsilonperturbation size

Definition at line 67 of file finite_diff_hessian.hpp.

template<typename T >
fvar<T> stan::math::floor ( const fvar< T > &  x)
inline

Definition at line 11 of file floor.hpp.

var stan::math::floor ( const var a)
inline

Return the floor of the specified variable (cmath).

The derivative of the floor function is defined and zero everywhere but at integers, so we set these derivatives to zero for convenience,

$\frac{d}{dx} {\lfloor x \rfloor} = 0$.

The floor function rounds down. For double values, this is the largest integral value that is not greater than the specified value. Although this function is not differentiable because it is discontinuous at integral values, its gradient is returned as zero everywhere.

\[ \mbox{floor}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{floor}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aInput variable.
Returns
Floor of the variable.

Definition at line 60 of file floor.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1,
const fvar< T2 > &  x2,
const fvar< T3 > &  x3 
)
inline

The fused multiply-add operation (C99).

This double-based operation delegates to fma.

The function is defined by

fma(a, b, c) = (a * b) + c.

\[ \mbox{fma}(x, y, z) = \begin{cases} x\cdot y+z & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fma}(x, y, z)}{\partial z} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y, z \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
x1First value.
x2Second value.
x3Third value.
Returns
Product of the first two values plus the third.

Definition at line 61 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1,
const fvar< T2 > &  x2,
const fvar< T3 > &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 74 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1,
const T2 &  x2,
const fvar< T3 > &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 86 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1,
const fvar< T2 > &  x2,
const T3 &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 98 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1,
const T2 &  x2,
const fvar< T3 > &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 110 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const fvar< T1 > &  x1,
const T2 &  x2,
const T3 &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 122 of file fma.hpp.

template<typename T1 , typename T2 , typename T3 >
fvar<typename stan::return_type<T1, T2, T3>::type> stan::math::fma ( const T1 &  x1,
const fvar< T2 > &  x2,
const T3 &  x3 
)
inline

See all-var input signature for details on the function and derivatives.

Definition at line 134 of file fma.hpp.

var stan::math::fma ( const stan::math::var a,
const stan::math::var b,
const stan::math::var c 
)
inline

The fused multiply-add function for three variables (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) + z = y$, and

$\frac{\partial}{\partial y} (x * y) + z = x$, and

$\frac{\partial}{\partial z} (x * y) + z = 1$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 136 of file fma.hpp.

var stan::math::fma ( const stan::math::var a,
const stan::math::var b,
const double &  c 
)
inline

The fused multiply-add function for two variables and a value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) + c = y$, and

$\frac{\partial}{\partial y} (x * y) + c = x$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 161 of file fma.hpp.

var stan::math::fma ( const stan::math::var a,
const double &  b,
const stan::math::var c 
)
inline

The fused multiply-add function for a variable, value, and variable (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The partial derivatives are

$\frac{\partial}{\partial x} (x * c) + z = c$, and

$\frac{\partial}{\partial z} (x * c) + z = 1$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 186 of file fma.hpp.

var stan::math::fma ( const stan::math::var a,
const double &  b,
const double &  c 
)
inline

The fused multiply-add function for a variable and two values (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The derivative is

$\frac{d}{d x} (x * c) + d = c$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 209 of file fma.hpp.

var stan::math::fma ( const double &  a,
const stan::math::var b,
const double &  c 
)
inline

The fused multiply-add function for a value, variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The derivative is

$\frac{d}{d y} (c * y) + d = c$, and

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 232 of file fma.hpp.

var stan::math::fma ( const double &  a,
const double &  b,
const stan::math::var c 
)
inline

The fused multiply-add function for two values and a variable, and value (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The derivative is

$\frac{\partial}{\partial z} (c * d) + z = 1$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 255 of file fma.hpp.

var stan::math::fma ( const double &  a,
const stan::math::var b,
const stan::math::var c 
)
inline

The fused multiply-add function for a value and two variables (C99).

This function returns the product of the first two arguments plus the third argument.

The double-based version ::fma(double, double, double) is defined in <cmath>.

The partial derivaties are

$\frac{\partial}{\partial y} (c * y) + z = c$, and

$\frac{\partial}{\partial z} (c * y) + z = 1$.

Parameters
aFirst multiplicand.
bSecond multiplicand.
cSummand.
Returns
Product of the multiplicands plus the summand, ($a * $b) + $c.

Definition at line 280 of file fma.hpp.

template<typename T >
fvar<T> stan::math::fmax ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 13 of file fmax.hpp.

template<typename T >
fvar<T> stan::math::fmax ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 33 of file fmax.hpp.

template<typename T >
fvar<T> stan::math::fmax ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 53 of file fmax.hpp.

var stan::math::fmax ( const stan::math::var a,
const stan::math::var b 
)
inline

Returns the maximum of the two variable arguments (C99).

No new variable implementations are created, with this function defined as if by

fmax(a, b) = a if a's value is greater than b's, and .

fmax(a, b) = b if b's value is greater than or equal to a's.

\[ \mbox{fmax}(x, y) = \begin{cases} x & \mbox{if } x \geq y \\ y & \mbox{if } x < y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \geq y \\ 0 & \mbox{if } x < y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmax}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
If the first variable's value is larger than the second's, the first variable, otherwise the second variable.

Definition at line 63 of file fmax.hpp.

var stan::math::fmax ( const stan::math::var a,
const double &  b 
)
inline

Returns the maximum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a's value is greater than b, then a is returned, otherwise a fesh variable implementation wrapping the value b is returned.

Parameters
aFirst variable.
bSecond value
Returns
If the first variable's value is larger than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 95 of file fmax.hpp.

var stan::math::fmax ( const double &  a,
const stan::math::var b 
)
inline

Returns the maximum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmax(a, b), if a is greater than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters
aFirst value.
bSecond variable.
Returns
If the first value is larger than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 127 of file fmax.hpp.

template<typename T >
fvar<T> stan::math::fmin ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 13 of file fmin.hpp.

template<typename T >
fvar<T> stan::math::fmin ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 33 of file fmin.hpp.

template<typename T >
fvar<T> stan::math::fmin ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 53 of file fmin.hpp.

var stan::math::fmin ( const stan::math::var a,
const stan::math::var b 
)
inline

Returns the minimum of the two variable arguments (C99).

For fmin(a, b), if a's value is less than b's, then a is returned, otherwise b is returned.

\[ \mbox{fmin}(x, y) = \begin{cases} x & \mbox{if } x \leq y \\ y & \mbox{if } x > y \\[6pt] x & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ y & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } x \leq y \\ 0 & \mbox{if } x > y \\[6pt] 1 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 0 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmin}(x, y)}{\partial y} = \begin{cases} 0 & \mbox{if } x \leq y \\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } -\infty\leq x\leq \infty, y = \textrm{NaN}\\ 1 & \mbox{if } -\infty\leq y\leq \infty, x = \textrm{NaN}\\ \textrm{NaN} & \mbox{if } x, y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
If the first variable's value is smaller than the second's, the first variable, otherwise the second variable.

Definition at line 59 of file fmin.hpp.

var stan::math::fmin ( const stan::math::var a,
double  b 
)
inline

Returns the minimum of the variable and scalar, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a's value is less than or equal to b, then a is returned, otherwise a fresh variable wrapping b is returned.

Parameters
aFirst variable.
bSecond value
Returns
If the first variable's value is less than or equal to the second value, the first variable, otherwise the second value promoted to a fresh variable.

Definition at line 89 of file fmin.hpp.

var stan::math::fmin ( double  a,
const stan::math::var b 
)
inline

Returns the minimum of a scalar and variable, promoting the scalar to a variable if it is larger (C99).

For fmin(a, b), if a is less than b's value, then a fresh variable implementation wrapping a is returned, otherwise b is returned.

Parameters
aFirst value.
bSecond variable.
Returns
If the first value is smaller than the second variable's value, return the first value promoted to a variable, otherwise return the second variable.

Definition at line 120 of file fmin.hpp.

template<typename T >
fvar<T> stan::math::fmod ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 16 of file fmod.hpp.

template<typename T >
fvar<T> stan::math::fmod ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 26 of file fmod.hpp.

template<typename T >
fvar<T> stan::math::fmod ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 39 of file fmod.hpp.

var stan::math::fmod ( const var a,
const var b 
)
inline

Return the floating point remainder after dividing the first variable by the second (cmath).

The partial derivatives with respect to the variables are defined everywhere but where $x = y$, but we set these to match other values, with

$\frac{\partial}{\partial x} \mbox{fmod}(x, y) = 1$, and

$\frac{\partial}{\partial y} \mbox{fmod}(x, y) = -\lfloor \frac{x}{y} \rfloor$.

\[ \mbox{fmod}(x, y) = \begin{cases} x - \lfloor \frac{x}{y}\rfloor y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{fmod}(x, y)}{\partial y} = \begin{cases} -\lfloor \frac{x}{y}\rfloor & \mbox{if } -\infty\leq x, y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
Floating pointer remainder of dividing the first variable by the second.

Definition at line 103 of file fmod.hpp.

var stan::math::fmod ( const var a,
const double  b 
)
inline

Return the floating point remainder after dividing the the first variable by the second scalar (cmath).

The derivative with respect to the variable is

$\frac{d}{d x} \mbox{fmod}(x, c) = \frac{1}{c}$.

Parameters
aFirst variable.
bSecond scalar.
Returns
Floating pointer remainder of dividing the first variable by the second scalar.

Definition at line 120 of file fmod.hpp.

var stan::math::fmod ( const double  a,
const var b 
)
inline

Return the floating point remainder after dividing the first scalar by the second variable (cmath).

The derivative with respect to the variable is

$\frac{d}{d y} \mbox{fmod}(c, y) = -\lfloor \frac{c}{y} \rfloor$.

Parameters
aFirst scalar.
bSecond variable.
Returns
Floating pointer remainder of dividing first scalar by the second variable.

Definition at line 137 of file fmod.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::frechet_ccdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 31 of file frechet_ccdf_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::frechet_cdf ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 31 of file frechet_cdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::frechet_cdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 31 of file frechet_cdf_log.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::frechet_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 34 of file frechet_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::frechet_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)
inline

Definition at line 140 of file frechet_log.hpp.

template<class RNG >
double stan::math::frechet_rng ( const double  alpha,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 30 of file frechet_rng.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >
return_type<T_y, T_shape, T_inv_scale>::type stan::math::gamma_ccdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 35 of file gamma_ccdf_log.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >
return_type<T_y, T_shape, T_inv_scale>::type stan::math::gamma_cdf ( const T_y &  y,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

The cumulative density function for a gamma distribution for y with the specified shape and inverse scale parameters.

Parameters
yA scalar variable.
alphaShape parameter.
betaInverse scale parameter.
Exceptions
std::domain_errorif alpha is not greater than 0.
std::domain_errorif beta is not greater than 0.
std::domain_errorif y is not greater than or equal to 0.
Template Parameters
T_yType of scalar.
T_shapeType of shape.
T_inv_scaleType of inverse scale.

Definition at line 49 of file gamma_cdf.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >
return_type<T_y, T_shape, T_inv_scale>::type stan::math::gamma_cdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 35 of file gamma_cdf_log.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_inv_scale >
return_type<T_y, T_shape, T_inv_scale>::type stan::math::gamma_log ( const T_y &  y,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

The log of a gamma density for y with the specified shape and inverse scale parameters.

Shape and inverse scale parameters must be greater than 0. y must be greater than or equal to 0.

\begin{eqnarray*} y &\sim& \mbox{\sf{Gamma}}(\alpha, \beta) \\ \log (p (y \, |\, \alpha, \beta) ) &=& \log \left( \frac{\beta^\alpha}{\Gamma(\alpha)} y^{\alpha - 1} \exp^{- \beta y} \right) \\ &=& \alpha \log(\beta) - \log(\Gamma(\alpha)) + (\alpha - 1) \log(y) - \beta y\\ & & \mathrm{where} \; y > 0 \end{eqnarray*}

Parameters
yA scalar variable.
alphaShape parameter.
betaInverse scale parameter.
Exceptions
std::domain_errorif alpha is not greater than 0.
std::domain_errorif beta is not greater than 0.
std::domain_errorif y is not greater than or equal to 0.
Template Parameters
T_yType of scalar.
T_shapeType of shape.
T_inv_scaleType of inverse scale.

Definition at line 52 of file gamma_log.hpp.

template<typename T_y , typename T_shape , typename T_inv_scale >
return_type<T_y, T_shape, T_inv_scale>::type stan::math::gamma_log ( const T_y &  y,
const T_shape &  alpha,
const T_inv_scale &  beta 
)
inline

Definition at line 164 of file gamma_log.hpp.

template<typename T >
fvar<T> stan::math::gamma_p ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 15 of file gamma_p.hpp.

template<typename T >
fvar<T> stan::math::gamma_p ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 51 of file gamma_p.hpp.

double stan::math::gamma_p ( double  x,
double  a 
)

\[ \mbox{gamma\_p}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ P(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_p}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, P(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ P(a, z)=\frac{1}{\Gamma(a)}\int_0^zt^{a-1}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_0^zt^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_0^z (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, P(a, z)}{\partial z} = \frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Definition at line 53 of file gamma_p.hpp.

template<typename T >
fvar<T> stan::math::gamma_p ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 86 of file gamma_p.hpp.

var stan::math::gamma_p ( const stan::math::var a,
const stan::math::var b 
)
inline

Definition at line 104 of file gamma_p.hpp.

var stan::math::gamma_p ( const stan::math::var a,
const double &  b 
)
inline

Definition at line 109 of file gamma_p.hpp.

var stan::math::gamma_p ( const double &  a,
const stan::math::var b 
)
inline

Definition at line 114 of file gamma_p.hpp.

template<typename T >
fvar<T> stan::math::gamma_q ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 15 of file gamma_q.hpp.

template<typename T >
fvar<T> stan::math::gamma_q ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 51 of file gamma_q.hpp.

double stan::math::gamma_q ( double  x,
double  a 
)

\[ \mbox{gamma\_q}(a, z) = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ Q(a, z) & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial a} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial a} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{gamma\_q}(a, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } a\leq 0 \textrm{ or } z < 0\\ \frac{\partial\, Q(a, z)}{\partial z} & \mbox{if } a > 0, z \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } a = \textrm{NaN or } z = \textrm{NaN} \end{cases} \]

\[ Q(a, z)=\frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial a} = -\frac{\Psi(a)}{\Gamma^2(a)}\int_z^\infty t^{a-1}e^{-t}dt + \frac{1}{\Gamma(a)}\int_z^\infty (a-1)t^{a-2}e^{-t}dt \]

\[ \frac{\partial \, Q(a, z)}{\partial z} = -\frac{z^{a-1}e^{-z}}{\Gamma(a)} \]

Definition at line 53 of file gamma_q.hpp.

var stan::math::gamma_q ( const stan::math::var a,
const stan::math::var b 
)
inline

Definition at line 58 of file gamma_q.hpp.

var stan::math::gamma_q ( const stan::math::var a,
const double &  b 
)
inline

Definition at line 63 of file gamma_q.hpp.

var stan::math::gamma_q ( const double &  a,
const stan::math::var b 
)
inline

Definition at line 68 of file gamma_q.hpp.

template<typename T >
fvar<T> stan::math::gamma_q ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 86 of file gamma_q.hpp.

template<class RNG >
double stan::math::gamma_rng ( const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 33 of file gamma_rng.hpp.

template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type<T_F, T_G, T_V, T_W, T_m0, T_C0>::type >::type stan::math::gaussian_dlm_obs_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F,
const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G,
const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &  V,
const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W,
const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0,
const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0 
)

The log of a Gaussian dynamic linear model (GDLM).

This distribution is equivalent to, for $t = 1:T$,

\begin{eqnarray*} y_t & \sim N(F' \theta_t, V) \\ \theta_t & \sim N(G \theta_{t-1}, W) \\ \theta_0 & \sim N(m_0, C_0) \end{eqnarray*}

If V is a vector, then the Kalman filter is applied sequentially.

Parameters
yA r x T matrix of observations. Rows are variables, columns are observations.
FA n x r matrix. The design matrix.
GA n x n matrix. The transition matrix.
VA r x r matrix. The observation covariance matrix.
WA n x n matrix. The state covariance matrix.
m0A n x 1 matrix. The mean vector of the distribution of the initial state.
C0A n x n matrix. The covariance matrix of the distribution of the initial state.
Returns
The log of the joint density of the GDLM.
Exceptions
std::domain_errorif a matrix in the Kalman filter is not positive semi-definite.
Template Parameters
T_yType of scalar.
T_FType of design matrix.
T_GType of transition matrix.
T_VType of observation covariance matrix.
T_WType of state covariance matrix.
T_m0Type of initial state mean vector.
T_C0Type of initial state covariance matrix.

Definition at line 79 of file gaussian_dlm_obs_log.hpp.

template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type<T_F, T_G, T_V, T_W, T_m0, T_C0>::type >::type stan::math::gaussian_dlm_obs_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F,
const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G,
const Eigen::Matrix< T_V, Eigen::Dynamic, Eigen::Dynamic > &  V,
const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W,
const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0,
const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0 
)
inline

Definition at line 225 of file gaussian_dlm_obs_log.hpp.

template<bool propto, typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type< T_y, typename return_type<T_F, T_G, T_V, T_W, T_m0, T_C0>::type >::type stan::math::gaussian_dlm_obs_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F,
const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G,
const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &  V,
const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W,
const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0,
const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0 
)

The log of a Gaussian dynamic linear model (GDLM) with uncorrelated observation disturbances.

This distribution is equivalent to, for $t = 1:T$,

\begin{eqnarray*} y_t & \sim N(F' \theta_t, diag(V)) \\ \theta_t & \sim N(G \theta_{t-1}, W) \\ \theta_0 & \sim N(m_0, C_0) \end{eqnarray*}

If V is a vector, then the Kalman filter is applied sequentially.

Parameters
yA r x T matrix of observations. Rows are variables, columns are observations.
FA n x r matrix. The design matrix.
GA n x n matrix. The transition matrix.
VA size r vector. The diagonal of the observation covariance matrix.
WA n x n matrix. The state covariance matrix.
m0A n x 1 matrix. The mean vector of the distribution of the initial state.
C0A n x n matrix. The covariance matrix of the distribution of the initial state.
Returns
The log of the joint density of the GDLM.
Exceptions
std::domain_errorif a matrix in the Kalman filter is not semi-positive definite.
Template Parameters
T_yType of scalar.
T_FType of design matrix.
T_GType of transition matrix.
T_VType of observation variances
T_WType of state covariance matrix.
T_m0Type of initial state mean vector.
T_C0Type of initial state covariance matrix.

Definition at line 285 of file gaussian_dlm_obs_log.hpp.

template<typename T_y , typename T_F , typename T_G , typename T_V , typename T_W , typename T_m0 , typename T_C0 >
return_type<T_y, typename return_type<T_F, T_G, T_V, T_W, T_m0, T_C0>::type>::type stan::math::gaussian_dlm_obs_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_F, Eigen::Dynamic, Eigen::Dynamic > &  F,
const Eigen::Matrix< T_G, Eigen::Dynamic, Eigen::Dynamic > &  G,
const Eigen::Matrix< T_V, Eigen::Dynamic, 1 > &  V,
const Eigen::Matrix< T_W, Eigen::Dynamic, Eigen::Dynamic > &  W,
const Eigen::Matrix< T_m0, Eigen::Dynamic, 1 > &  m0,
const Eigen::Matrix< T_C0, Eigen::Dynamic, Eigen::Dynamic > &  C0 
)
inline

Definition at line 442 of file gaussian_dlm_obs_log.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< T > &  x,
size_t  i,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
iIndex into vector plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at i - 1
Template Parameters
Ttype of value.

Definition at line 27 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< T > > &  x,
size_t  i1,
size_t  i2,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 53 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< T > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 81 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< T > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 111 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 143 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 178 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
size_t  i7,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
i7Seventh index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 216 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
size_t  i7,
size_t  i8,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
i7Seventh index plus 1.
i8Eigth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 256 of file get_base1.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x,
size_t  m,
const char *  error_msg,
size_t  idx 
)
inline

Return a copy of the row of the specified vector at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Parameters
xMatrix from which to get a row
mIndex into matrix plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Row of matrix at i - 1.
Template Parameters
Ttype of value.

Definition at line 297 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x,
size_t  m,
size_t  n,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xMatrix from which to get a row
mRow index plus 1.
nColumn index plus 1.
error_msgError message if either index is out of range.
idxNested index level to report in error message if either index is out of range.
Returns
Value of matrix at row m - 1 and column n - 1.
Template Parameters
Ttype of value.

Definition at line 324 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
size_t  m,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified column vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xColumn vector from which to get a value.
mRow index plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of column vector at row m - 1.
Template Parameters
Ttype of value.

Definition at line 351 of file get_base1.hpp.

template<typename T >
const T& stan::math::get_base1 ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  x,
size_t  n,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified row vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xRow vector from which to get a value.
nColumn index plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of row vector at column n - 1.
Template Parameters
Ttype of value.

Definition at line 376 of file get_base1.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< T > &  x,
size_t  i,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
iIndex into vector plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at i - 1
Template Parameters
Ttype of value.

Definition at line 27 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< T > > &  x,
size_t  i1,
size_t  i2,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 53 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< T > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 81 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< T > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 111 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 144 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 179 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
size_t  i7,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
i7Seventh index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 217 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< std::vector< T > > > > > > > > &  x,
size_t  i1,
size_t  i2,
size_t  i3,
size_t  i4,
size_t  i5,
size_t  i6,
size_t  i7,
size_t  i8,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified vector at the specified base-one indexes.

If an index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xVector from which to get a value.
i1First index plus 1.
i2Second index plus 1.
i3Third index plus 1.
i4Fourth index plus 1.
i5Fifth index plus 1.
i6Sixth index plus 1.
i7Seventh index plus 1.
i8Eigth index plus 1.
error_msgError message if an index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of vector at indexes.
Template Parameters
Ttype of value.

Definition at line 258 of file get_base1_lhs.hpp.

template<typename T >
Eigen::Block<Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> > stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x,
size_t  m,
const char *  error_msg,
size_t  idx 
)
inline

Return a copy of the row of the specified vector at the specified base-one row index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Warning: Because a copy is involved, it is inefficient to access element of matrices by first using this method to get a row then using a second call to get the value at a specified column.

Parameters
xMatrix from which to get a row
mIndex into matrix plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Row of matrix at i - 1.
Template Parameters
Ttype of value.

Definition at line 301 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x,
size_t  m,
size_t  n,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified matrix at the specified base-one row and column indexes.

If either index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xMatrix from which to get a row
mRow index plus 1.
nColumn index plus 1.
error_msgError message if either index is out of range.
idxNested index level to report in error message if either index is out of range.
Returns
Value of matrix at row m - 1 and column n - 1.
Template Parameters
Ttype of value.

Definition at line 328 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
size_t  m,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified column vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xColumn vector from which to get a value.
mRow index plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of column vector at row m - 1.
Template Parameters
Ttype of value.

Definition at line 355 of file get_base1_lhs.hpp.

template<typename T >
T& stan::math::get_base1_lhs ( Eigen::Matrix< T, 1, Eigen::Dynamic > &  x,
size_t  n,
const char *  error_msg,
size_t  idx 
)
inline

Return a reference to the value of the specified row vector at the specified base-one index.

If the index is out of range, throw a std::out_of_range exception with the specified error message and index indicated.

Parameters
xRow vector from which to get a value.
nColumn index plus 1.
error_msgError message if the index is out of range.
idxNested index level to report in error message if the index is out of range.
Returns
Value of row vector at column n - 1.
Template Parameters
Ttype of value.

Definition at line 380 of file get_base1_lhs.hpp.

template<typename T_lp , typename T_lp_accum >
boost::math::tools::promote_args<T_lp, T_lp_accum>::type stan::math::get_lp ( const T_lp &  lp,
const stan::math::accumulator< T_lp_accum > &  lp_accum 
)
inline

Definition at line 14 of file get_lp.hpp.

static void stan::math::grad ( vari vi)
static
void stan::math::grad ( var v,
Eigen::Matrix< var, Eigen::Dynamic, 1 > &  x,
Eigen::VectorXd &  g 
)

Propagate chain rule to calculate gradients starting from the specified variable.

Resizes the input vector to be the correct size.

The grad() function does not itself recover any memory. use recover_memory() or recover_memory_nested() to recover memory.

Parameters
[in]vValue of function being differentiated
[in]xVariables being differentiated with respect to
[out]gGradient, d/dx v, evaluated at x.

Definition at line 26 of file grad.hpp.

static void stan::math::grad ( vari vi)
static

Compute the gradient for all variables starting from the specified root variable implementation.

Does not recover memory. This chainable variable's adjoint is initialized using the method init_dependent() and then the chain rule is applied working down the stack from this vari and calling each vari's chain() method in turn.

This function computes a nested gradient only going back as far as the last nesting.

This function does not recover any memory from the computation.

Parameters
viVariable implementation for root of partial derivative propagation.

Definition at line 30 of file grad.hpp.

template<typename T >
void stan::math::grad_2F1 ( T &  gradA,
T &  gradC,
a,
b,
c,
z,
precision = 1e-6 
)

Definition at line 13 of file grad_2F1.hpp.

template<typename T >
void stan::math::grad_F32 ( T *  g,
a,
b,
c,
d,
e,
z,
precision = 1e-6 
)

Definition at line 11 of file grad_F32.hpp.

template<typename F >
void stan::math::grad_hessian ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
double &  fx,
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  H,
std::vector< Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > > &  grad_H 
)

Calculate the value, the Hessian, and the gradient of the Hessian of the specified function at the specified argument.

The functor must implement

stan::math::fvar<stan::math::fvar<stan::math::var> > operator()(const Eigen::Matrix<stan::math::fvar<stan::math::fvar<stan::math::var> >, Eigen::Dynamic, 1>&)

using only operations that are defined for stan::math::fvar and stan::math::var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]HHessian of function at argument
[out]grad_HGradient of the Hessian of function at argument

Definition at line 44 of file grad_hessian.hpp.

void stan::math::grad_inc_beta ( double &  g1,
double &  g2,
double  a,
double  b,
double  z 
)

Definition at line 17 of file grad_inc_beta.hpp.

void stan::math::grad_inc_beta ( var g1,
var g2,
const var a,
const var b,
const var z 
)

Definition at line 24 of file grad_inc_beta.hpp.

template<typename T >
void stan::math::grad_inc_beta ( stan::math::fvar< T > &  g1,
stan::math::fvar< T > &  g2,
stan::math::fvar< T >  a,
stan::math::fvar< T >  b,
stan::math::fvar< T >  z 
)

Definition at line 24 of file grad_inc_beta.hpp.

template<typename T >
void stan::math::grad_reg_inc_beta ( T &  g1,
T &  g2,
a,
b,
z,
digammaA,
digammaB,
digammaSum,
betaAB 
)

Definition at line 14 of file grad_reg_inc_beta.hpp.

template<typename T >
T stan::math::grad_reg_inc_gamma ( a,
z,
g,
dig,
precision = 1e-6 
)

Definition at line 15 of file grad_reg_inc_gamma.hpp.

template<typename F >
void stan::math::grad_tr_mat_times_hessian ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  M,
Eigen::Matrix< double, Eigen::Dynamic, 1 > &  grad_tr_MH 
)

Definition at line 19 of file grad_tr_mat_times_hessian.hpp.

template<typename F >
void stan::math::gradient ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
double &  fx,
Eigen::Matrix< double, Eigen::Dynamic, 1 > &  grad_fx 
)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

stan::math::var operator()(const Eigen::Matrix<stan::math::var, Eigen::Dynamic, 1>&)

using only operations that are defined for stan::math::var. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]grad_fxGradient of function at argument

Definition at line 42 of file gradient.hpp.

template<typename T , typename F >
void stan::math::gradient ( const F &  f,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
T &  fx,
Eigen::Matrix< T, Eigen::Dynamic, 1 > &  grad_fx 
)

Calculate the value and the gradient of the specified function at the specified argument.

The functor must implement

stan::math::fvar operator()(const Eigen::Matrix<stan::math::var, Eigen::Dynamic, 1>&)

using only operations that are defined for stan::math::fvar. This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Time and memory usage is on the order of the size of the fully unfolded expression for the function applied to the argument, independently of dimension.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]grad_fxGradient of function at argument

Definition at line 42 of file gradient.hpp.

template<typename T1 , typename T2 , typename F >
void stan::math::gradient_dot_vector ( const F &  f,
const Eigen::Matrix< T1, Eigen::Dynamic, 1 > &  x,
const Eigen::Matrix< T2, Eigen::Dynamic, 1 > &  v,
T1 &  fx,
T1 &  grad_fx_dot_v 
)

Definition at line 17 of file gradient_dot_vector.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::gumbel_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  beta 
)

Definition at line 28 of file gumbel_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::gumbel_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  beta 
)

Definition at line 28 of file gumbel_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::gumbel_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  beta 
)

Definition at line 28 of file gumbel_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::gumbel_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  beta 
)

Definition at line 28 of file gumbel_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::gumbel_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  beta 
)
inline

Definition at line 118 of file gumbel_log.hpp.

template<class RNG >
double stan::math::gumbel_rng ( const double  mu,
const double  beta,
RNG &  rng 
)
inline

Definition at line 27 of file gumbel_rng.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::head ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v,
size_t  n 
)
inline

Return the specified number of elements as a vector from the front of the specified vector.

Template Parameters
TType of value in vector
Parameters
vVector input
nSize of return
Returns
The first n elements of v

Definition at line 24 of file head.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::head ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv,
size_t  n 
)
inline

Return the specified number of elements as a row vector from the front of the specified row vector.

Template Parameters
TType of value in vector
Parameters
rvRow vector
nSize of return row vector
Returns
The first n elements of rv

Definition at line 42 of file head.hpp.

template<typename T >
std::vector<T> stan::math::head ( const std::vector< T > &  sv,
size_t  n 
)

Return the specified number of elements as a standard vector from the front of the specified standard vector.

Template Parameters
TType of value in vector
Parameters
svStandard vector
nSize of return
Returns
The first n elements of sv

Definition at line 58 of file head.hpp.

template<typename F >
void stan::math::hessian ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
double &  fx,
Eigen::Matrix< double, Eigen::Dynamic, 1 > &  grad,
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  H 
)

Calculate the value, the gradient, and the Hessian, of the specified function at the specified argument in O(N^2) time and O(N^2) space.

The functor must implement

stan::math::fvar<stan::math::var> operator()(const Eigen::Matrix<stan::math::fvar<stan::math::var>, Eigen::Dynamic, 1>&)

using only operations that are defined for stan::math::fvar and stan::math::var.

This latter constraint usually requires the functions to be defined in terms of the libraries defined in Stan or in terms of functions with appropriately general namespace imports that eventually depend on functions defined in Stan.

Template Parameters
FType of function
Parameters
[in]fFunction
[in]xArgument to function
[out]fxFunction applied to argument
[out]gradgradient of function at argument
[out]HHessian of function at argument

Definition at line 44 of file hessian.hpp.

template<typename T , typename F >
void stan::math::hessian ( const F &  f,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
T &  fx,
Eigen::Matrix< T, Eigen::Dynamic, 1 > &  grad,
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  H 
)

Definition at line 73 of file hessian.hpp.

template<typename F >
void stan::math::hessian_times_vector ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  v,
double &  fx,
Eigen::Matrix< double, Eigen::Dynamic, 1 > &  Hv 
)

Definition at line 15 of file hessian_times_vector.hpp.

template<typename T , typename F >
void stan::math::hessian_times_vector ( const F &  f,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v,
T &  fx,
Eigen::Matrix< T, Eigen::Dynamic, 1 > &  Hv 
)

Definition at line 44 of file hessian_times_vector.hpp.

template<bool propto, typename T_n , typename T_N , typename T_a , typename T_b >
double stan::math::hypergeometric_log ( const T_n &  n,
const T_N &  N,
const T_a &  a,
const T_b &  b 
)

Definition at line 31 of file hypergeometric_log.hpp.

template<typename T_n , typename T_N , typename T_a , typename T_b >
double stan::math::hypergeometric_log ( const T_n &  n,
const T_N &  N,
const T_a &  a,
const T_b &  b 
)
inline

Definition at line 88 of file hypergeometric_log.hpp.

template<class RNG >
int stan::math::hypergeometric_rng ( int  N,
int  a,
int  b,
RNG &  rng 
)
inline

Definition at line 16 of file hypergeometric_rng.hpp.

template<typename T >
fvar<T> stan::math::hypot ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 13 of file hypot.hpp.

template<typename T >
fvar<T> stan::math::hypot ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 22 of file hypot.hpp.

template<typename T >
fvar<T> stan::math::hypot ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 31 of file hypot.hpp.

var stan::math::hypot ( const var a,
const var b 
)
inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See hypot() for double-based function.

The partial derivatives are given by

$\frac{\partial}{\partial x} \sqrt{x^2 + y^2} = \frac{x}{\sqrt{x^2 + y^2}}$, and

$\frac{\partial}{\partial y} \sqrt{x^2 + y^2} = \frac{y}{\sqrt{x^2 + y^2}}$.

Parameters
aLength of first side.
bLength of second side.
Returns
Length of hypoteneuse.

Definition at line 53 of file hypot.hpp.

var stan::math::hypot ( const var a,
double  b 
)
inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See hypot() for double-based function.

The derivative is

$\frac{d}{d x} \sqrt{x^2 + c^2} = \frac{x}{\sqrt{x^2 + c^2}}$.

Parameters
aLength of first side.
bLength of second side.
Returns
Length of hypoteneuse.

Definition at line 71 of file hypot.hpp.

var stan::math::hypot ( double  a,
const var b 
)
inline

Returns the length of the hypoteneuse of a right triangle with sides of the specified lengths (C99).

See hypot() for double-based function.

The derivative is

$\frac{d}{d y} \sqrt{c^2 + y^2} = \frac{y}{\sqrt{c^2 + y^2}}$.

\[ \mbox{hypot}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \sqrt{x^2+y^2} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{x}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{hypot}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \text{ or } y < 0 \\ \frac{y}{\sqrt{x^2+y^2}} & \mbox{if } x, y\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aLength of first side.
bLength of second side.
Returns
Length of hypoteneuse.

Definition at line 116 of file hypot.hpp.

double stan::math::ibeta ( const double  a,
const double  b,
const double  x 
)
inline

The normalized incomplete beta function of a, b, and x.

Used to compute the cumulative density function for the beta distribution.

Parameters
aShape parameter a <= 0; a and b can't both be 0
bShape parameter b <= 0
xRandom variate. 0 <= x <= 1
Exceptions
ifconstraints are violated or if any argument is NaN
Returns
The normalized incomplete beta function.

Definition at line 23 of file ibeta.hpp.

var stan::math::ibeta ( const var a,
const var b,
const var x 
)
inline

The normalized incomplete beta function of a, b, and x.

Used to compute the cumulative density function for the beta distribution.

Partial derivatives are those specified by wolfram alpha. The values were checked using both finite differences and by independent code for calculating the derivatives found in JSS (paper by Boik and Robison-Cox).

Parameters
aShape parameter.
bShape parameter.
xRandom variate.
Returns
The normalized incomplete beta function.
Exceptions
ifany argument is NaN.

Definition at line 238 of file ibeta.hpp.

template<typename T >
T stan::math::identity_constrain ( x)
inline

Returns the result of applying the identity constraint transform to the input.

This method is effectively a no-op and is mainly useful as a placeholder in auto-generated code.

Parameters
xFree scalar.
Returns
Transformed input.
Template Parameters
TType of scalar.

Definition at line 22 of file identity_constrain.hpp.

template<typename T >
T stan::math::identity_constrain ( const T  x,
T &   
)
inline

Returns the result of applying the identity constraint transform to the input and increments the log probability reference with the log absolute Jacobian determinant.

This method is effectively a no-op and mainly useful as a placeholder in auto-generated code.

Parameters
xFree scalar. lp Reference to log probability.
Returns
Transformed input.
Template Parameters
TType of scalar.

Definition at line 41 of file identity_constrain.hpp.

template<typename T >
T stan::math::identity_free ( const T  y)
inline

Returns the result of applying the inverse of the identity constraint transform to the input.

This method is effectively a no-op and mainly useful as a placeholder in auto-generated code.

Parameters
yConstrained scalar.
Returns
The input.
Template Parameters
TType of scalar.

Definition at line 21 of file identity_free.hpp.

var stan::math::if_else ( bool  c,
const var y_true,
const var y_false 
)
inline

If the specified condition is true, return the first variable, otherwise return the second variable.

Parameters
cBoolean condition.
y_trueVariable to return if condition is true.
y_falseVariable to return if condition is false.

Definition at line 17 of file if_else.hpp.

template<typename T_true , typename T_false >
boost::math::tools::promote_args<T_true, T_false>::type stan::math::if_else ( const bool  c,
const T_true  y_true,
const T_false  y_false 
)
inline

Return the second argument if the first argument is true and otherwise return the second argument.

This is just a convenience method to provide a function with the same behavior as the built-in ternary operator. In general, this function behaves as if defined by

if_else(c, y1, y0) = c ? y1 : y0.

Parameters
cBoolean condition value.
y_trueValue to return if condition is true.
y_falseValue to return if condition is false.

Definition at line 25 of file if_else.hpp.

var stan::math::if_else ( bool  c,
double  y_true,
const var y_false 
)
inline

If the specified condition is true, return a new variable constructed from the first scalar, otherwise return the second variable.

Parameters
cBoolean condition.
y_trueValue to promote to variable and return if condition is true.
y_falseVariable to return if condition is false.

Definition at line 29 of file if_else.hpp.

var stan::math::if_else ( bool  c,
const var y_true,
const double  y_false 
)
inline

If the specified condition is true, return the first variable, otherwise return a new variable constructed from the second scalar.

Parameters
cBoolean condition.
y_trueVariable to return if condition is true.
y_falseValue to promote to variable and return if condition is false.

Definition at line 44 of file if_else.hpp.

double stan::math::inc_beta ( const double &  a,
const double &  b,
const double &  x 
)
inline

Definition at line 10 of file inc_beta.hpp.

template<typename T >
fvar<T> stan::math::inc_beta ( const fvar< T > &  a,
const fvar< T > &  b,
const fvar< T > &  x 
)
inline

Definition at line 20 of file inc_beta.hpp.

var stan::math::inc_beta ( const stan::math::var a,
const stan::math::var b,
const stan::math::var c 
)
inline

Definition at line 45 of file inc_beta.hpp.

template<typename T >
T stan::math::inc_beta_dda ( a,
b,
z,
digamma_a,
digamma_ab 
)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to a.

The power series used to compute the deriative tends to converge slowly when a and b are large, especially if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters
Tscalar types of arguments
Parameters
aa
bb
zupper bound of the integral
digamma_avalue of digamma(a)
digamma_abvalue of digamma(b)
Returns
partial derivative of the incomplete beta with respect to a
Precondition
a >= 0
b >= 0
0 <= z <= 1

Definition at line 38 of file inc_beta_dda.hpp.

template<typename T >
T stan::math::inc_beta_ddb ( a,
b,
z,
digamma_b,
digamma_ab 
)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to b.

The power series used to compute the deriative tends to converge slowly when a and b are large, especailly if z approaches 1. The implementation will throw an exception if the series have not converged within 100,000 iterations. The current implementation has been tested for values of a and b up to 12500 and z = 0.999.

Template Parameters
Tscalar types of arguments
Parameters
aa
bb
zupper bound of the integral
digamma_bvalue of digamma(b)
digamma_abvalue of digamma(b)
Returns
partial derivative of the incomplete beta with respect to b
Precondition
a >= 0
b >= 0
0 <= z <= 1

Definition at line 38 of file inc_beta_ddb.hpp.

template<typename T >
T stan::math::inc_beta_ddz ( a,
b,
z 
)

Returns the partial derivative of the regularized incomplete beta function, I_{z}(a, b) with respect to z.

Template Parameters
Tscalar types of arguments
Parameters
aa
bb
zupper bound of the integral
Returns
partial derivative of the incomplete beta with respect to z
Precondition
a > 0
b > 0
0 < z <= 1

Definition at line 27 of file inc_beta_ddz.hpp.

template<>
double stan::math::inc_beta_ddz ( double  a,
double  b,
double  z 
)

Definition at line 35 of file inc_beta_ddz.hpp.

template<typename T >
void stan::math::initialize ( T &  x,
const T &  v 
)
inline

Definition at line 17 of file initialize.hpp.

template<typename T , typename V >
boost::enable_if_c<boost::is_arithmetic<V>::value, void>::type stan::math::initialize ( T &  x,
v 
)
inline

Definition at line 23 of file initialize.hpp.

template<typename T , int R, int C, typename V >
void stan::math::initialize ( Eigen::Matrix< T, R, C > &  x,
const V &  v 
)
inline

Definition at line 27 of file initialize.hpp.

template<typename T , typename V >
void stan::math::initialize ( std::vector< T > &  x,
const V &  v 
)
inline

Definition at line 32 of file initialize.hpp.

void stan::math::initialize_variable ( var variable,
const var value 
)
inline

Initialize variable to value.

(Function may look pointless, but its needed to bottom out recursion.)

Definition at line 15 of file initialize_variable.hpp.

template<int R, int C>
void stan::math::initialize_variable ( Eigen::Matrix< var, R, C > &  matrix,
const var value 
)
inline

Initialize every cell in the matrix to the specified value.

Definition at line 24 of file initialize_variable.hpp.

template<typename T >
void stan::math::initialize_variable ( std::vector< T > &  variables,
const var value 
)
inline

Initialize the variables in the standard vector recursively.

Definition at line 34 of file initialize_variable.hpp.

template<typename T >
unsigned int stan::math::int_step ( const T  y)

The integer step, or Heaviside, function.

For double NaN input, int_step(NaN) returns 0.

\[ \mbox{int\_step}(x) = \begin{cases} 0 & \mbox{if } x \leq 0 \\ 1 & \mbox{if } x > 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
yValue to test.
Returns
1 if value is greater than 0 and 0 otherwise
Template Parameters
TScalar argument type.

Definition at line 25 of file int_step.hpp.

template<typename F , typename T1 , typename T2 >
std::vector<std::vector<typename stan::return_type<T1, T2>::type> > stan::math::integrate_ode ( const F &  f,
const std::vector< T1 >  y0,
const double  t0,
const std::vector< double > &  ts,
const std::vector< T2 > &  theta,
const std::vector< double > &  x,
const std::vector< int > &  x_int,
std::ostream *  msgs 
)

Return the solutions for the specified system of ordinary differential equations given the specified initial state, initial times, times of desired solution, and parameters and data, writing error and warning messages to the specified stream.

Warning: If the system of equations is stiff, roughly defined by having varying time scales across dimensions, then this solver is likely to be slow.

This function is templated to allow the initial times to be either data or autodiff variables and the parameters to be data or autodiff variables. The autodiff-based implementation for reverse-mode are defined in namespace stan::math and may be invoked via argument-dependent lookup by including their headers.

This function uses the Dormand-Prince method as implemented in Boost's boost::numeric::odeint::runge_kutta_dopri5 integrator.

Template Parameters
Ftype of ODE system function.
T1type of scalars for initial values.
T2type of scalars for parameters.
Parameters
[in]ffunctor for the base ordinary differential equation.
[in]y0initial state.
[in]t0initial time.
[in]tstimes of the desired solutions, in strictly increasing order, all greater than the initial time.
[in]thetaparameter vector for the ODE.
[in]xcontinuous data vector for the ODE.
[in]x_intinteger data vector for the ODE.
[in,out]msgsthe print stream for warning messages.
Returns
a vector of states, each state being a vector of the same size as the state variable, corresponding to a time in ts.

Definition at line 60 of file integrate_ode.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::inv ( const T  x)
inline

Definition at line 12 of file inv.hpp.

template<typename T >
fvar<T> stan::math::inv ( const fvar< T > &  x)
inline

Definition at line 15 of file inv.hpp.

var stan::math::inv ( const var a)
inline

\[ \mbox{inv}(x) = \begin{cases} \frac{1}{x} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv}(x)}{\partial x} = \begin{cases} -\frac{1}{x^2} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 42 of file inv.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::inv_chi_square_ccdf_log ( const T_y &  y,
const T_dof &  nu 
)

Definition at line 33 of file inv_chi_square_ccdf_log.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::inv_chi_square_cdf ( const T_y &  y,
const T_dof &  nu 
)

Definition at line 33 of file inv_chi_square_cdf.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::inv_chi_square_cdf_log ( const T_y &  y,
const T_dof &  nu 
)

Definition at line 33 of file inv_chi_square_cdf_log.hpp.

template<bool propto, typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::inv_chi_square_log ( const T_y &  y,
const T_dof &  nu 
)

The log of an inverse chi-squared density for y with the specified degrees of freedom parameter.

The degrees of freedom prarameter must be greater than 0. y must be greater than 0.

\begin{eqnarray*} y &\sim& \mbox{\sf{Inv-}}\chi^2_\nu \\ \log (p (y \, |\, \nu)) &=& \log \left( \frac{2^{-\nu / 2}}{\Gamma (\nu / 2)} y^{- (\nu / 2 + 1)} \exp^{-1 / (2y)} \right) \\ &=& - \frac{\nu}{2} \log(2) - \log (\Gamma (\nu / 2)) - (\frac{\nu}{2} + 1) \log(y) - \frac{1}{2y} \\ & & \mathrm{ where } \; y > 0 \end{eqnarray*}

Parameters
yA scalar variable.
nuDegrees of freedom.
Exceptions
std::domain_errorif nu is not greater than or equal to 0
std::domain_errorif y is not greater than or equal to 0.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.

Definition at line 52 of file inv_chi_square_log.hpp.

template<typename T_y , typename T_dof >
return_type<T_y, T_dof>::type stan::math::inv_chi_square_log ( const T_y &  y,
const T_dof &  nu 
)
inline

Definition at line 142 of file inv_chi_square_log.hpp.

template<class RNG >
double stan::math::inv_chi_square_rng ( const double  nu,
RNG &  rng 
)
inline

Definition at line 31 of file inv_chi_square_rng.hpp.

template<typename T >
fvar<T> stan::math::inv_cloglog ( const fvar< T > &  x)
inline

Definition at line 15 of file inv_cloglog.hpp.

var stan::math::inv_cloglog ( const stan::math::var a)
inline

Return the inverse complementary log-log function applied specified variable (stan).

See stan::math::inv_cloglog() for the double-based version.

The derivative is given by

$\frac{d}{dx} \mbox{cloglog}^{-1}(x) = \exp (x - \exp (x))$.

Parameters
aVariable argument.
Returns
The inverse complementary log-log of the specified argument.

Definition at line 36 of file inv_cloglog.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::inv_cloglog ( x)
inline

The inverse complementary log-log function.

The function is defined by

inv_cloglog(x) = 1 - exp(-exp(x)).

This function can be used to implement the inverse link function for complementary-log-log regression.

\[ \mbox{inv\_cloglog}(y) = \begin{cases} \mbox{cloglog}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_cloglog}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{cloglog}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{cloglog}^{-1}(y) = 1 - \exp \left( - \exp(y) \right) \]

\[ \frac{\partial \, \mbox{cloglog}^{-1}(y)}{\partial y} = \exp(y-\exp(y)) \]

Parameters
xArgument.
Returns
Inverse complementary log-log of the argument.

Definition at line 49 of file inv_cloglog.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::inv_gamma_ccdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  beta 
)

Definition at line 35 of file inv_gamma_ccdf_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::inv_gamma_cdf ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  beta 
)

The CDF of an inverse gamma density for y with the specified shape and scale parameters.

y, shape, and scale parameters must be greater than 0.

Parameters
yA scalar variable.
alphaShape parameter.
betaScale parameter.
Exceptions
std::domain_errorif alpha is not greater than 0.
std::domain_errorif beta is not greater than 0.
std::domain_errorif y is not greater than 0.
Template Parameters
T_yType of scalar.
T_shapeType of shape.
T_scaleType of scale.

Definition at line 51 of file inv_gamma_cdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::inv_gamma_cdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  beta 
)

Definition at line 35 of file inv_gamma_cdf_log.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::inv_gamma_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  beta 
)

The log of an inverse gamma density for y with the specified shape and scale parameters.

Shape and scale parameters must be greater than 0. y must be greater than 0.

Parameters
yA scalar variable.
alphaShape parameter.
betaScale parameter.
Exceptions
std::domain_errorif alpha is not greater than 0.
std::domain_errorif beta is not greater than 0.
std::domain_errorif y is not greater than 0.
Template Parameters
T_yType of scalar.
T_shapeType of shape.
T_scaleType of scale.

Definition at line 51 of file inv_gamma_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::inv_gamma_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  beta 
)
inline

Definition at line 163 of file inv_gamma_log.hpp.

template<class RNG >
double stan::math::inv_gamma_rng ( const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 33 of file inv_gamma_rng.hpp.

template<typename T >
fvar<T> stan::math::inv_logit ( const fvar< T > &  x)
inline

Definition at line 15 of file inv_logit.hpp.

var stan::math::inv_logit ( const stan::math::var a)
inline

The inverse logit function for variables (stan).

See stan::math::inv_logit() for the double-based version.

The derivative of inverse logit is

$\frac{d}{dx} \mbox{logit}^{-1}(x) = \mbox{logit}^{-1}(x) (1 - \mbox{logit}^{-1}(x))$.

Parameters
aArgument variable.
Returns
Inverse logit of argument.

Definition at line 34 of file inv_logit.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::inv_logit ( const T  a)
inline

Returns the inverse logit function applied to the argument.

The inverse logit function is defined by

$\mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(-x)}$.

This function can be used to implement the inverse link function for logistic regression.

The inverse to this function is stan::math::logit.

\[ \mbox{inv\_logit}(y) = \begin{cases} \mbox{logit}^{-1}(y) & \mbox{if } -\infty\leq y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_logit}(y)}{\partial y} = \begin{cases} \frac{\partial\, \mbox{logit}^{-1}(y)}{\partial y} & \mbox{if } -\infty\leq y\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } y = \textrm{NaN} \end{cases} \]

\[ \mbox{logit}^{-1}(y) = \frac{1}{1 + \exp(-y)} \]

\[ \frac{\partial \, \mbox{logit}^{-1}(y)}{\partial y} = \frac{\exp(y)}{(\exp(y)+1)^2} \]

Parameters
aArgument.
Returns
Inverse logit of argument.

Definition at line 52 of file inv_logit.hpp.

template<typename T >
fvar<T> stan::math::inv_Phi ( const fvar< T > &  p)
inline

Definition at line 15 of file inv_Phi.hpp.

double stan::math::inv_Phi ( double  p)
inline

The inverse of the unit normal cumulative distribution function.

The return value for a specified input probability, $p$, is the unit normal variate, $x$, such that

$\Phi(x) = \int_{-\infty}^x \mbox{\sf Norm}(x|0, 1) \ dx = p$

Algorithm first derived in 2003 by Peter Jon Aklam at http://home.online.no/~pjacklam/notes/invnorm/

Parameters
pArgument between 0 and 1.
Returns
Real number

Definition at line 26 of file inv_Phi.hpp.

var stan::math::inv_Phi ( const stan::math::var p)
inline

The inverse of unit normal cumulative density function.

See stan::math::inv_Phi() for the double-based version.

The derivative is the reciprocal of unit normal density function,

Parameters
pProbability
Returns
The unit normal inverse cdf evaluated at p

Definition at line 37 of file inv_Phi.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::inv_sqrt ( const T  x)
inline

Definition at line 12 of file inv_sqrt.hpp.

template<typename T >
fvar<T> stan::math::inv_sqrt ( const fvar< T > &  x)
inline

Definition at line 15 of file inv_sqrt.hpp.

var stan::math::inv_sqrt ( const var a)
inline

\[ \mbox{inv\_sqrt}(x) = \begin{cases} \frac{1}{\sqrt{x}} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_sqrt}(x)}{\partial x} = \begin{cases} -\frac{1}{2\sqrt{x^3}} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 42 of file inv_sqrt.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::inv_square ( const T  x)
inline

Definition at line 12 of file inv_square.hpp.

template<typename T >
fvar<T> stan::math::inv_square ( const fvar< T > &  x)
inline

Definition at line 15 of file inv_square.hpp.

var stan::math::inv_square ( const var a)
inline

\[ \mbox{inv\_square}(x) = \begin{cases} \frac{1}{x^2} & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{inv\_square}(x)}{\partial x} = \begin{cases} -\frac{2}{x^3} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 42 of file inv_square.hpp.

template<bool propto, typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args<T_y, T_dof, T_scale>::type stan::math::inv_wishart_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W,
const T_dof &  nu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S 
)

The log of the Inverse-Wishart density for the given W, degrees of freedom, and scale matrix.

The scale matrix, S, must be k x k, symmetric, and semi-positive definite.

\begin{eqnarray*} W &\sim& \mbox{\sf{Inv-Wishart}}_{\nu} (S) \\ \log (p (W \, |\, \nu, S) ) &=& \log \left( \left(2^{\nu k/2} \pi^{k (k-1) /4} \prod_{i=1}^k{\Gamma (\frac{\nu + 1 - i}{2})} \right)^{-1} \times \left| S \right|^{\nu/2} \left| W \right|^{-(\nu + k + 1) / 2} \times \exp (-\frac{1}{2} \mbox{tr} (S W^{-1})) \right) \\ &=& -\frac{\nu k}{2}\log(2) - \frac{k (k-1)}{4} \log(\pi) - \sum_{i=1}^{k}{\log (\Gamma (\frac{\nu+1-i}{2}))} +\frac{\nu}{2} \log(\det(S)) - \frac{\nu+k+1}{2}\log (\det(W)) - \frac{1}{2} \mbox{tr}(S W^{-1}) \end{eqnarray*}

Parameters
WA scalar matrix
nuDegrees of freedom
SThe scale matrix
Returns
The log of the Inverse-Wishart density at W given nu and S.
Exceptions
std::domain_errorif nu is not greater than k-1
std::domain_errorif S is not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.
T_scaleType of scale.

Definition at line 52 of file inv_wishart_log.hpp.

template<typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args<T_y, T_dof, T_scale>::type stan::math::inv_wishart_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W,
const T_dof &  nu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S 
)
inline

Definition at line 125 of file inv_wishart_log.hpp.

template<class RNG >
Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::inv_wishart_rng ( const double  nu,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  S,
RNG &  rng 
)
inline

Definition at line 21 of file inv_wishart_rng.hpp.

template<typename T >
void stan::math::invalid_argument ( const char *  function,
const char *  name,
const T &  y,
const char *  msg1,
const char *  msg2 
)
inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once.

The message is: "<function>: <name> <msg1><y><msg2>"

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
msg1Message to print before the variable
msg2Message to print after the variable
Exceptions
std::invalid_argument

Definition at line 31 of file invalid_argument.hpp.

template<typename T >
void stan::math::invalid_argument ( const char *  function,
const char *  name,
const T &  y,
const char *  msg1 
)
inline

Throw an invalid_argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid argument. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name> <msg1><y>"

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
msg1Message to print before the variable
Exceptions
std::invalid_argument

Definition at line 66 of file invalid_argument.hpp.

template<typename T >
void stan::math::invalid_argument_vec ( const char *  function,
const char *  name,
const T &  y,
const size_t  i,
const char *  msg1,
const char *  msg2 
)
inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
iIndex
msg1Message to print before the variable
msg2Message to print after the variable
Exceptions
std::invalid_argument

Definition at line 38 of file invalid_argument_vec.hpp.

template<typename T >
void stan::math::invalid_argument_vec ( const char *  function,
const char *  name,
const T &  y,
const size_t  i,
const char *  msg 
)
inline

Throw an invalid argument exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing invalid arguments. This will allow us to change the behavior for all functions at once. (We've already changed behavior mulitple times up to Stan v2.5.0.)

The message is: "<function>: <name>[<i+error_index>] <msg1><y>" where error_index is the value of stan::error_index::value which indicates whether the message should be 0 or 1 indexed.

Template Parameters
TType of variable
Parameters
functionName of the function
nameName of the variable
yVariable
iIndex
msgMessage to print before the variable
Exceptions
std::invalid_argument

Definition at line 74 of file invalid_argument_vec.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::inverse ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the inverse of the specified matrix.

Parameters
mSpecified matrix.
Returns
Inverse of the matrix.

Definition at line 18 of file inverse.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::inverse ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 20 of file inverse.hpp.

template<typename Vector >
void stan::math::inverse_softmax ( const Vector &  simplex,
Vector &  y 
)

Writes the inverse softmax of the simplex argument into the second argument.

See stan::math::softmax for the inverse function and a definition of the relation.

The inverse softmax function is defined by

$\mbox{inverse\_softmax}(x)[i] = \log x[i]$.

This function defines the inverse of stan::math::softmax up to a scaling factor.

Because of the definition, values of 0.0 in the simplex are converted to negative infinity, and values of 1.0 are converted to 0.0.

There is no check that the input vector is a valid simplex vector.

Parameters
simplexSimplex vector input.
yVector into which the inverse softmax is written.
Exceptions
std::invalid_argumentif size of the input and output vectors differ.

Definition at line 34 of file inverse_softmax.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::inverse_spd ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)
inline

Returns the inverse of the specified symmetric, pos/neg-definite matrix.

Parameters
mSpecified matrix.
Returns
Inverse of the matrix.

Definition at line 19 of file inverse_spd.hpp.

template<typename T >
bool stan::math::is_aligned ( T *  ptr,
unsigned int  bytes_aligned 
)

Return true if the specified pointer is aligned on the number of bytes.

This doesn't really make sense other than for powers of 2.

Parameters
ptrPointer to test.
bytes_alignedNumber of bytes of alignment required.
Returns
true if pointer is aligned.
Template Parameters
Typeof object to which pointer points.

Definition at line 30 of file stack_alloc.hpp.

int stan::math::is_inf ( const double  x)
inline

Returns 1 if the input is infinite and 0 otherwise.

Delegates to boost::math::isinf.

Parameters
xValue to test.
Returns
1 if the value is infinite.

Definition at line 19 of file is_inf.hpp.

template<typename T >
int stan::math::is_inf ( const fvar< T > &  x)
inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to stan::math::is_inf.

Parameters
xValue to test.
Returns
1 if the value is infinite and 0 otherwise.

Definition at line 22 of file is_inf.hpp.

int stan::math::is_inf ( const var v)
inline

Returns 1 if the input's value is infinite and 0 otherwise.

Delegates to stan::math::is_inf.

Parameters
vValue to test.
Returns
1 if the value is infinite and 0 otherwise.

Definition at line 23 of file is_inf.hpp.

bool stan::math::is_nan ( double  x)
inline

Returns 1 if the input is NaN and 0 otherwise.

Delegates to boost::math::isnan.

Parameters
xValue to test.
Returns
1 if the value is NaN.

Definition at line 18 of file is_nan.hpp.

bool stan::math::is_nan ( const var v)
inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to stan::math::is_nan(double).

Parameters
vValue to test.
Returns
1 if the value is NaN and 0 otherwise.

Definition at line 21 of file is_nan.hpp.

template<typename T >
int stan::math::is_nan ( const fvar< T > &  x)
inline

Returns 1 if the input's value is NaN and 0 otherwise.

Delegates to stan::math::is_nan.

Parameters
xValue to test.
Returns
1 if the value is NaN and 0 otherwise.

Definition at line 22 of file is_nan.hpp.

template<typename T >
bool stan::math::is_uninitialized ( x)
inline

Returns true if the specified variable is uninitialized.

Arithmetic types are always initialized by definition (the value is not specified).

Template Parameters
TType of object to test.
Parameters
xObject to test.
Returns
true if the specified object is uninitialized.
false if input is NaN.

Definition at line 19 of file is_uninitialized.hpp.

bool stan::math::is_uninitialized ( var  x)
inline

Returns true if the specified variable is uninitialized.

This overload of the stan::math::is_uninitialized() function delegates the return to the is_uninitialized() method on the specified variable.

Parameters
xObject to test.
Returns
true if the specified object is uninitialized.

Definition at line 23 of file is_uninitialized.hpp.

template<typename F >
void stan::math::jacobian ( const F &  f,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  x,
Eigen::Matrix< double, Eigen::Dynamic, 1 > &  fx,
Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  J 
)

Definition at line 14 of file jacobian.hpp.

template<typename T , typename F >
void stan::math::jacobian ( const F &  f,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
Eigen::Matrix< T, Eigen::Dynamic, 1 > &  fx,
Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  J 
)

Definition at line 14 of file jacobian.hpp.

template<typename T , typename TL >
T stan::math::lb_constrain ( const T  x,
const TL  lb 
)
inline

Return the lower-bounded value for the specified unconstrained input and specified lower bound.

The transform applied is

$f(x) = \exp(x) + L$

where $L$ is the constant lower bound.

If the lower bound is negative infinity, this function reduces to identity_constrain(x).

Parameters
xUnconstrained scalar input.
lbLower-bound on constrained ouptut.
Returns
Lower-bound constrained value correspdonding to inputs.
Template Parameters
TType of scalar.
TLType of lower bound.

Definition at line 35 of file lb_constrain.hpp.

template<typename T , typename TL >
boost::math::tools::promote_args<T, TL>::type stan::math::lb_constrain ( const T  x,
const TL  lb,
T &  lp 
)
inline

Return the lower-bounded value for the speicifed unconstrained input and specified lower bound, incrementing the specified reference with the log absolute Jacobian determinant of the transform.

If the lower bound is negative infinity, this function reduces to identity_constraint(x, lp).

Parameters
xUnconstrained scalar input.
lbLower-bound on output.
lpReference to log probability to increment.
Returns
Loer-bound constrained value corresponding to inputs.
Template Parameters
TType of scalar.
TLType of lower bound.

Definition at line 61 of file lb_constrain.hpp.

template<typename T , typename TL >
boost::math::tools::promote_args<T, TL>::type stan::math::lb_free ( const T  y,
const TL  lb 
)
inline

Return the unconstrained value that produces the specified lower-bound constrained value.

If the lower bound is negative infinity, it is ignored and the function reduces to identity_free(y).

Parameters
yInput scalar.
lbLower bound.
Returns
Unconstrained value that produces the input when constrained.
Template Parameters
TType of scalar.
TLType of lower bound.
Exceptions
std::domain_errorif y is lower than the lower bound.

Definition at line 32 of file lb_free.hpp.

template<typename T >
fvar<T> stan::math::lbeta ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 16 of file lbeta.hpp.

template<typename T >
fvar<T> stan::math::lbeta ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 28 of file lbeta.hpp.

template<typename T >
fvar<T> stan::math::lbeta ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 38 of file lbeta.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::lbeta ( const T1  a,
const T2  b 
)
inline

Return the log of the beta function applied to the specified arguments.

The beta function is defined for $a > 0$ and $b > 0$ by

$\mbox{B}(a, b) = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$.

This function returns its log,

$\log \mbox{B}(a, b) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)$.

See boost::math::lgamma() for the double-based and stan::math for the variable-based log Gamma function.

\[ \mbox{lbeta}(\alpha, \beta) = \begin{cases} \ln\int_0^1 u^{\alpha - 1} (1 - u)^{\beta - 1} \, du & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \alpha} = \begin{cases} \Psi(\alpha)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lbeta}(\alpha, \beta)}{\partial \beta} = \begin{cases} \Psi(\beta)-\Psi(\alpha+\beta) & \mbox{if } \alpha, \beta>0 \\[6pt] \textrm{NaN} & \mbox{if } \alpha = \textrm{NaN or } \beta = \textrm{NaN} \end{cases} \]

Parameters
aFirst value
bSecond value
Returns
Log of the beta function applied to the two values.
Template Parameters
T1Type of first value.
T2Type of second value.

Definition at line 59 of file lbeta.hpp.

template<typename T >
fvar<T> stan::math::lgamma ( const fvar< T > &  x)
inline

Definition at line 15 of file lgamma.hpp.

double stan::math::lgamma ( double  x)

\[ \mbox{lgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 31 of file lgamma.hpp.

var stan::math::lgamma ( const stan::math::var a)
inline

The log gamma function for variables (C99).

The derivatie is the digamma function,

$\frac{d}{dx} \Gamma(x) = \psi^{(0)}(x)$.

Parameters
aThe variable.
Returns
Log gamma of the variable.

Definition at line 35 of file lgamma.hpp.

template<bool propto, typename T_covar , typename T_shape >
boost::math::tools::promote_args<T_covar, T_shape>::type stan::math::lkj_corr_cholesky_log ( const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L,
const T_shape &  eta 
)

Definition at line 56 of file lkj_corr_cholesky_log.hpp.

template<typename T_covar , typename T_shape >
boost::math::tools::promote_args<T_covar, T_shape>::type stan::math::lkj_corr_cholesky_log ( const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L,
const T_shape &  eta 
)
inline

Definition at line 99 of file lkj_corr_cholesky_log.hpp.

template<class RNG >
Eigen::MatrixXd stan::math::lkj_corr_cholesky_rng ( const size_t  K,
const double  eta,
RNG &  rng 
)
inline

Definition at line 52 of file lkj_corr_cholesky_rng.hpp.

template<bool propto, typename T_y , typename T_shape >
boost::math::tools::promote_args<T_y, T_shape>::type stan::math::lkj_corr_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const T_shape &  eta 
)

Definition at line 86 of file lkj_corr_log.hpp.

template<typename T_y , typename T_shape >
boost::math::tools::promote_args<T_y, T_shape>::type stan::math::lkj_corr_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const T_shape &  eta 
)
inline

Definition at line 122 of file lkj_corr_log.hpp.

template<class RNG >
Eigen::MatrixXd stan::math::lkj_corr_rng ( const size_t  K,
const double  eta,
RNG &  rng 
)
inline

Definition at line 52 of file lkj_corr_rng.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args<T_y, T_loc, T_scale, T_shape>::type stan::math::lkj_cov_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &  mu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &  sigma,
const T_shape &  eta 
)

Definition at line 24 of file lkj_cov_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args<T_y, T_loc, T_scale, T_shape>::type stan::math::lkj_cov_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_loc, Eigen::Dynamic, 1 > &  mu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, 1 > &  sigma,
const T_shape &  eta 
)
inline

Definition at line 74 of file lkj_cov_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args<T_y, T_loc, T_scale, T_shape>::type stan::math::lkj_cov_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  eta 
)

Definition at line 87 of file lkj_cov_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
boost::math::tools::promote_args<T_y, T_loc, T_scale, T_shape>::type stan::math::lkj_cov_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  eta 
)
inline

Definition at line 124 of file lkj_cov_log.hpp.

template<typename T >
fvar<typename stan::return_type<T, int>::type> stan::math::lmgamma ( int  x1,
const fvar< T > &  x2 
)
inline

Definition at line 16 of file lmgamma.hpp.

var stan::math::lmgamma ( int  a,
const stan::math::var b 
)
inline

Definition at line 28 of file lmgamma.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::lmgamma ( const int  k,
x 
)
inline

Return the natural logarithm of the multivariate gamma function with the speciifed dimensions and argument.

The multivariate gamma function $\Gamma_k(x)$ for dimensionality $k$ and argument $x$ is defined by

$\Gamma_k(x) = \pi^{k(k-1)/4} \, \prod_{j=1}^k \Gamma(x + (1 - j)/2)$,

where $\Gamma()$ is the gamma function.

\[ \mbox{lmgamma}(n, x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma_n(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{lmgamma}(n, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \ln\Gamma_n(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \ln\Gamma_n(x) = \pi^{n(n-1)/4} \, \prod_{j=1}^n \Gamma(x + (1 - j)/2) \]

\[ \frac{\partial \, \ln\Gamma_n(x)}{\partial x} = \sum_{j=1}^n \Psi(x + (1 - j) / 2) \]

Parameters
kNumber of dimensions.
xFunction argument.
Returns
Natural log of the multivariate gamma function.
Template Parameters
TType of scalar.

Definition at line 57 of file lmgamma.hpp.

template<typename T >
fvar<T> stan::math::log ( const fvar< T > &  x)
inline

Definition at line 15 of file log.hpp.

template<typename T , int Rows, int Cols>
Eigen::Matrix<T, Rows, Cols> stan::math::log ( const Eigen::Matrix< T, Rows, Cols > &  m)
inline

Return the element-wise logarithm of the matrix or vector.

Parameters
mThe matrix or vector.
Returns
ret(i, j) = log(m(i, j))

Definition at line 17 of file log.hpp.

var stan::math::log ( const var a)
inline

Return the natural log of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \log x = \frac{1}{x}$.

\[ \mbox{log}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \ln(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable whose log is taken.
Returns
Natural log of variable.

Definition at line 50 of file log.hpp.

template<typename T >
fvar<T> stan::math::log10 ( const fvar< T > &  x)
inline

Definition at line 15 of file log10.hpp.

var stan::math::log10 ( const var a)
inline

Return the base 10 log of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \log_{10} x = \frac{1}{x \log 10}$.

\[ \mbox{log10}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \log_{10}(x) & \mbox{if } x \geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log10}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0\\ \frac{1}{x \ln10} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable whose log is taken.
Returns
Base 10 log of variable.

Definition at line 54 of file log10.hpp.

double stan::math::log10 ( )
inline

Return natural logarithm of ten.

Returns
Natural logarithm of ten.

Definition at line 114 of file constants.hpp.

template<typename T >
fvar<T> stan::math::log1m ( const fvar< T > &  x)
inline

Definition at line 16 of file log1m.hpp.

var stan::math::log1m ( const stan::math::var a)
inline

The log (1 - x) function for variables.

The derivative is given by

$\frac{d}{dx} \log (1 - x) = -\frac{1}{1 - x}$.

Parameters
aThe variable.
Returns
The variable representing log of 1 minus the variable.

Definition at line 32 of file log1m.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log1m ( x)
inline

Return the natural logarithm of one minus the specified value.

The main use of this function is to cut down on intermediate values during algorithmic differentiation.

\[ \mbox{log1m}(x) = \begin{cases} \ln(1-x) & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m}(x)}{\partial x} = \begin{cases} -\frac{1}{1-x} & \mbox{if } x \leq 1 \\ \textrm{NaN} & \mbox{if } x > 1\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
xSpecified value.
Returns
Natural log of one minus x.

Definition at line 40 of file log1m.hpp.

template<typename T >
fvar<T> stan::math::log1m_exp ( const fvar< T > &  x)
inline

Definition at line 16 of file log1m_exp.hpp.

var stan::math::log1m_exp ( const stan::math::var a)
inline

Return the log of 1 minus the exponential of the specified variable.

Definition at line 38 of file log1m_exp.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log1m_exp ( const T  a)
inline

Calculates the log of 1 minus the exponential of the specified value without overflow log1m_exp(x) = log(1-exp(x)).

This function is only defined for x<0

\[ \mbox{log1m\_exp}(x) = \begin{cases} \ln(1-\exp(x)) & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{asinh}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{1-\exp(x)} & \mbox{if } x < 0 \\ \textrm{NaN} & \mbox{if } x \geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 41 of file log1m_exp.hpp.

template<typename T >
fvar<T> stan::math::log1m_inv_logit ( const fvar< T > &  x)
inline

Definition at line 15 of file log1m_inv_logit.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log1m_inv_logit ( const T  u)
inline

Returns the natural logarithm of 1 minus the inverse logit of the specified argument.

\[ \mbox{log1m\_inv\_logit}(x) = \begin{cases} -\ln(\exp(x)+1) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1m\_inv\_logit}(x)}{\partial x} = \begin{cases} -\frac{\exp(x)}{\exp(x)+1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters
TScalar type
Parameters
uInput.
Returns
log of 1 minus the inverse logit of the input.

Definition at line 36 of file log1m_inv_logit.hpp.

template<typename T >
fvar<T> stan::math::log1p ( const fvar< T > &  x)
inline

Definition at line 16 of file log1p.hpp.

var stan::math::log1p ( const stan::math::var a)
inline

The log (1 + x) function for variables (C99).

The derivative is given by

$\frac{d}{dx} \log (1 + x) = \frac{1}{1 + x}$.

Parameters
aThe variable.
Returns
The log of 1 plus the variable.

Definition at line 34 of file log1p.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log1p ( const T  x)
inline

Return the natural logarithm of one plus the specified value.

The main use of this function is to cut down on intermediate values during algorithmic differentiation.

\[ \mbox{log1p}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \ln(1+x)& \mbox{if } x\geq -1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1p}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < -1\\ \frac{1}{1+x} & \mbox{if } x\geq -1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
xSpecified value.
Returns
Natural log of one plus x.

Definition at line 39 of file log1p.hpp.

template<typename T >
fvar<T> stan::math::log1p_exp ( const fvar< T > &  x)
inline

Definition at line 13 of file log1p_exp.hpp.

var stan::math::log1p_exp ( const stan::math::var a)
inline

Return the log of 1 plus the exponential of the specified variable.

Definition at line 28 of file log1p_exp.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log1p_exp ( const T  a)
inline

Calculates the log of 1 plus the exponential of the specified value without overflow.

This function is related to other special functions by:

log1p_exp(x)

= log1p(exp(a))

= log(1 + exp(x))

= log_sum_exp(0, x).

\[ \mbox{log1p\_exp}(x) = \begin{cases} \ln(1+\exp(x)) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log1p\_exp}(x)}{\partial x} = \begin{cases} \frac{\exp(x)}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Definition at line 44 of file log1p_exp.hpp.

template<typename T >
fvar<T> stan::math::log2 ( const fvar< T > &  x)
inline

Definition at line 17 of file log2.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log2 ( const T  a)
inline

Returns the base 2 logarithm of the argument (C99).

The function is defined by:

log2(a) = log(a) / std::log(2.0).

Template Parameters
Ttype of scalar
Parameters
aValue.
Returns
Base 2 logarithm of the value.

Definition at line 25 of file log2.hpp.

double stan::math::log2 ( )
inline

Return natural logarithm of two.

Returns
Natural logarithm of two.

Definition at line 35 of file log2.hpp.

var stan::math::log2 ( const stan::math::var a)
inline

Returns the base 2 logarithm of the specified variable (C99).

See stan::math::log2() for the double-based version.

The derivative is

$\frac{d}{dx} \log_2 x = \frac{1}{x \log 2}$.

\[ \mbox{log2}(x) = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \log_2(x) & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log2}(x)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x < 0 \\ \frac{1}{x\ln2} & \mbox{if } x\geq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aSpecified variable.
Returns
Base 2 logarithm of the variable.

Definition at line 53 of file log2.hpp.

template<int R, int C>
var stan::math::log_determinant ( const Eigen::Matrix< var, R, C > &  m)
inline

Definition at line 13 of file log_determinant.hpp.

template<typename T , int R, int C>
T stan::math::log_determinant ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the log absolute determinant of the specified square matrix.

Parameters
mSpecified matrix.
Returns
log absolute determinant of the matrix.
Exceptions
std::domain_errorif matrix is not square.

Definition at line 18 of file log_determinant.hpp.

template<typename T , int R, int C>
fvar<T> stan::math::log_determinant ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 20 of file log_determinant.hpp.

template<int R, int C, typename T >
T stan::math::log_determinant_ldlt ( stan::math::LDLT_factor< T, R, C > &  A)
inline

Definition at line 12 of file log_determinant_ldlt.hpp.

template<int R, int C>
var stan::math::log_determinant_ldlt ( stan::math::LDLT_factor< var, R, C > &  A)

Definition at line 48 of file log_determinant_ldlt.hpp.

template<int R, int C>
var stan::math::log_determinant_spd ( const Eigen::Matrix< var, R, C > &  m)
inline

Definition at line 15 of file log_determinant_spd.hpp.

template<typename T , int R, int C>
T stan::math::log_determinant_spd ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the log absolute determinant of the specified square matrix.

Parameters
mSpecified matrix.
Returns
log absolute determinant of the matrix.
Exceptions
std::domain_errorif matrix is not square.

Definition at line 19 of file log_determinant_spd.hpp.

template<typename T >
fvar<T> stan::math::log_diff_exp ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 14 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >
fvar<T2> stan::math::log_diff_exp ( const T1 &  x1,
const fvar< T2 > &  x2 
)
inline

Definition at line 26 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >
fvar<T1> stan::math::log_diff_exp ( const fvar< T1 > &  x1,
const T2 &  x2 
)
inline

Definition at line 37 of file log_diff_exp.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::log_diff_exp ( const T1  x,
const T2  y 
)
inline

The natural logarithm of the difference of the natural exponentiation of x1 and the natural exponentiation of x2.

This function is only defined for x<0

\[ \mbox{log\_diff\_exp}(x, y) = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ \ln(\exp(x)-\exp(y)) & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial x} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ \frac{\exp(x)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_diff\_exp}(x, y)}{\partial y} = \begin{cases} \textrm{NaN} & \mbox{if } x \leq y\\ -\frac{\exp(y)}{\exp(x)-\exp(y)} & \mbox{if } x > y \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Definition at line 50 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const stan::math::var a,
const stan::math::var b 
)
inline

Returns the log sum of exponentials.

Definition at line 54 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const stan::math::var a,
const double &  b 
)
inline

Returns the log sum of exponentials.

Definition at line 61 of file log_diff_exp.hpp.

var stan::math::log_diff_exp ( const double &  a,
const stan::math::var b 
)
inline

Returns the log sum of exponentials.

Definition at line 68 of file log_diff_exp.hpp.

template<typename T >
fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x,
const fvar< T > &  n 
)
inline

Definition at line 15 of file log_falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::log_falling_factorial ( const double  x,
const fvar< T > &  n 
)
inline

Definition at line 25 of file log_falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::log_falling_factorial ( const fvar< T > &  x,
const double  n 
)
inline

Definition at line 35 of file log_falling_factorial.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::log_falling_factorial ( const T1  x,
const T2  n 
)
inline

\[ \mbox{log\_falling\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln (x)_n & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_falling\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ -\Psi(n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Definition at line 41 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var a,
const double &  b 
)
inline

Definition at line 62 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const var a,
const var b 
)
inline

Definition at line 67 of file log_falling_factorial.hpp.

var stan::math::log_falling_factorial ( const double &  a,
const var b 
)
inline

Definition at line 72 of file log_falling_factorial.hpp.

template<typename T >
fvar<T> stan::math::log_inv_logit ( const fvar< T > &  x)
inline

Definition at line 15 of file log_inv_logit.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::log_inv_logit ( const T &  u)
inline

Returns the natural logarithm of the inverse logit of the specified argument.

\[ \mbox{log\_inv\_logit}(x) = \begin{cases} \ln\left(\frac{1}{1+\exp(-x)}\right)& \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_inv\_logit}(x)}{\partial x} = \begin{cases} \frac{1}{1+\exp(x)} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Template Parameters
TScalar type
Parameters
uInput.
Returns
log of the inverse logit of the input.

Definition at line 36 of file log_inv_logit.hpp.

template<typename T >
T stan::math::log_inv_logit_diff ( const T &  alpha,
const T &  beta 
)
inline

Definition at line 26 of file ordered_logistic_log.hpp.

double stan::math::log_mix ( double  theta,
double  lambda1,
double  lambda2 
)

Return the log mixture density with specified mixing proportion and log densities.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \lambda_1 + (1 - \theta) \lambda_2 \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = FIXME \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = FIXME \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = FIXME \]

Parameters
[in]thetamixing proportion in [0, 1].
lambda1first log density.
lambda2second log density.
Returns
log mixture of densities in specified proportion

Definition at line 46 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 >
return_type<T_theta, T_lambda1, T_lambda2>::type stan::math::log_mix ( const T_theta &  theta,
const T_lambda1 &  lambda1,
const T_lambda2 &  lambda2 
)
inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters
T_thetatheta scalar type.
T_lambda1lambda1 scalar type.
T_lambda2lambda2 scalar type.
Parameters
[in]thetamixing proportion in [0, 1].
[in]lambda1first log density.
[in]lambda2second log density.
Returns
log mixture of densities in specified proportion

Definition at line 88 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const fvar< T > &  theta,
const fvar< T > &  lambda1,
const fvar< T > &  lambda2 
)
inline

Return the log mixture density with specified mixing proportion and log densities and its derivative at each.

\[ \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \log \left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right). \]

\[ \frac{\partial}{\partial \theta} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\exp(\lambda_1) - \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_1} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_1)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

\[ \frac{\partial}{\partial \lambda_2} \mbox{log\_mix}(\theta, \lambda_1, \lambda_2) = \dfrac{\theta \exp(\lambda_2)} {\left( \theta \exp(\lambda_1) + (1 - \theta) \exp(\lambda_2) \right)} \]

Template Parameters
Tscalar type.
Parameters
[in]thetamixing proportion in [0, 1].
[in]lambda1first log density.
[in]lambda2second log density.
Returns
log mixture of densities in specified proportion

Definition at line 117 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const fvar< T > &  theta,
const fvar< T > &  lambda1,
const double  lambda2 
)
inline

Definition at line 143 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const fvar< T > &  theta,
const double  lambda1,
const fvar< T > &  lambda2 
)
inline

Definition at line 168 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const double  theta,
const fvar< T > &  lambda1,
const fvar< T > &  lambda2 
)
inline

Definition at line 193 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const fvar< T > &  theta,
const double  lambda1,
const double  lambda2 
)
inline

Definition at line 217 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const double  theta,
const fvar< T > &  lambda1,
const double  lambda2 
)
inline

Definition at line 238 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_mix ( const double  theta,
const double  lambda1,
const fvar< T > &  lambda2 
)
inline

Definition at line 259 of file log_mix.hpp.

void stan::math::log_mix_partial_helper ( const double &  theta_val,
const double &  lambda1_val,
const double &  lambda2_val,
double &  one_m_exp_lam2_m_lam1,
double &  one_m_t_prod_exp_lam2_m_lam1,
double &  one_d_t_plus_one_m_t_prod_exp_lam2_m_lam1 
)
inline

Definition at line 28 of file log_mix.hpp.

template<typename T_theta , typename T_lambda1 , typename T_lambda2 , int N>
void stan::math::log_mix_partial_helper ( const T_theta &  theta,
const T_lambda1 &  lambda1,
const T_lambda2 &  lambda2,
typename boost::math::tools::promote_args< T_theta, T_lambda1, T_lambda2 >::type(&)  partials_array[N] 
)
inline

Definition at line 29 of file log_mix.hpp.

template<typename T >
fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x,
const fvar< T > &  n 
)
inline

Definition at line 16 of file log_rising_factorial.hpp.

template<typename T >
fvar<T> stan::math::log_rising_factorial ( const fvar< T > &  x,
const double  n 
)
inline

Definition at line 28 of file log_rising_factorial.hpp.

template<typename T >
fvar<T> stan::math::log_rising_factorial ( const double  x,
const fvar< T > &  n 
)
inline

Definition at line 39 of file log_rising_factorial.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::log_rising_factorial ( const T1  x,
const T2  n 
)
inline

\[ \mbox{log\_rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \ln x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) - \Psi(x) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \Psi(x+n) & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

Definition at line 41 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var a,
const double &  b 
)
inline

Definition at line 49 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const var a,
const var b 
)
inline

Definition at line 54 of file log_rising_factorial.hpp.

var stan::math::log_rising_factorial ( const double &  a,
const var b 
)
inline

Definition at line 59 of file log_rising_factorial.hpp.

template<typename T >
Eigen::Matrix<fvar<T>, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &  alpha)
inline

Definition at line 16 of file log_softmax.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v)
inline

Return the natural logarithm of the softmax of the specified vector.

$ \log \mbox{softmax}(y) \ = \ y - \log \sum_{k=1}^K \exp(y_k) \ = \ y - \mbox{log\_sum\_exp}(y). $

For the log softmax function, the entries in the Jacobian are $ \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] = \left\{ \begin{array}{ll} 1 - \mbox{softmax}(y)[m] & \mbox{ if } m = k, \mbox{ and} \\[6pt] \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. $

Template Parameters
TScalar type of values in vector.
Parameters
[in]vVector to transform.
Returns
Unit simplex result of the softmax transform of the vector.

Definition at line 44 of file log_softmax.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::log_softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  alpha)
inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

The gradient calculations are unfolded.

Parameters
alphaUnconstrained input vector.
Returns
Softmax of the input.
Exceptions
std::domain_errorIf the input vector is size 0.

Definition at line 61 of file log_softmax.hpp.

template<typename T >
fvar<T> stan::math::log_sum_exp ( const std::vector< fvar< T > > &  v)

Definition at line 14 of file log_sum_exp.hpp.

template<typename T >
fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 15 of file log_sum_exp.hpp.

template<typename T , int R, int C>
fvar<T> stan::math::log_sum_exp ( const Eigen::Matrix< fvar< T >, R, C > &  v)

Definition at line 19 of file log_sum_exp.hpp.

double stan::math::log_sum_exp ( const std::vector< double > &  x)

Return the log of the sum of the exponentiated values of the specified sequence of values.

The function is defined as follows to prevent overflow in exponential calculations.

$\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))$.

Parameters
[in]xarray of specified values
Returns
The log of the sum of the exponentiated vector values.

Definition at line 24 of file log_sum_exp.hpp.

template<typename T >
fvar<T> stan::math::log_sum_exp ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 26 of file log_sum_exp.hpp.

template<int R, int C>
double stan::math::log_sum_exp ( const Eigen::Matrix< double, R, C > &  x)

Return the log of the sum of the exponentiated values of the specified matrix of values.

The matrix may be a full matrix, a vector, or a row vector.

The function is defined as follows to prevent overflow in exponential calculations.

$\log \sum_{n=1}^N \exp(x_n) = \max(x) + \log \sum_{n=1}^N \exp(x_n - \max(x))$.

Parameters
[in]xMatrix of specified values
Returns
The log of the sum of the exponentiated vector values.

Definition at line 28 of file log_sum_exp.hpp.

template<typename T >
fvar<T> stan::math::log_sum_exp ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 36 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const std::vector< var > &  x)
inline

Returns the log sum of exponentials.

Definition at line 45 of file log_sum_exp.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::log_sum_exp ( const T2 &  a,
const T1 &  b 
)
inline

Calculates the log sum of exponetials without overflow.

$\log (\exp(a) + \exp(b)) = m + \log(\exp(a-m) + \exp(b-m))$,

where $m = max(a, b)$.

\[ \mbox{log\_sum\_exp}(x, y) = \begin{cases} \ln(\exp(x)+\exp(y)) & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial x} = \begin{cases} \frac{\exp(x)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{log\_sum\_exp}(x, y)}{\partial y} = \begin{cases} \frac{\exp(y)}{\exp(x)+\exp(y)} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
athe first variable
bthe second variable

Definition at line 48 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const stan::math::var a,
const stan::math::var b 
)
inline

Returns the log sum of exponentials.

Definition at line 50 of file log_sum_exp.hpp.

template<int R, int C>
var stan::math::log_sum_exp ( const Eigen::Matrix< var, R, C > &  x)
inline

Returns the log sum of exponentials.

Parameters
xmatrix

Definition at line 54 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const stan::math::var a,
const double &  b 
)
inline

Returns the log sum of exponentials.

Definition at line 57 of file log_sum_exp.hpp.

var stan::math::log_sum_exp ( const double &  a,
const stan::math::var b 
)
inline

Returns the log sum of exponentials.

Definition at line 64 of file log_sum_exp.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_and ( const T1  x1,
const T2  x2 
)
inline

The logical and function which returns 1 if both arguments are unequal to zero and 0 otherwise.

Equivalent to x1 != 0 && x2 != 0.

\[ \mbox{operator\&\&}(x, y) = \begin{cases} 0 & \mbox{if } x = 0 \textrm{ or } y=0 \\ 1 & \mbox{if } x, y \neq 0 \\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true if both x1 and x2 are not equal to 0.

Definition at line 30 of file logical_and.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_eq ( const T1  x1,
const T2  x2 
)
inline

Return 1 if the first argument is equal to the second.

Equivalent to x1 == x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 == x2

Definition at line 19 of file logical_eq.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_gt ( const T1  x1,
const T2  x2 
)
inline

Return 1 if the first argument is strictly greater than the second.

Equivalent to x1 < x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 > x2

Definition at line 19 of file logical_gt.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_gte ( const T1  x1,
const T2  x2 
)
inline

Return 1 if the first argument is greater than or equal to the second.

Equivalent to x1 >= x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 >= x2

Definition at line 19 of file logical_gte.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_lt ( T1  x1,
T2  x2 
)
inline

Return 1 if the first argument is strictly less than the second.

Equivalent to x1 < x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 < x2

Definition at line 20 of file logical_lt.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_lte ( const T1  x1,
const T2  x2 
)
inline

Return 1 if the first argument is less than or equal to the second.

Equivalent to x1 <= x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 <= x2

Definition at line 19 of file logical_lte.hpp.

template<typename T >
int stan::math::logical_negation ( const T  x)
inline

The logical negation function which returns 1 if the input is equal to zero and 0 otherwise.

Template Parameters
TType to compare to zero.
Parameters
xValue to compare to zero.
Returns
1 if input is equal to zero.

Definition at line 17 of file logical_negation.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_neq ( const T1  x1,
const T2  x2 
)
inline

Return 1 if the first argument is unequal to the second.

Equivalent to x1 != x2.

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true iff x1 != x2

Definition at line 19 of file logical_neq.hpp.

template<typename T1 , typename T2 >
int stan::math::logical_or ( T1  x1,
T2  x2 
)
inline

The logical or function which returns 1 if either argument is unequal to zero and 0 otherwise.

Equivalent to x1 != 0 || x2 != 0.

\[ \mbox{operator||}(x, y) = \begin{cases} 0 & \mbox{if } x, y=0 \\ 1 & \mbox{if } x \neq 0 \textrm{ or } y\neq0\\[6pt] 1 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
x1First argument
x2Second argument
Returns
true if either x1 or x2 is not equal to 0.

Definition at line 29 of file logical_or.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::logistic_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 31 of file logistic_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::logistic_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 31 of file logistic_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::logistic_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 30 of file logistic_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::logistic_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 32 of file logistic_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::logistic_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)
inline

Definition at line 142 of file logistic_log.hpp.

template<class RNG >
double stan::math::logistic_rng ( const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 27 of file logistic_rng.hpp.

template<typename T >
fvar<T> stan::math::logit ( const fvar< T > &  x)
inline

Definition at line 17 of file logit.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::logit ( const T  a)
inline

Returns the logit function applied to the argument.

The logit function is defined as for $x \in [0, 1]$ by returning the log odds of $x$ treated as a probability,

$\mbox{logit}(x) = \log \left( \frac{x}{1 - x} \right)$.

The inverse to this function is stan::math::inv_logit.

\[ \mbox{logit}(x) = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \ln\frac{x}{1-x} & \mbox{if } 0\leq x \leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{logit}(x)}{\partial x} = \begin{cases} \textrm{NaN}& \mbox{if } x < 0 \textrm{ or } x > 1\\ \frac{1}{x-x^2}& \mbox{if } 0\leq x\leq 1 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aArgument.
Returns
Logit of the argument.

Definition at line 44 of file logit.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::lognormal_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 23 of file lognormal_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::lognormal_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 23 of file lognormal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::lognormal_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 23 of file lognormal_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::lognormal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 33 of file lognormal_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::lognormal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)
inline

Definition at line 158 of file lognormal_log.hpp.

template<class RNG >
double stan::math::lognormal_rng ( const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 22 of file lognormal_rng.hpp.

template<typename T , typename TL , typename TU >
boost::math::tools::promote_args<T, TL, TU>::type stan::math::lub_constrain ( const T  x,
TL  lb,
TU  ub 
)
inline

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds.

The transform is the transformed and scaled inverse logit,

$f(x) = L + (U - L) \mbox{logit}^{-1}(x)$

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_constrain(x, ub). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_constrain(x, lb). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_constrain(x).

Parameters
xFree scalar to transform.
lbLower bound.
ubUpper bound.
Returns
Lower- and upper-bounded scalar derived from transforming the free scalar.
Template Parameters
TType of scalar.
TLType of lower bound.
TUType of upper bound.
Exceptions
std::domain_errorif ub <= lb

Definition at line 44 of file lub_constrain.hpp.

template<typename T , typename TL , typename TU >
boost::math::tools::promote_args<T, TL, TU>::type stan::math::lub_constrain ( const T  x,
const TL  lb,
const TU  ub,
T &  lp 
)

Return the lower- and upper-bounded scalar derived by transforming the specified free scalar given the specified lower and upper bounds and increment the specified log probability with the log absolute Jacobian determinant.

The transform is as defined in lub_constrain(T, double, double). The log absolute Jacobian determinant is given by

$\log \left| \frac{d}{dx} \left( L + (U-L) \mbox{logit}^{-1}(x) \right) \right|$

$ {} = \log | (U-L) \, (\mbox{logit}^{-1}(x)) \, (1 - \mbox{logit}^{-1}(x)) |$

$ {} = \log (U - L) + \log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))$

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_constrain(x, ub, lp). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_constrain(x, lb, lp). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_constrain(x, lp).

Parameters
xFree scalar to transform.
lbLower bound.
ubUpper bound.
lpLog probability scalar reference.
Returns
Lower- and upper-bounded scalar derived from transforming the free scalar.
Template Parameters
TType of scalar.
TLType of lower bound.
TUType of upper bound.
Exceptions
std::domain_errorif ub <= lb

Definition at line 114 of file lub_constrain.hpp.

template<typename T , typename TL , typename TU >
boost::math::tools::promote_args<T, TL, TU>::type stan::math::lub_free ( const T  y,
TL  lb,
TU  ub 
)
inline

Return the unconstrained scalar that transforms to the specified lower- and upper-bounded scalar given the specified bounds.

The transfrom in lub_constrain(T, double, double), is reversed by a transformed and scaled logit,

$f^{-1}(y) = \mbox{logit}(\frac{y - L}{U - L})$

where $U$ and $L$ are the lower and upper bounds.

If the lower bound is negative infinity and upper bound finite, this function reduces to ub_free(y, ub). If the upper bound is positive infinity and the lower bound finite, this function reduces to lb_free(x, lb). If the upper bound is positive infinity and the lower bound negative infinity, this function reduces to identity_free(y).

Template Parameters
TType of scalar.
Parameters
yScalar input.
lbLower bound.
ubUpper bound.
Returns
The free scalar that transforms to the input scalar given the bounds.
Exceptions
std::invalid_argumentif the lower bound is greater than the upper bound, y is less than the lower bound, or y is greater than the upper bound

Definition at line 47 of file lub_free.hpp.

double stan::math::machine_precision ( )
inline

Returns the difference between 1.0 and the next value representable.

Returns
Minimum positive number.

Definition at line 151 of file constants.hpp.

template<typename T >
const Eigen::Array<T, Eigen::Dynamic, 1> stan::math::make_nu ( const T  eta,
const size_t  K 
)

This function calculates the degrees of freedom for the t distribution that corresponds to the shape parameter in the Lewandowski et.

al. distribution

Parameters
etahyperparameter on (0, inf), eta = 1 <-> correlation matrix is uniform
Knumber of variables in covariance matrix

Definition at line 22 of file make_nu.hpp.

template<bool propto, typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
boost::math::tools::promote_args<T_y, T_Mu, T_Sigma, T_D>::type stan::math::matrix_normal_prec_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &  Mu,
const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &  Sigma,
const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &  D 
)

The log of the matrix normal density for the given y, mu, Sigma and D where Sigma and D are given as precision matrices, not covariance matrices.

Parameters
yAn mxn matrix.
MuThe mean matrix.
SigmaThe mxm inverse covariance matrix (i.e., the precision matrix) of the rows of y.
DThe nxn inverse covariance matrix (i.e., the precision matrix) of the columns of y.
Returns
The log of the matrix normal density.
Exceptions
std::domain_errorif Sigma or D are not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_MuType of location.
T_SigmaType of Sigma.
T_DType of D.

Definition at line 43 of file matrix_normal_prec_log.hpp.

template<typename T_y , typename T_Mu , typename T_Sigma , typename T_D >
boost::math::tools::promote_args<T_y, T_Mu, T_Sigma, T_D>::type stan::math::matrix_normal_prec_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_Mu, Eigen::Dynamic, Eigen::Dynamic > &  Mu,
const Eigen::Matrix< T_Sigma, Eigen::Dynamic, Eigen::Dynamic > &  Sigma,
const Eigen::Matrix< T_D, Eigen::Dynamic, Eigen::Dynamic > &  D 
)

Definition at line 112 of file matrix_normal_prec_log.hpp.

int stan::math::max ( const std::vector< int > &  x)
inline

Returns the maximum coefficient in the specified column vector.

Parameters
xSpecified vector.
Returns
Maximum coefficient value in the vector.
Template Parameters
Typeof values being compared and returned
Exceptions
std::domain_errorIf the size of the vector is zero.

Definition at line 21 of file max.hpp.

template<typename T >
T stan::math::max ( const std::vector< T > &  x)
inline

Returns the maximum coefficient in the specified column vector.

Parameters
xSpecified vector.
Returns
Maximum coefficient value in the vector.
Template Parameters
TType of values being compared and returned

Definition at line 39 of file max.hpp.

template<typename T , int R, int C>
T stan::math::max ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the maximum coefficient in the specified vector, row vector, or matrix.

Parameters
mSpecified vector, row vector, or matrix.
Returns
Maximum coefficient value in the vector.

Definition at line 56 of file max.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 24 of file mdivide_left.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b.

Parameters
AMatrix.
bRight hand side matrix or vector.
Returns
x = A^-1 b, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 25 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 68 of file mdivide_left.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 94 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 274 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 301 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 328 of file mdivide_left.hpp.

template<int R1, int C1, int R2, int C2, typename T2 >
Eigen::Matrix<fvar<T2>, R1, C2> stan::math::mdivide_left_ldlt ( const stan::math::LDLT_factor< double, R1, C1 > &  A,
const Eigen::Matrix< fvar< T2 >, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 25 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_left_ldlt ( const stan::math::LDLT_factor< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 26 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const stan::math::LDLT_factor< var, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 246 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const stan::math::LDLT_factor< var, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 274 of file mdivide_left_ldlt.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_ldlt ( const stan::math::LDLT_factor< double, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 302 of file mdivide_left_ldlt.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b where A is symmetric positive definite.

Parameters
AMatrix.
bRight hand side matrix or vector.
Returns
x = A^-1 b, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 28 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 248 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 275 of file mdivide_left_spd.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_spd ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 302 of file mdivide_left_spd.hpp.

template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  b 
)
inline

Returns the solution of the system Ax=b when A is triangular.

Parameters
ATriangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
bRight hand side matrix or vector.
Returns
x = A^-1 b, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 27 of file mdivide_left_tri.hpp.

template<int TriView, typename T , int R1, int C1>
Eigen::Matrix<T, R1, C1> stan::math::mdivide_left_tri ( const Eigen::Matrix< T, R1, C1 > &  A)
inline

Returns the solution of the system Ax=b when A is triangular and b=I.

Parameters
ATriangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
Returns
x = A^-1 .
Exceptions
std::domain_errorif A is not square

Definition at line 50 of file mdivide_left_tri.hpp.

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 304 of file mdivide_left_tri.hpp.

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< var, R2, C2 > &  b 
)
inline

Definition at line 330 of file mdivide_left_tri.hpp.

template<int TriView, int R1, int C1, int R2, int C2>
Eigen::Matrix<var, R1, C2> stan::math::mdivide_left_tri ( const Eigen::Matrix< var, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 356 of file mdivide_left_tri.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< T1, R1, C1 > &  A,
const Eigen::Matrix< T2, R2, C2 > &  b 
)
inline

Definition at line 16 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 22 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<T, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< T, R1, C1 > &  A)
inline

Definition at line 32 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 68 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_left_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 109 of file mdivide_left_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 24 of file mdivide_right.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< T1, R1, C1 > &  b,
const Eigen::Matrix< T2, R2, C2 > &  A 
)
inline

Returns the solution of the system Ax=b.

Parameters
AMatrix.
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 26 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 68 of file mdivide_right.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 95 of file mdivide_right.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_right_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  b,
const stan::math::LDLT_factor< T2, R2, C2 > &  A 
)
inline

Returns the solution of the system xA=b given an LDLT_factor of A.

Parameters
ALDLT_factor
bRight hand side matrix or vector.
Returns
x = A^-1 b, solution of the linear system.
Exceptions
std::domain_errorif rows of b don't match the size of A.

Definition at line 26 of file mdivide_right_ldlt.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<double, R1, C2> stan::math::mdivide_right_ldlt ( const Eigen::Matrix< double, R1, C1 > &  b,
const stan::math::LDLT_factor< double, R2, C2 > &  A 
)
inline

Definition at line 38 of file mdivide_right_ldlt.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_right_spd ( const Eigen::Matrix< T1, R1, C1 > &  b,
const Eigen::Matrix< T2, R2, C2 > &  A 
)
inline

Returns the solution of the system Ax=b where A is symmetric positive definite.

Parameters
AMatrix.
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 29 of file mdivide_right_spd.hpp.

template<int TriView, typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_right_tri ( const Eigen::Matrix< T1, R1, C1 > &  b,
const Eigen::Matrix< T2, R2, C2 > &  A 
)
inline

Returns the solution of the system Ax=b when A is triangular.

Parameters
ATriangular matrix. Specify upper or lower with TriView being Eigen::Upper or Eigen::Lower.
bRight hand side matrix or vector.
Returns
x = b A^-1, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 29 of file mdivide_right_tri.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 22 of file mdivide_right_tri_low.hpp.

template<typename T1 , typename T2 , int R1, int C1, int R2, int C2>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< T1, R1, C1 > &  b,
const Eigen::Matrix< T2, R2, C2 > &  A 
)
inline

Returns the solution of the system tri(A)x=b when tri(A) is a lower triangular view of the matrix A.

Parameters
AMatrix.
bRight hand side matrix or vector.
Returns
x = b * tri(A)^-1, solution of the linear system.
Exceptions
std::domain_errorif A is not square or the rows of b don't match the size of A.

Definition at line 25 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< fvar< T >, R1, C1 > &  A,
const Eigen::Matrix< double, R2, C2 > &  b 
)
inline

Definition at line 68 of file mdivide_right_tri_low.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::mdivide_right_tri_low ( const Eigen::Matrix< double, R1, C1 > &  A,
const Eigen::Matrix< fvar< T >, R2, C2 > &  b 
)
inline

Definition at line 103 of file mdivide_right_tri_low.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::mean ( const std::vector< T > &  v)
inline

Returns the sample mean (i.e., average) of the coefficients in the specified standard vector.

Parameters
vSpecified vector.
Returns
Sample mean of vector coefficients.
Exceptions
std::domain_errorif the size of the vector is less than 1.

Definition at line 23 of file mean.hpp.

template<typename T , int R, int C>
boost::math::tools::promote_args<T>::type stan::math::mean ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the sample mean (i.e., average) of the coefficients in the specified vector, row vector, or matrix.

Parameters
mSpecified vector, row vector, or matrix.
Returns
Sample mean of vector coefficients.

Definition at line 40 of file mean.hpp.

int stan::math::min ( const std::vector< int > &  x)
inline

Returns the minimum coefficient in the specified column vector.

Parameters
xSpecified vector.
Returns
Minimum coefficient value in the vector.
Template Parameters
Typeof values being compared and returned

Definition at line 20 of file min.hpp.

template<typename T >
T stan::math::min ( const std::vector< T > &  x)
inline

Returns the minimum coefficient in the specified column vector.

Parameters
xSpecified vector.
Returns
Minimum coefficient value in the vector.
Template Parameters
Typeof values being compared and returned

Definition at line 38 of file min.hpp.

template<typename T , int R, int C>
T stan::math::min ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the minimum coefficient in the specified matrix, vector, or row vector.

Parameters
mSpecified matrix, vector, or row vector.
Returns
Minimum coefficient value in the vector.

Definition at line 55 of file min.hpp.

template<typename T >
T stan::math::minus ( const T &  x)
inline

Returns the negation of the specified scalar or matrix.

Template Parameters
TType of subtrahend.
Parameters
xSubtrahend.
Returns
Negation of subtrahend.

Definition at line 16 of file minus.hpp.

template<typename T >
fvar<T> stan::math::modified_bessel_first_kind ( int  v,
const fvar< T > &  z 
)
inline

Definition at line 15 of file modified_bessel_first_kind.hpp.

var stan::math::modified_bessel_first_kind ( const int &  v,
const var a 
)
inline

Definition at line 27 of file modified_bessel_first_kind.hpp.

template<typename T2 >
T2 stan::math::modified_bessel_first_kind ( const int  v,
const T2  z 
)
inline

\[ \mbox{modified\_bessel\_first\_kind}(v, z) = \begin{cases} I_v(z) & \mbox{if } -\infty\leq z \leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_first\_kind}(v, z)}{\partial z} = \begin{cases} \frac{\partial\, I_v(z)}{\partial z} & \mbox{if } -\infty\leq z\leq \infty \\[6pt] \textrm{error} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {I_v}(z) = \left(\frac{1}{2}z\right)^v\sum_{k=0}^\infty \frac{\left(\frac{1}{4}z^2\right)^k}{k!\Gamma(v+k+1)} \]

\[ \frac{\partial \, I_v(z)}{\partial z} = I_{v-1}(z)-\frac{v}{z}I_v(z) \]

Definition at line 39 of file modified_bessel_first_kind.hpp.

template<typename T >
fvar<T> stan::math::modified_bessel_second_kind ( int  v,
const fvar< T > &  z 
)
inline

Definition at line 15 of file modified_bessel_second_kind.hpp.

var stan::math::modified_bessel_second_kind ( const int &  v,
const var a 
)
inline

Definition at line 27 of file modified_bessel_second_kind.hpp.

template<typename T2 >
T2 stan::math::modified_bessel_second_kind ( const int  v,
const T2  z 
)
inline

\[ \mbox{modified\_bessel\_second\_kind}(v, z) = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ K_v(z) & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{modified\_bessel\_second\_kind}(v, z)}{\partial z} = \begin{cases} \textrm{error} & \mbox{if } z \leq 0 \\ \frac{\partial\, K_v(z)}{\partial z} & \mbox{if } z > 0 \\[6pt] \textrm{NaN} & \mbox{if } z = \textrm{NaN} \end{cases} \]

\[ {K_v}(z) = \frac{\pi}{2}\cdot\frac{I_{-v}(z) - I_{v}(z)}{\sin(v\pi)} \]

\[ \frac{\partial \, K_v(z)}{\partial z} = -\frac{v}{z}K_v(z)-K_{v-1}(z) \]

Definition at line 42 of file modified_bessel_second_kind.hpp.

int stan::math::modulus ( const int  x,
const int  y 
)
inline

Definition at line 10 of file modulus.hpp.

template<bool propto, typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args<T_y, T_covar, T_w>::type stan::math::multi_gp_cholesky_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L,
const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w 
)

The log of a multivariate Gaussian Process for the given y, w, and a Cholesky factor L of the kernel matrix Sigma.

Sigma = LL', a square, semi-positive definite matrix.. y is a dxN matrix, where each column is a different observation and each row is a different output dimension. The Gaussian Process is assumed to have a scaled kernel matrix with a different scale for each output dimension. This distribution is equivalent to: for (i in 1:d) row(y, i) ~ multi_normal(0, (1/w[i])*LL').

Parameters
yA dxN matrix
LThe Cholesky decomposition of a kernel matrix
wA d-dimensional vector of positve inverse scale parameters for each output.
Returns
The log of the multivariate GP density.
Exceptions
std::domain_errorif Sigma is not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_covarType of kernel.
T_wType of weight.

Definition at line 43 of file multi_gp_cholesky_log.hpp.

template<typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args<T_y, T_covar, T_w>::type stan::math::multi_gp_cholesky_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  L,
const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w 
)
inline

Definition at line 106 of file multi_gp_cholesky_log.hpp.

template<bool propto, typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args<T_y, T_covar, T_w>::type stan::math::multi_gp_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  Sigma,
const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w 
)

The log of a multivariate Gaussian Process for the given y, Sigma, and w.

y is a dxN matrix, where each column is a different observation and each row is a different output dimension. The Gaussian Process is assumed to have a scaled kernel matrix with a different scale for each output dimension. This distribution is equivalent to: for (i in 1:d) row(y, i) ~ multi_normal(0, (1/w[i])*Sigma).

Parameters
yA dxN matrix
SigmaThe NxN kernel matrix
wA d-dimensional vector of positve inverse scale parameters for each output.
Returns
The log of the multivariate GP density.
Exceptions
std::domain_errorif Sigma is not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_covarType of kernel.
T_wType of weight.

Definition at line 45 of file multi_gp_log.hpp.

template<typename T_y , typename T_covar , typename T_w >
boost::math::tools::promote_args<T_y, T_covar, T_w>::type stan::math::multi_gp_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  y,
const Eigen::Matrix< T_covar, Eigen::Dynamic, Eigen::Dynamic > &  Sigma,
const Eigen::Matrix< T_w, Eigen::Dynamic, 1 > &  w 
)
inline

Definition at line 112 of file multi_gp_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_cholesky_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  L 
)

The log of the multivariate normal density for the given y, mu, and a Cholesky factor L of the variance matrix.

Sigma = LL', a square, semi-positive definite matrix.

Parameters
yA scalar vector
muThe mean vector of the multivariate normal distribution.
LThe Cholesky decomposition of a variance matrix of the multivariate normal distribution
Returns
The log of the multivariate normal density.
Exceptions
std::domain_errorif LL' is not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_locType of location.
T_covarType of scale.

Definition at line 48 of file multi_normal_cholesky_log.hpp.

template<typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_cholesky_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  L 
)
inline

Definition at line 154 of file multi_normal_cholesky_log.hpp.

template<class RNG >
Eigen::VectorXd stan::math::multi_normal_cholesky_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  mu,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  S,
RNG &  rng 
)
inline

Definition at line 29 of file multi_normal_cholesky_rng.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  Sigma 
)

Definition at line 27 of file multi_normal_log.hpp.

template<typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  Sigma 
)
inline

Definition at line 128 of file multi_normal_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_prec_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  Sigma 
)

Definition at line 35 of file multi_normal_prec_log.hpp.

template<typename T_y , typename T_loc , typename T_covar >
return_type<T_y, T_loc, T_covar>::type stan::math::multi_normal_prec_log ( const T_y &  y,
const T_loc &  mu,
const T_covar &  Sigma 
)
inline

Definition at line 143 of file multi_normal_prec_log.hpp.

template<class RNG >
Eigen::VectorXd stan::math::multi_normal_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  mu,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  S,
RNG &  rng 
)
inline

Definition at line 24 of file multi_normal_rng.hpp.

template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::multi_student_t_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  Sigma 
)

Return the log of the multivariate Student t distribution at the specified arguments.

Template Parameters
proptoCarry out calculations up to a proportion

Definition at line 36 of file multi_student_t_log.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::multi_student_t_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  Sigma 
)
inline

Definition at line 171 of file multi_student_t_log.hpp.

template<class RNG >
Eigen::VectorXd stan::math::multi_student_t_rng ( const double  nu,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  mu,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  s,
RNG &  rng 
)
inline

Definition at line 29 of file multi_student_t_rng.hpp.

template<bool propto, typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::multinomial_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Definition at line 24 of file multinomial_log.hpp.

template<typename T_prob >
boost::math::tools::promote_args<T_prob>::type stan::math::multinomial_log ( const std::vector< int > &  ns,
const Eigen::Matrix< T_prob, Eigen::Dynamic, 1 > &  theta 
)

Definition at line 59 of file multinomial_log.hpp.

template<class RNG >
std::vector<int> stan::math::multinomial_rng ( const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  theta,
const int  N,
RNG &  rng 
)
inline

Definition at line 23 of file multinomial_rng.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m,
const fvar< T > &  c 
)
inline

Definition at line 20 of file multiply.hpp.

template<int R, int C, typename T >
boost::enable_if_c<boost::is_arithmetic<T>::value, Eigen::Matrix<double, R, C> >::type stan::math::multiply ( const Eigen::Matrix< double, R, C > &  m,
c 
)
inline

Return specified matrix multiplied by specified scalar.

Template Parameters
RRow type for matrix.
CColumn type for matrix.
Parameters
mMatrix.
cScalar.
Returns
Product of matrix and scalar.

Definition at line 25 of file multiply.hpp.

template<typename T1 , typename T2 >
boost::enable_if_c< (boost::is_scalar<T1>::value || boost::is_same<T1, var>::value) && (boost::is_scalar<T2>::value || boost::is_same<T2, var>::value), typename boost::math::tools::promote_args<T1, T2>::type>::type stan::math::multiply ( const T1 &  v,
const T2 &  c 
)
inline

Return the product of two scalars.

Parameters
[in]vFirst scalar.
[in]cSpecified scalar.
Returns
Product of scalars.

Definition at line 31 of file multiply.hpp.

template<typename T , int R2, int C2>
Eigen::Matrix<fvar<T>, R2, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R2, C2 > &  m,
const double  c 
)
inline

Definition at line 32 of file multiply.hpp.

template<typename T1 , typename T2 , int R2, int C2>
Eigen::Matrix<var, R2, C2> stan::math::multiply ( const T1 &  c,
const Eigen::Matrix< T2, R2, C2 > &  m 
)
inline

Return the product of scalar and matrix.

Parameters
[in]cSpecified scalar.
[in]mMatrix.
Returns
Product of scalar and matrix.

Definition at line 43 of file multiply.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const Eigen::Matrix< double, R1, C1 > &  m,
const fvar< T > &  c 
)
inline

Definition at line 44 of file multiply.hpp.

template<int R, int C, typename T >
boost::enable_if_c<boost::is_arithmetic<T>::value, Eigen::Matrix<double, R, C> >::type stan::math::multiply ( c,
const Eigen::Matrix< double, R, C > &  m 
)
inline

Return specified scalar multiplied by specified matrix.

Template Parameters
RRow type for matrix.
CColumn type for matrix.
Parameters
cScalar.
mMatrix.
Returns
Product of scalar and matrix.

Definition at line 45 of file multiply.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const fvar< T > &  c,
const Eigen::Matrix< fvar< T >, R1, C1 > &  m 
)
inline

Definition at line 56 of file multiply.hpp.

template<typename T1 , int R1, int C1, typename T2 >
Eigen::Matrix<var, R1, C1> stan::math::multiply ( const Eigen::Matrix< T1, R1, C1 > &  m,
const T2 &  c 
)
inline

Return the product of scalar and matrix.

Parameters
[in]mMatrix.
[in]cSpecified scalar.
Returns
Product of scalar and matrix.

Definition at line 57 of file multiply.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<double, R1, C2> stan::math::multiply ( const Eigen::Matrix< double, R1, C1 > &  m1,
const Eigen::Matrix< double, R2, C2 > &  m2 
)
inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Parameters
m1First matrix.
m2Second matrix.
Returns
The product of the first and second matrices.
Exceptions
std::domain_errorif the number of columns of m1 does not match the number of rows of m2.

Definition at line 62 of file multiply.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const double  c,
const Eigen::Matrix< fvar< T >, R1, C1 > &  m 
)
inline

Definition at line 63 of file multiply.hpp.

template<typename T , int R1, int C1>
Eigen::Matrix<fvar<T>, R1, C1> stan::math::multiply ( const fvar< T > &  c,
const Eigen::Matrix< double, R1, C1 > &  m 
)
inline

Definition at line 70 of file multiply.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c< boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, Eigen::Matrix<var, R1, C2> >::type stan::math::multiply ( const Eigen::Matrix< T1, R1, C1 > &  m1,
const Eigen::Matrix< T2, R2, C2 > &  m2 
)
inline

Return the product of the specified matrices.

The number of columns in the first matrix must be the same as the number of rows in the second matrix.

Parameters
[in]m1First matrix.
[in]m2Second matrix.
Returns
The product of the first and second matrices.
Exceptions
std::domain_errorif the number of columns of m1 does not match the number of rows of m2.

Definition at line 76 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  m2 
)
inline

Definition at line 77 of file multiply.hpp.

template<int C1, int R2>
double stan::math::multiply ( const Eigen::Matrix< double, 1, C1 > &  rv,
const Eigen::Matrix< double, R2, 1 > &  v 
)
inline

Return the scalar product of the specified row vector and specified column vector.

The return is the same as the dot product. The two vectors must be the same size.

Parameters
rvRow vector.
vColumn vector.
Returns
Scalar result of multiplying row vector by column vector.
Exceptions
std::domain_errorif rv and v are not the same size.

Definition at line 80 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< fvar< T >, R1, C1 > &  m1,
const Eigen::Matrix< double, R2, C2 > &  m2 
)
inline

Definition at line 96 of file multiply.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, C2> stan::math::multiply ( const Eigen::Matrix< double, R1, C1 > &  m1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  m2 
)
inline

Definition at line 115 of file multiply.hpp.

template<typename T1 , int C1, typename T2 , int R2>
boost::enable_if_c< boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, var >::type stan::math::multiply ( const Eigen::Matrix< T1, 1, C1 > &  rv,
const Eigen::Matrix< T2, R2, 1 > &  v 
)
inline

Return the scalar product of the specified row vector and specified column vector.

The return is the same as the dot product. The two vectors must be the same size.

Parameters
[in]rvRow vector.
[in]vColumn vector.
Returns
Scalar result of multiplying row vector by column vector.
Exceptions
std::domain_errorif rv and v are not the same size

Definition at line 128 of file multiply.hpp.

template<typename T , int C1, int R2>
fvar<T> stan::math::multiply ( const Eigen::Matrix< fvar< T >, 1, C1 > &  rv,
const Eigen::Matrix< fvar< T >, R2, 1 > &  v 
)
inline

Definition at line 134 of file multiply.hpp.

template<typename T , int C1, int R2>
fvar<T> stan::math::multiply ( const Eigen::Matrix< fvar< T >, 1, C1 > &  rv,
const Eigen::Matrix< double, R2, 1 > &  v 
)
inline

Definition at line 145 of file multiply.hpp.

template<typename T , int C1, int R2>
fvar<T> stan::math::multiply ( const Eigen::Matrix< double, 1, C1 > &  rv,
const Eigen::Matrix< fvar< T >, R2, 1 > &  v 
)
inline

Definition at line 156 of file multiply.hpp.

template<typename T >
fvar<T> stan::math::multiply_log ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 15 of file multiply_log.hpp.

template<typename T >
fvar<T> stan::math::multiply_log ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 25 of file multiply_log.hpp.

template<typename T >
fvar<T> stan::math::multiply_log ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 35 of file multiply_log.hpp.

template<typename T_a , typename T_b >
boost::math::tools::promote_args<T_a, T_b>::type stan::math::multiply_log ( const T_a  a,
const T_b  b 
)
inline

Calculated the value of the first argument times log of the second argument while behaving properly with 0 inputs.

$ a * \log b $.

\[ \mbox{multiply\_log}(x, y) = \begin{cases} 0 & \mbox{if } x=y=0\\ x\ln y & \mbox{if } x, y\neq0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial x} = \begin{cases} \infty & \mbox{if } x=y=0\\ \ln y & \mbox{if } x, y\neq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{multiply\_log}(x, y)}{\partial y} = \begin{cases} \infty & \mbox{if } x=y=0\\ \frac{x}{y} & \mbox{if } x, y\neq 0 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
athe first variable
bthe second variable
Returns
a * log(b)

Definition at line 51 of file multiply_log.hpp.

var stan::math::multiply_log ( const var a,
const var b 
)
inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b). The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters
aFirst variable.
bSecond variable.
Returns
Value of a*log(b)

Definition at line 74 of file multiply_log.hpp.

var stan::math::multiply_log ( const var a,
const double  b 
)
inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to a is log(b).

Parameters
aFirst variable.
bSecond scalar.
Returns
Value of a*log(b)

Definition at line 87 of file multiply_log.hpp.

var stan::math::multiply_log ( const double  a,
const var b 
)
inline

Return the value of a*log(b).

When both a and b are 0, the value returned is 0. The partial deriviative with respect to b is a/b. When a and b are both 0, this is set to Inf.

Parameters
aFirst scalar.
bSecond variable.
Returns
Value of a*log(b)

Definition at line 101 of file multiply_log.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, R> stan::math::multiply_lower_tri_self_transpose ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 17 of file multiply_lower_tri_self_transpose.hpp.

matrix_d stan::math::multiply_lower_tri_self_transpose ( const matrix_d L)
inline

Returns the result of multiplying the lower triangular portion of the input matrix by its own transpose.

Parameters
LMatrix to multiply.
Returns
The lower triangular values in L times their own transpose.
Exceptions
std::domain_errorIf the input matrix is not square.

Definition at line 18 of file multiply_lower_tri_self_transpose.hpp.

matrix_v stan::math::multiply_lower_tri_self_transpose ( const matrix_v L)
inline

Definition at line 19 of file multiply_lower_tri_self_transpose.hpp.

template<typename T_n , typename T_location , typename T_precision >
return_type<T_location, T_precision>::type stan::math::neg_binomial_2_ccdf_log ( const T_n &  n,
const T_location &  mu,
const T_precision &  phi 
)

Definition at line 33 of file neg_binomial_2_ccdf_log.hpp.

template<typename T_n , typename T_location , typename T_precision >
return_type<T_location, T_precision>::type stan::math::neg_binomial_2_cdf ( const T_n &  n,
const T_location &  mu,
const T_precision &  phi 
)

Definition at line 37 of file neg_binomial_2_cdf.hpp.

template<typename T_n , typename T_location , typename T_precision >
return_type<T_location, T_precision>::type stan::math::neg_binomial_2_cdf_log ( const T_n &  n,
const T_location &  mu,
const T_precision &  phi 
)

Definition at line 31 of file neg_binomial_2_cdf_log.hpp.

template<bool propto, typename T_n , typename T_location , typename T_precision >
return_type<T_location, T_precision>::type stan::math::neg_binomial_2_log ( const T_n &  n,
const T_location &  mu,
const T_precision &  phi 
)

Definition at line 37 of file neg_binomial_2_log.hpp.

template<typename T_n , typename T_location , typename T_precision >
return_type<T_location, T_precision>::type stan::math::neg_binomial_2_log ( const T_n &  n,
const T_location &  mu,
const T_precision &  phi 
)
inline

Definition at line 141 of file neg_binomial_2_log.hpp.

template<bool propto, typename T_n , typename T_log_location , typename T_precision >
return_type<T_log_location, T_precision>::type stan::math::neg_binomial_2_log_log ( const T_n &  n,
const T_log_location &  eta,
const T_precision &  phi 
)

Definition at line 33 of file neg_binomial_2_log_log.hpp.

template<typename T_n , typename T_log_location , typename T_precision >
return_type<T_log_location, T_precision>::type stan::math::neg_binomial_2_log_log ( const T_n &  n,
const T_log_location &  eta,
const T_precision &  phi 
)
inline

Definition at line 140 of file neg_binomial_2_log_log.hpp.

template<class RNG >
int stan::math::neg_binomial_2_log_rng ( const double  eta,
const double  phi,
RNG &  rng 
)
inline

Definition at line 29 of file neg_binomial_2_log_rng.hpp.

template<class RNG >
int stan::math::neg_binomial_2_rng ( const double  mu,
const double  phi,
RNG &  rng 
)
inline

Definition at line 29 of file neg_binomial_2_rng.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >
return_type<T_shape, T_inv_scale>::type stan::math::neg_binomial_ccdf_log ( const T_n &  n,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 32 of file neg_binomial_ccdf_log.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >
return_type<T_shape, T_inv_scale>::type stan::math::neg_binomial_cdf ( const T_n &  n,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 29 of file neg_binomial_cdf.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >
return_type<T_shape, T_inv_scale>::type stan::math::neg_binomial_cdf_log ( const T_n &  n,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 32 of file neg_binomial_cdf_log.hpp.

template<bool propto, typename T_n , typename T_shape , typename T_inv_scale >
return_type<T_shape, T_inv_scale>::type stan::math::neg_binomial_log ( const T_n &  n,
const T_shape &  alpha,
const T_inv_scale &  beta 
)

Definition at line 39 of file neg_binomial_log.hpp.

template<typename T_n , typename T_shape , typename T_inv_scale >
return_type<T_shape, T_inv_scale>::type stan::math::neg_binomial_log ( const T_n &  n,
const T_shape &  alpha,
const T_inv_scale &  beta 
)
inline

Definition at line 183 of file neg_binomial_log.hpp.

template<class RNG >
int stan::math::neg_binomial_rng ( const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 30 of file neg_binomial_rng.hpp.

double stan::math::negative_infinity ( )
inline

Return negative infinity.

Returns
Negative infinity.

Definition at line 132 of file constants.hpp.

static size_t stan::math::nested_size ( )
inlinestatic

Definition at line 10 of file nested_size.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::normal_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file normal_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::normal_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Calculates the normal cumulative distribution function for the given variate, location, and scale.

$\Phi(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\inf}^x e^{-t^2/2} dt$.

Parameters
yA scalar variate.
muThe location of the normal distribution.
sigmaThe scale of the normal distriubtion
Returns
The unit normal cdf evaluated at the specified arguments.
Template Parameters
T_yType of y.
T_locType of mean parameter.
T_scaleType of standard deviation paramater.

Definition at line 38 of file normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::normal_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 24 of file normal_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)

The log of the normal density for the specified scalar(s) given the specified mean(s) and deviation(s).

y, mu, or sigma can each be either a scalar or a vector. Any vector inputs must be the same length.

The result log probability is defined to be the sum of the log probabilities for each observation/mean/deviation triple.

Parameters
y(Sequence of) scalar(s).
mu(Sequence of) location parameter(s) for the normal distribution.
sigma(Sequence of) scale parameters for the normal distribution.
Returns
The log of the product of the densities.
Exceptions
std::domain_errorif the scale is not positive.
Template Parameters
T_yUnderlying type of scalar in sequence.
T_locType of location parameter.

Definition at line 43 of file normal_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma 
)
inline

Definition at line 136 of file normal_log.hpp.

template<class RNG >
double stan::math::normal_rng ( const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 20 of file normal_rng.hpp.

double stan::math::not_a_number ( )
inline

Return (quiet) not-a-number.

Returns
Quiet not-a-number.

Definition at line 141 of file constants.hpp.

template<typename T >
int stan::math::num_elements ( const T &  x)
inline

Returns 1, the number of elements in a primitive type.

Parameters
xArgument of primitive type.
Returns
1

Definition at line 19 of file num_elements.hpp.

template<typename T , int R, int C>
int stan::math::num_elements ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the size of the specified matrix.

Parameters
margument matrix
Returns
size of matrix

Definition at line 31 of file num_elements.hpp.

template<typename T >
int stan::math::num_elements ( const std::vector< T > &  v)
inline

Returns the number of elements in the specified vector.

This assumes it is not ragged and that each of its contained elements has the same number of elements.

Parameters
vargument vector
Returns
number of contained arguments

Definition at line 45 of file num_elements.hpp.

bool stan::math::operator! ( const var a)
inline

Prefix logical negation for the value of variables (C++).

The expression (!a) is equivalent to negating the scalar value of the variable a.

Note that this is the only logical operator defined for variables. Overridden logical conjunction (&&) and disjunction (||) operators do not apply the same "short circuit" rules as the built-in logical operators.

\[ \mbox{operator!}(x) = \begin{cases} 0 & \mbox{if } x \neq 0 \\ 1 & \mbox{if } x = 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable to negate.
Returns
True if variable is non-zero.

Definition at line 31 of file operator_unary_not.hpp.

template<typename T >
bool stan::math::operator!= ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 14 of file operator_not_equal.hpp.

template<typename T >
bool stan::math::operator!= ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 21 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var a,
const var b 
)
inline

Inequality operator comparing two variables' values (C++).

\[ \mbox{operator!=}(x, y) = \begin{cases} 0 & \mbox{if } x = y\\ 1 & \mbox{if } x \neq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if the first variable's value is not the same as the second's.

Definition at line 26 of file operator_not_equal.hpp.

template<typename T >
bool stan::math::operator!= ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 28 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const var a,
const double  b 
)
inline

Inequality operator comparing a variable's value and a double (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if the first variable's value is not the same as the second value.

Definition at line 39 of file operator_not_equal.hpp.

bool stan::math::operator!= ( const double  a,
const var b 
)
inline

Inequality operator comparing a double and a variable's value (C++).

Parameters
aFirst value.
bSecond variable.
Returns
True if the first value is not the same as the second variable's value.

Definition at line 52 of file operator_not_equal.hpp.

template<typename T >
fvar<T> stan::math::operator* ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 14 of file operator_multiplication.hpp.

template<typename T >
fvar<T> stan::math::operator* ( double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 22 of file operator_multiplication.hpp.

template<typename T >
fvar<T> stan::math::operator* ( const fvar< T > &  x1,
double  x2 
)
inline

Definition at line 29 of file operator_multiplication.hpp.

var stan::math::operator* ( const var a,
const var b 
)
inline

Multiplication operator for two variables (C++).

The partial derivatives are

$\frac{\partial}{\partial x} (x * y) = y$, and

$\frac{\partial}{\partial y} (x * y) = x$.

\[ \mbox{operator*}(x, y) = \begin{cases} xy & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial x} = \begin{cases} y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator*}(x, y)}{\partial y} = \begin{cases} x & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable operand.
bSecond variable operand.
Returns
Variable result of multiplying operands.

Definition at line 83 of file operator_multiplication.hpp.

var stan::math::operator* ( const var a,
const double  b 
)
inline

Multiplication operator for a variable and a scalar (C++).

The partial derivative for the variable is

$\frac{\partial}{\partial x} (x * c) = c$, and

Parameters
aVariable operand.
bScalar operand.
Returns
Variable result of multiplying operands.

Definition at line 98 of file operator_multiplication.hpp.

var stan::math::operator* ( const double  a,
const var b 
)
inline

Multiplication operator for a scalar and a variable (C++).

The partial derivative for the variable is

$\frac{\partial}{\partial y} (c * y) = c$.

Parameters
aScalar operand.
bVariable operand.
Returns
Variable result of multiplying the operands.

Definition at line 115 of file operator_multiplication.hpp.

template<typename T >
fvar<T> stan::math::operator+ ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 13 of file operator_addition.hpp.

template<typename T >
fvar<T> stan::math::operator+ ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 20 of file operator_addition.hpp.

template<typename T >
fvar<T> stan::math::operator+ ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 27 of file operator_addition.hpp.

var stan::math::operator+ ( const var a)
inline

Unary plus operator for variables (C++).

The function simply returns its input, because

$\frac{d}{dx} +x = \frac{d}{dx} x = 1$.

The effect of unary plus on a built-in C++ scalar type is integer promotion. Because variables are all double-precision floating point already, promotion is not necessary.

\[ \mbox{operator+}(x) = \begin{cases} x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aArgument variable.
Returns
The input reference.

Definition at line 43 of file operator_unary_plus.hpp.

var stan::math::operator+ ( const var a,
const var b 
)
inline

Addition operator for variables (C++).

The partial derivatives are defined by

$\frac{\partial}{\partial x} (x+y) = 1$, and

$\frac{\partial}{\partial y} (x+y) = 1$.

\[ \mbox{operator+}(x, y) = \begin{cases} x+y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator+}(x, y)}{\partial y} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable operand.
bSecond variable operand.
Returns
Variable result of adding two variables.

Definition at line 84 of file operator_addition.hpp.

var stan::math::operator+ ( const var a,
const double  b 
)
inline

Addition operator for variable and scalar (C++).

The derivative with respect to the variable is

$\frac{d}{dx} (x + c) = 1$.

Parameters
aFirst variable operand.
bSecond scalar operand.
Returns
Result of adding variable and scalar.

Definition at line 99 of file operator_addition.hpp.

var stan::math::operator+ ( const double  a,
const var b 
)
inline

Addition operator for scalar and variable (C++).

The derivative with respect to the variable is

$\frac{d}{dy} (c + y) = 1$.

Parameters
aFirst scalar operand.
bSecond variable operand.
Returns
Result of adding variable and scalar.

Definition at line 116 of file operator_addition.hpp.

var& stan::math::operator++ ( var a)
inline

Prefix increment operator for variables (C++).

Following C++, (++a) is defined to behave exactly as (a = a + 1.0) does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters
aVariable to increment.
Returns
Reference the result of incrementing this input variable.

Definition at line 36 of file operator_unary_increment.hpp.

var stan::math::operator++ ( var a,
int   
)
inline

Postfix increment operator for variables (C++).

Following C++, the expression (a++) is defined to behave like the sequence of operations

var temp = a; a = a + 1.0; return temp;

Parameters
aVariable to increment.
Returns
Input variable.

Definition at line 52 of file operator_unary_increment.hpp.

template<typename T >
fvar<T> stan::math::operator- ( const fvar< T > &  x)
inline

Definition at line 14 of file operator_unary_minus.hpp.

template<typename T >
fvar<T> stan::math::operator- ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 14 of file operator_subtraction.hpp.

template<typename T >
fvar<T> stan::math::operator- ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 21 of file operator_subtraction.hpp.

template<typename T >
fvar<T> stan::math::operator- ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 28 of file operator_subtraction.hpp.

var stan::math::operator- ( const var a)
inline

Unary negation operator for variables (C++).

$\frac{d}{dx} -x = -1$.

\[ \mbox{operator-}(x) = \begin{cases} -x & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x)}{\partial x} = \begin{cases} -1 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aArgument variable.
Returns
Negation of variable.

Definition at line 51 of file operator_unary_negative.hpp.

var stan::math::operator- ( const var a,
const var b 
)
inline

Subtraction operator for variables (C++).

The partial derivatives are defined by

$\frac{\partial}{\partial x} (x-y) = 1$, and

$\frac{\partial}{\partial y} (x-y) = -1$.

\[ \mbox{operator-}(x, y) = \begin{cases} x-y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial x} = \begin{cases} 1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator-}(x, y)}{\partial y} = \begin{cases} -1 & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable operand.
bSecond variable operand.
Returns
Variable result of subtracting the second variable from the first.

Definition at line 99 of file operator_subtraction.hpp.

var stan::math::operator- ( const var a,
const double  b 
)
inline

Subtraction operator for variable and scalar (C++).

The derivative for the variable is

$\frac{\partial}{\partial x} (x-c) = 1$, and

Parameters
aFirst variable operand.
bSecond scalar operand.
Returns
Result of subtracting the scalar from the variable.

Definition at line 114 of file operator_subtraction.hpp.

var stan::math::operator- ( const double  a,
const var b 
)
inline

Subtraction operator for scalar and variable (C++).

The derivative for the variable is

$\frac{\partial}{\partial y} (c-y) = -1$, and

Parameters
aFirst scalar operand.
bSecond variable operand.
Returns
Result of sutracting a variable from a scalar.

Definition at line 131 of file operator_subtraction.hpp.

var& stan::math::operator-- ( var a)
inline

Prefix decrement operator for variables (C++).

Following C++, (–a) is defined to behave exactly as

a = a - 1.0)

does, but is faster and uses less memory. In particular, the result is an assignable lvalue.

Parameters
aVariable to decrement.
Returns
Reference the result of decrementing this input variable.

Definition at line 40 of file operator_unary_decrement.hpp.

var stan::math::operator-- ( var a,
int   
)
inline

Postfix decrement operator for variables (C++).

Following C++, the expression (a–) is defined to behave like the sequence of operations

var temp = a; a = a - 1.0; return temp;

Parameters
aVariable to decrement.
Returns
Input variable.

Definition at line 56 of file operator_unary_decrement.hpp.

template<typename T >
fvar<T> stan::math::operator/ ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 14 of file operator_division.hpp.

template<typename T >
fvar<T> stan::math::operator/ ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 22 of file operator_division.hpp.

template<typename T >
fvar<T> stan::math::operator/ ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 30 of file operator_division.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< fvar< T >, R, C > &  v,
const fvar< T > &  c 
)
inline

Definition at line 58 of file divide.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< fvar< T >, R, C > &  v,
const double  c 
)
inline

Definition at line 64 of file divide.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::operator/ ( const Eigen::Matrix< double, R, C > &  v,
const fvar< T > &  c 
)
inline

Definition at line 70 of file divide.hpp.

var stan::math::operator/ ( const var a,
const var b 
)
inline

Division operator for two variables (C++).

The partial derivatives for the variables are

$\frac{\partial}{\partial x} (x/y) = 1/y$, and

$\frac{\partial}{\partial y} (x/y) = -x / y^2$.

\[ \mbox{operator/}(x, y) = \begin{cases} \frac{x}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial x} = \begin{cases} \frac{1}{y} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{operator/}(x, y)}{\partial y} = \begin{cases} -\frac{x}{y^2} & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable operand.
bSecond variable operand.
Returns
Variable result of dividing the first variable by the second.

Definition at line 96 of file operator_division.hpp.

var stan::math::operator/ ( const var a,
const double  b 
)
inline

Division operator for dividing a variable by a scalar (C++).

The derivative with respect to the variable is

$\frac{\partial}{\partial x} (x/c) = 1/c$.

Parameters
aVariable operand.
bScalar operand.
Returns
Variable result of dividing the variable by the scalar.

Definition at line 111 of file operator_division.hpp.

var stan::math::operator/ ( const double  a,
const var b 
)
inline

Division operator for dividing a scalar by a variable (C++).

The derivative with respect to the variable is

$\frac{d}{d y} (c/y) = -c / y^2$.

Parameters
aScalar operand.
bVariable operand.
Returns
Variable result of dividing the scalar by the variable.

Definition at line 128 of file operator_division.hpp.

template<typename T >
bool stan::math::operator< ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 12 of file operator_less_than.hpp.

template<typename T >
bool stan::math::operator< ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 18 of file operator_less_than.hpp.

template<typename T >
bool stan::math::operator< ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 24 of file operator_less_than.hpp.

bool stan::math::operator< ( const var a,
const var b 
)
inline

Less than operator comparing variables' values (C++).

\[ \mbox{operator\textless}(x, y) = \begin{cases} 0 & \mbox{if } x \geq y \\ 1 & \mbox{if } x < y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if first variable's value is less than second's.

Definition at line 24 of file operator_less_than.hpp.

bool stan::math::operator< ( const var a,
const double  b 
)
inline

Less than operator comparing variable's value and a double (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if first variable's value is less than second value.

Definition at line 36 of file operator_less_than.hpp.

bool stan::math::operator< ( const double  a,
const var b 
)
inline

Less than operator comparing a double and variable's value (C++).

Parameters
aFirst value.
bSecond variable.
Returns
True if first value is less than second variable's value.

Definition at line 48 of file operator_less_than.hpp.

template<typename T >
bool stan::math::operator<= ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 14 of file operator_less_than_or_equal.hpp.

template<typename T >
bool stan::math::operator<= ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 21 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const var a,
const var b 
)
inline

Less than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textless=}(x, y) = \begin{cases} 0 & \mbox{if } x > y\\ 1 & \mbox{if } x \leq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if first variable's value is less than or equal to the second's.

Definition at line 26 of file operator_less_than_or_equal.hpp.

template<typename T >
bool stan::math::operator<= ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 28 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const var a,
const double  b 
)
inline

Less than or equal operator comparing a variable's value and a scalar (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if first variable's value is less than or equal to the second value.

Definition at line 39 of file operator_less_than_or_equal.hpp.

bool stan::math::operator<= ( const double  a,
const var b 
)
inline

Less than or equal operator comparing a double and variable's value (C++).

Parameters
aFirst value.
bSecond variable.
Returns
True if first value is less than or equal to the second variable's value.

Definition at line 52 of file operator_less_than_or_equal.hpp.

template<typename T >
bool stan::math::operator== ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 14 of file operator_equal.hpp.

template<typename T >
bool stan::math::operator== ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 21 of file operator_equal.hpp.

bool stan::math::operator== ( const var a,
const var b 
)
inline

Equality operator comparing two variables' values (C++).

\[ \mbox{operator==}(x, y) = \begin{cases} 0 & \mbox{if } x \neq y\\ 1 & \mbox{if } x = y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if the first variable's value is the same as the second's.

Definition at line 26 of file operator_equal.hpp.

template<typename T >
bool stan::math::operator== ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 28 of file operator_equal.hpp.

bool stan::math::operator== ( const var a,
const double  b 
)
inline

Equality operator comparing a variable's value and a double (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if the first variable's value is the same as the second value.

Definition at line 39 of file operator_equal.hpp.

bool stan::math::operator== ( const double  a,
const var b 
)
inline

Equality operator comparing a scalar and a variable's value (C++).

Parameters
aFirst scalar.
bSecond variable.
Returns
True if the variable's value is equal to the scalar.

Definition at line 51 of file operator_equal.hpp.

template<typename T >
bool stan::math::operator> ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 14 of file operator_greater_than.hpp.

template<typename T >
bool stan::math::operator> ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 21 of file operator_greater_than.hpp.

bool stan::math::operator> ( const var a,
const var b 
)
inline

Greater than operator comparing variables' values (C++).

\[ \mbox{operator\textgreater}(x, y) = \begin{cases} 0 & \mbox{if } x \leq y\\ 1 & \mbox{if } x > y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if first variable's value is greater than second's.

Definition at line 25 of file operator_greater_than.hpp.

template<typename T >
bool stan::math::operator> ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 28 of file operator_greater_than.hpp.

bool stan::math::operator> ( const var a,
const double  b 
)
inline

Greater than operator comparing variable's value and double (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if first variable's value is greater than second value.

Definition at line 37 of file operator_greater_than.hpp.

bool stan::math::operator> ( const double  a,
const var b 
)
inline

Greater than operator comparing a double and a variable's value (C++).

Parameters
aFirst value.
bSecond variable.
Returns
True if first value is greater than second variable's value.

Definition at line 49 of file operator_greater_than.hpp.

template<typename T >
bool stan::math::operator>= ( const fvar< T > &  x,
const fvar< T > &  y 
)
inline

Definition at line 14 of file operator_greater_than_or_equal.hpp.

template<typename T >
bool stan::math::operator>= ( const fvar< T > &  x,
double  y 
)
inline

Definition at line 21 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const var a,
const var b 
)
inline

Greater than or equal operator comparing two variables' values (C++).

\[ \mbox{operator\textgreater=}(x, y) = \begin{cases} 0 & \mbox{if } x < y\\ 1 & \mbox{if } x \geq y \\[6pt] 0 & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
aFirst variable.
bSecond variable.
Returns
True if first variable's value is greater than or equal to the second's.

Definition at line 27 of file operator_greater_than_or_equal.hpp.

template<typename T >
bool stan::math::operator>= ( double  x,
const fvar< T > &  y 
)
inline

Definition at line 28 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const var a,
const double  b 
)
inline

Greater than or equal operator comparing variable's value and double (C++).

Parameters
aFirst variable.
bSecond value.
Returns
True if first variable's value is greater than or equal to second value.

Definition at line 40 of file operator_greater_than_or_equal.hpp.

bool stan::math::operator>= ( const double  a,
const var b 
)
inline

Greater than or equal operator comparing double and variable's value (C++).

Parameters
aFirst value.
bSecond variable.
Returns
True if the first value is greater than or equal to the second variable's value.

Definition at line 53 of file operator_greater_than_or_equal.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x)

Return an increasing ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters
xFree vector of scalars.
Returns
Positive, increasing ordered vector.
Template Parameters
TType of scalar.

Definition at line 23 of file ordered_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
T &  lp 
)
inline

Return a positive valued, increasing ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters
xFree vector of scalars.
lpLog probability reference.
Returns
Positive, increasing ordered vector.
Template Parameters
TType of scalar.

Definition at line 56 of file ordered_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::ordered_free ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in ordered_constrain(Matrix),

Parameters
yVector of positive, ordered scalars.
Returns
Free vector that transforms into the input vector.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif y is not a vector of positive, ordered scalars.

Definition at line 28 of file ordered_free.hpp.

template<bool propto, typename T_lambda , typename T_cut >
boost::math::tools::promote_args<T_lambda, T_cut>::type stan::math::ordered_logistic_log ( int  y,
const T_lambda &  lambda,
const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &  c 
)

Returns the (natural) log probability of the specified integer outcome given the continuous location and specified cutpoints in an ordered logistic model.

Typically the continous location will be the dot product of a vector of regression coefficients and a vector of predictors for the outcome.

Template Parameters
proptoTrue if calculating up to a proportion.
T_locLocation type.
T_cutCut-point type.
Parameters
yOutcome.
lambdaLocation.
cPositive increasing vector of cutpoints.
Returns
Log probability of outcome given location and cutpoints.
Exceptions
std::domain_errorIf the outcome is not between 1 and the number of cutpoints plus 2; if the cutpoint vector is empty; if the cutpoint vector contains a non-positive, non-finite value; or if the cutpoint vector is not sorted in ascending order.

Definition at line 61 of file ordered_logistic_log.hpp.

template<typename T_lambda , typename T_cut >
boost::math::tools::promote_args<T_lambda, T_cut>::type stan::math::ordered_logistic_log ( int  y,
const T_lambda &  lambda,
const Eigen::Matrix< T_cut, Eigen::Dynamic, 1 > &  c 
)

Definition at line 107 of file ordered_logistic_log.hpp.

template<class RNG >
int stan::math::ordered_logistic_rng ( const double  eta,
const Eigen::Matrix< double, Eigen::Dynamic, 1 > &  c,
RNG &  rng 
)
inline

Definition at line 24 of file ordered_logistic_rng.hpp.

void stan::math::out_of_range ( const char *  function,
const int  max,
const int  index,
const char *  msg1 = "",
const char *  msg2 = "" 
)
inline

Throw an out_of_range exception with a consistently formatted message.

This is an abstraction for all Stan functions to use when throwing out of range. This will allow us to change the behavior for all functions at once.

The message is: "<function>: index <index> out of range; expecting index to be between " "1 and <max><msg1><msg2>"

Parameters
functionName of the function
maxMax
indexIndex
msg1Message to print. Default is "".
msg2Message to print. Default is "".

Definition at line 30 of file out_of_range.hpp.

template<typename T >
fvar<T> stan::math::owens_t ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 14 of file owens_t.hpp.

template<typename T >
fvar<T> stan::math::owens_t ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 34 of file owens_t.hpp.

template<typename T >
fvar<T> stan::math::owens_t ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 48 of file owens_t.hpp.

double stan::math::owens_t ( const double  h,
const double  a 
)
inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

\[ \mbox{owens\_t}(h, a) = \begin{cases} \mbox{owens\_t}(h, a) & \mbox{if } -\infty\leq h, a \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial h} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} = \begin{cases} \frac{\partial\, \mbox{owens\_t}(h, a)}{\partial a} & \mbox{if } -\infty\leq h, a\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } h = \textrm{NaN or } a = \textrm{NaN} \end{cases} \]

\[ \mbox{owens\_t}(h, a) = \frac{1}{2\pi} \int_0^a \frac{\exp(-\frac{1}{2}h^2(1+x^2))}{1+x^2}dx \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial h} = -\frac{1}{2\sqrt{2\pi}} \operatorname{erf}\left(\frac{ha}{\sqrt{2}}\right) \exp\left(-\frac{h^2}{2}\right) \]

\[ \frac{\partial \, \mbox{owens\_t}(h, a)}{\partial a} = \frac{\exp\left(-\frac{1}{2}h^2(1+a^2)\right)}{2\pi (1+a^2)} \]

Template Parameters
T1Type of first argument.
T2Type of second argument.
Parameters
hFirst argument
aSecond argument
Returns
The Owen's T function.

Definition at line 62 of file owens_t.hpp.

var stan::math::owens_t ( const var h,
const var a 
)
inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters
hvar parameter.
avar parameter.
Returns
The Owen's T function.

Definition at line 71 of file owens_t.hpp.

var stan::math::owens_t ( const var h,
double  a 
)
inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters
hvar parameter.
adouble parameter.
Returns
The Owen's T function.

Definition at line 85 of file owens_t.hpp.

var stan::math::owens_t ( double  h,
const var a 
)
inline

The Owen's T function of h and a.

Used to compute the cumulative density function for the skew normal distribution.

Parameters
hdouble parameter.
avar parameter.
Returns
The Owen's T function.

Definition at line 99 of file owens_t.hpp.

template<typename T_y , typename T_scale , typename T_shape >
return_type<T_y, T_scale, T_shape>::type stan::math::pareto_ccdf_log ( const T_y &  y,
const T_scale &  y_min,
const T_shape &  alpha 
)

Definition at line 23 of file pareto_ccdf_log.hpp.

template<typename T_y , typename T_scale , typename T_shape >
return_type<T_y, T_scale, T_shape>::type stan::math::pareto_cdf ( const T_y &  y,
const T_scale &  y_min,
const T_shape &  alpha 
)

Definition at line 24 of file pareto_cdf.hpp.

template<typename T_y , typename T_scale , typename T_shape >
return_type<T_y, T_scale, T_shape>::type stan::math::pareto_cdf_log ( const T_y &  y,
const T_scale &  y_min,
const T_shape &  alpha 
)

Definition at line 23 of file pareto_cdf_log.hpp.

template<bool propto, typename T_y , typename T_scale , typename T_shape >
return_type<T_y, T_scale, T_shape>::type stan::math::pareto_log ( const T_y &  y,
const T_scale &  y_min,
const T_shape &  alpha 
)

Definition at line 27 of file pareto_log.hpp.

template<typename T_y , typename T_scale , typename T_shape >
return_type<T_y, T_scale, T_shape>::type stan::math::pareto_log ( const T_y &  y,
const T_scale &  y_min,
const T_shape &  alpha 
)
inline

Definition at line 130 of file pareto_log.hpp.

template<class RNG >
double stan::math::pareto_rng ( const double  y_min,
const double  alpha,
RNG &  rng 
)
inline

Definition at line 22 of file pareto_rng.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::pareto_type_2_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  lambda,
const T_shape &  alpha 
)

Definition at line 25 of file pareto_type_2_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::pareto_type_2_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  lambda,
const T_shape &  alpha 
)

Definition at line 25 of file pareto_type_2_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::pareto_type_2_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  lambda,
const T_shape &  alpha 
)

Definition at line 25 of file pareto_type_2_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::pareto_type_2_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  lambda,
const T_shape &  alpha 
)

Definition at line 28 of file pareto_type_2_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::pareto_type_2_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  lambda,
const T_shape &  alpha 
)
inline

Definition at line 147 of file pareto_type_2_log.hpp.

template<class RNG >
double stan::math::pareto_type_2_rng ( const double  mu,
const double  lambda,
const double  alpha,
RNG &  rng 
)
inline

Definition at line 23 of file pareto_type_2_rng.hpp.

template<typename T , typename F >
void stan::math::partial_derivative ( const F &  f,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
int  n,
T &  fx,
T &  dfx_dxn 
)

Return the partial derivative of the specified multiivariate function at the specified argument.

Template Parameters
TArgument type
FFunction type
Parameters
fFunction
[in]xArgument vector
[in]nIndex of argument with which to take derivative
[out]fxValue of function applied to argument
[out]dfx_dxnValue of partial derivative

Definition at line 27 of file partial_derivative.hpp.

template<typename T >
fvar<T> stan::math::Phi ( const fvar< T > &  x)
inline

Definition at line 14 of file Phi.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::Phi ( const T  x)
inline

The unit normal cumulative distribution function.

The return value for a specified input is the probability that a random unit normal variate is less than or equal to the specified value, defined by

$\Phi(x) = \int_{-\infty}^x \mbox{\sf Norm}(x|0, 1) \ dx$

This function can be used to implement the inverse link function for probit regression.

Phi will underflow to 0 below -37.5 and overflow to 1 above 8

Parameters
xArgument.
Returns
Probability random sample is less than or equal to argument.

Definition at line 31 of file Phi.hpp.

var stan::math::Phi ( const stan::math::var a)
inline

The unit normal cumulative density function for variables (stan).

See stan::math::Phi() for the double-based version.

The derivative is the unit normal density function,

$\frac{d}{dx} \Phi(x) = \mbox{\sf Norm}(x|0, 1) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2)$.

\[ \mbox{Phi}(x) = \begin{cases} 0 & \mbox{if } x < -37.5 \\ \Phi(x) & \mbox{if } -37.5 \leq x \leq 8.25 \\ 1 & \mbox{if } x > 8.25 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } x < -27.5 \\ \frac{\partial\, \Phi(x)}{\partial x} & \mbox{if } -27.5 \leq x \leq 27.5 \\ 0 & \mbox{if } x > 27.5 \\[6pt] \textrm{error} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{0}^{x} e^{-t^2/2} dt \]

\[ \frac{\partial \, \Phi(x)}{\partial x} = \frac{e^{-x^2/2}}{\sqrt{2\pi}} \]

Parameters
aVariable argument.
Returns
The unit normal cdf evaluated at the specified argument.

Definition at line 66 of file Phi.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::Phi_approx ( x)
inline

Approximation of the unit normal CDF.

http://www.jiem.org/index.php/jiem/article/download/60/27

This function can be used to implement the inverse link function for probit regression.

Parameters
xArgument.
Returns
Probability random sample is less than or equal to argument.

Definition at line 23 of file Phi_approx.hpp.

var stan::math::Phi_approx ( const stan::math::var a)
inline

Approximation of the unit normal CDF for variables (stan).

http://www.jiem.org/index.php/jiem/article/download/60/27

\[ \mbox{Phi\_approx}(x) = \begin{cases} \Phi_{\mbox{\footnotesize approx}}(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{Phi\_approx}(x)}{\partial x} = \begin{cases} \frac{\partial\, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Phi_{\mbox{\footnotesize approx}}(x) = \mbox{logit}^{-1}(0.07056 \, x^3 + 1.5976 \, x) \]

\[ \frac{\partial \, \Phi_{\mbox{\footnotesize approx}}(x)}{\partial x} = -\Phi_{\mbox{\footnotesize approx}}^2(x) e^{-0.07056x^3 - 1.5976x}(-0.21168x^2-1.5976) \]

Parameters
aVariable argument.
Returns
The corresponding unit normal cdf approximation.

Definition at line 47 of file Phi_approx.hpp.

double stan::math::pi ( )
inline

Return the value of pi.

Returns
Pi.

Definition at line 86 of file constants.hpp.

template<typename T_n , typename T_rate >
return_type<T_rate>::type stan::math::poisson_ccdf_log ( const T_n &  n,
const T_rate &  lambda 
)

Definition at line 25 of file poisson_ccdf_log.hpp.

template<typename T_n , typename T_rate >
return_type<T_rate>::type stan::math::poisson_cdf ( const T_n &  n,
const T_rate &  lambda 
)

Definition at line 26 of file poisson_cdf.hpp.

template<typename T_n , typename T_rate >
return_type<T_rate>::type stan::math::poisson_cdf_log ( const T_n &  n,
const T_rate &  lambda 
)

Definition at line 25 of file poisson_cdf_log.hpp.

template<bool propto, typename T_n , typename T_rate >
return_type<T_rate>::type stan::math::poisson_log ( const T_n &  n,
const T_rate &  lambda 
)

Definition at line 27 of file poisson_log.hpp.

template<typename T_n , typename T_rate >
return_type<T_rate>::type stan::math::poisson_log ( const T_n &  n,
const T_rate &  lambda 
)
inline

Definition at line 99 of file poisson_log.hpp.

template<bool propto, typename T_n , typename T_log_rate >
return_type<T_log_rate>::type stan::math::poisson_log_log ( const T_n &  n,
const T_log_rate &  alpha 
)

Definition at line 30 of file poisson_log_log.hpp.

template<typename T_n , typename T_log_rate >
return_type<T_log_rate>::type stan::math::poisson_log_log ( const T_n &  n,
const T_log_rate &  alpha 
)
inline

Definition at line 110 of file poisson_log_log.hpp.

template<class RNG >
int stan::math::poisson_log_rng ( const double  alpha,
RNG &  rng 
)
inline

Definition at line 24 of file poisson_log_rng.hpp.

template<class RNG >
int stan::math::poisson_rng ( const double  lambda,
RNG &  rng 
)
inline

Definition at line 25 of file poisson_rng.hpp.

template<typename T >
T stan::math::positive_constrain ( const T  x)
inline

Return the positive value for the specified unconstrained input.

The transform applied is

$f(x) = \exp(x)$.

Parameters
xArbitrary input scalar.
Returns
Input transformed to be positive.

Definition at line 22 of file positive_constrain.hpp.

template<typename T >
T stan::math::positive_constrain ( const T  x,
T &  lp 
)
inline

Return the positive value for the specified unconstrained input, incrementing the scalar reference with the log absolute Jacobian determinant.

See positive_constrain(T) for details of the transform. The log absolute Jacobian determinant is

$\log | \frac{d}{dx} \mbox{exp}(x) | = \log | \mbox{exp}(x) | = x$.

Parameters
xArbitrary input scalar.
lpLog probability reference.
Returns
Input transformed to be positive.
Template Parameters
TType of scalar.

Definition at line 44 of file positive_constrain.hpp.

template<typename T >
T stan::math::positive_free ( const T  y)
inline

Return the unconstrained value corresponding to the specified positive-constrained value.

The transform is the inverse of the transform $f$ applied by positive_constrain(T), namely

$f^{-1}(x) = \log(x)$.

The input is validated using stan::math::check_positive().

Parameters
yInput scalar.
Returns
Unconstrained value that produces the input when constrained.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif the variable is negative.

Definition at line 29 of file positive_free.hpp.

double stan::math::positive_infinity ( )
inline

Return positive infinity.

Returns
Positive infinity.

Definition at line 123 of file constants.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x)

Return an increasing positive ordered vector derived from the specified free vector.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters
xFree vector of scalars.
Returns
Positive, increasing ordered vector.
Template Parameters
TType of scalar.

Definition at line 23 of file positive_ordered_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x,
T &  lp 
)
inline

Return a positive valued, increasing positive ordered vector derived from the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The returned constrained vector will have the same dimensionality as the specified free vector.

Parameters
xFree vector of scalars.
lpLog probability reference.
Returns
Positive, increasing ordered vector.
Template Parameters
TType of scalar.

Definition at line 55 of file positive_ordered_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::positive_ordered_free ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y)

Return the vector of unconstrained scalars that transform to the specified positive ordered vector.

This function inverts the constraining operation defined in positive_ordered_constrain(Matrix),

Parameters
yVector of positive, ordered scalars.
Returns
Free vector that transforms into the input vector.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif y is not a vector of positive, ordered scalars.

Definition at line 28 of file positive_ordered_free.hpp.

template<typename T >
fvar<T> stan::math::pow ( const fvar< T > &  x1,
const fvar< T > &  x2 
)
inline

Definition at line 18 of file pow.hpp.

template<typename T >
fvar<T> stan::math::pow ( const double  x1,
const fvar< T > &  x2 
)
inline

Definition at line 30 of file pow.hpp.

template<typename T >
fvar<T> stan::math::pow ( const fvar< T > &  x1,
const double  x2 
)
inline

Definition at line 40 of file pow.hpp.

var stan::math::pow ( const var base,
const var exponent 
)
inline

Return the base raised to the power of the exponent (cmath).

The partial derivatives are

$\frac{\partial}{\partial x} \mbox{pow}(x, y) = y x^{y-1}$, and

$\frac{\partial}{\partial y} \mbox{pow}(x, y) = x^y \ \log x$.

\[ \mbox{pow}(x, y) = \begin{cases} x^y & \mbox{if } -\infty\leq x, y \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial x} = \begin{cases} yx^{y-1} & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{pow}(x, y)}{\partial y} = \begin{cases} x^y\ln x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } y = \textrm{NaN} \end{cases} \]

Parameters
baseBase variable.
exponentExponent variable.
Returns
Base raised to the exponent.

Definition at line 103 of file pow.hpp.

var stan::math::pow ( const var base,
const double  exponent 
)
inline

Return the base variable raised to the power of the exponent scalar (cmath).

The derivative for the variable is

$\frac{d}{dx} \mbox{pow}(x, c) = c x^{c-1}$.

Parameters
baseBase variable.
exponentExponent scalar.
Returns
Base raised to the exponent.

Definition at line 119 of file pow.hpp.

var stan::math::pow ( const double  base,
const var exponent 
)
inline

Return the base scalar raised to the power of the exponent variable (cmath).

The derivative for the variable is

$\frac{d}{d y} \mbox{pow}(c, y) = c^y \log c $.

Parameters
baseBase scalar.
exponentExponent variable.
Returns
Base raised to the exponent.

Definition at line 141 of file pow.hpp.

var stan::math::precomputed_gradients ( const double  value,
const std::vector< var > &  operands,
const std::vector< double > &  gradients 
)

This function returns a var for an expression that has the specified value, vector of operands, and vector of partial derivatives of value with respect to the operands.

Parameters
[in]valueThe value of the resulting dependent variable.
[in]operandsoperands.
[in]gradientsvector of partial derivatives of result with respect to operands.
Returns
An auto-diff variable that uses the precomputed gradients provided.

Definition at line 98 of file precomputed_gradients.hpp.

double stan::math::primitive_value ( const var v)
inline

Return the primitive double value for the specified auto-diff variable.

Parameters
vinput variable.
Returns
value of input.

Definition at line 17 of file primitive_value.hpp.

template<typename T >
double stan::math::primitive_value ( const fvar< T > &  v)
inline

Return the primitive value of the specified forward-mode autodiff variable.

This function applies recursively to higher-order autodiff types to return a primitive double value.

Template Parameters
Tscalar type for autodiff variable.
Parameters
vinput variable.
Returns
primitive value of input.

Definition at line 22 of file primitive_value.hpp.

template<typename T >
boost::enable_if<boost::is_arithmetic<T>, T>::type stan::math::primitive_value ( x)
inline

Return the value of the specified arithmetic argument unmodified with its own declared type.

This template function can only be instantiated with arithmetic types as defined by Boost's is_arithmetic trait metaprogram.

This function differs from stan::math::value_of in that it does not cast all return types to double.

Template Parameters
Ttype of arithmetic input.
Parameters
xinput.
Returns
input unmodified.

Definition at line 30 of file primitive_value.hpp.

template<typename T >
boost::disable_if<boost::is_arithmetic<T>, double>::type stan::math::primitive_value ( const T &  x)
inline

Return the primitive value of the specified argument.

This implementation only applies to non-arithmetic types as defined by Boost's is_arithmetic trait metaprogram.

Template Parameters
Ttype of non-arithmetic input.
Parameters
xinput.
Returns
value of input.

Definition at line 47 of file primitive_value.hpp.

void stan::math::print_mat_size ( int  n,
std::ostream &  o 
)

Helper function to return the matrix size as either "dynamic" or "1".

Parameters
nEigen matrix size specification.
oOutput stream.
Returns
String representing size.

Definition at line 26 of file assign.hpp.

void stan::math::print_stack ( std::ostream &  o)
inline

Prints the auto-dif variable stack.

This function is used for debugging purposes.

Only works if all members of stack are vari* as it casts to vari*.

Parameters
oostream to modify

Definition at line 20 of file print_stack.hpp.

template<typename T >
T stan::math::prob_constrain ( const T  x)
inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar.

The transform is the inverse logit,

$f(x) = \mbox{logit}^{-1}(x) = \frac{1}{1 + \exp(x)}$.

Parameters
xFree scalar.
Returns
Probability-constrained result of transforming the free scalar.
Template Parameters
TType of scalar.

Definition at line 27 of file prob_constrain.hpp.

template<typename T >
T stan::math::prob_constrain ( const T  x,
T &  lp 
)
inline

Return a probability value constrained to fall between 0 and 1 (inclusive) for the specified free scalar and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is as defined for prob_constrain(T). The log absolute Jacobian determinant is

The log absolute Jacobian determinant is

$\log | \frac{d}{dx} \mbox{logit}^{-1}(x) |$

$\log ((\mbox{logit}^{-1}(x)) (1 - \mbox{logit}^{-1}(x))$

$\log (\mbox{logit}^{-1}(x)) + \log (1 - \mbox{logit}^{-1}(x))$.

Parameters
xFree scalar.
lpLog probability reference.
Returns
Probability-constrained result of transforming the free scalar.
Template Parameters
TType of scalar.

Definition at line 55 of file prob_constrain.hpp.

template<typename T >
T stan::math::prob_free ( const T  y)
inline

Return the free scalar that when transformed to a probability produces the specified scalar.

The function that reverses the constraining transform specified in prob_constrain(T) is the logit function,

$f^{-1}(y) = \mbox{logit}(y) = \frac{1 - y}{y}$.

Parameters
yScalar input.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif y is less than 0 or greater than 1.

Definition at line 27 of file prob_free.hpp.

template<typename T >
T stan::math::prod ( const std::vector< T > &  v)
inline

Returns the product of the coefficients of the specified standard vector.

Parameters
vSpecified vector.
Returns
Product of coefficients of vector.

Definition at line 17 of file prod.hpp.

template<typename T , int R, int C>
T stan::math::prod ( const Eigen::Matrix< T, R, C > &  v)
inline

Returns the product of the coefficients of the specified column vector.

Parameters
vSpecified vector.
Returns
Product of coefficients of vector.

Definition at line 32 of file prod.hpp.

template<typename T1 , typename T2 , typename F >
common_type<T1, T2>::type stan::math::promote_common ( const F &  u)
inline

Definition at line 14 of file promote_common.hpp.

template<typename T , typename S >
promote_scalar_type<T, S>::type stan::math::promote_scalar ( const S &  x)

This is the top-level function to call to promote the scalar types of an input of type S to type T.

Template Parameters
Tscalar type of output.
Sinput type.
Parameters
xinput vector.
Returns
input vector with scalars promoted to type T.

Definition at line 100 of file promote_scalar.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_Q ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Definition at line 14 of file qr_Q.hpp.

template<typename T >
Eigen::Matrix<fvar<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_Q ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &  m)

Definition at line 15 of file qr_Q.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_R ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Definition at line 14 of file qr_R.hpp.

template<typename T >
Eigen::Matrix<fvar<T>, Eigen::Dynamic, Eigen::Dynamic> stan::math::qr_R ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, Eigen::Dynamic > &  m)

Definition at line 15 of file qr_R.hpp.

template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix<T, CB, CB> stan::math::quad_form ( const Eigen::Matrix< T, RA, CA > &  A,
const Eigen::Matrix< T, RB, CB > &  B 
)
inline

Compute B^T A B.

Definition at line 21 of file quad_form.hpp.

template<int RA, int CA, int RB, typename T >
T stan::math::quad_form ( const Eigen::Matrix< T, RA, CA > &  A,
const Eigen::Matrix< T, RB, 1 > &  B 
)
inline

Definition at line 33 of file quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, Eigen::Matrix<var, CB, CB> >::type stan::math::quad_form ( const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, CB > &  B 
)
inline

Definition at line 124 of file quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c< boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, var >::type stan::math::quad_form ( const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, 1 > &  B 
)
inline

Definition at line 141 of file quad_form.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix< typename boost::math::tools::promote_args<T1, T2>::type, Eigen::Dynamic, Eigen::Dynamic> stan::math::quad_form_diag ( const Eigen::Matrix< T1, Eigen::Dynamic, Eigen::Dynamic > &  mat,
const Eigen::Matrix< T2, R, C > &  vec 
)
inline

Definition at line 17 of file quad_form_diag.hpp.

template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix<fvar<T>, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< fvar< T >, RA, CA > &  A,
const Eigen::Matrix< double, RB, CB > &  B 
)
inline

Definition at line 14 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix<T, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< T, RA, CA > &  A,
const Eigen::Matrix< T, RB, CB > &  B 
)
inline

Definition at line 19 of file quad_form_sym.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, Eigen::Matrix<var, CB, CB> >::type stan::math::quad_form_sym ( const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, CB > &  B 
)
inline

Definition at line 25 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >
fvar<T> stan::math::quad_form_sym ( const Eigen::Matrix< fvar< T >, RA, CA > &  A,
const Eigen::Matrix< double, RB, 1 > &  B 
)
inline

Definition at line 28 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >
T stan::math::quad_form_sym ( const Eigen::Matrix< T, RA, CA > &  A,
const Eigen::Matrix< T, RB, 1 > &  B 
)
inline

Definition at line 34 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, int CB, typename T >
Eigen::Matrix<fvar<T>, CB, CB> stan::math::quad_form_sym ( const Eigen::Matrix< double, RA, CA > &  A,
const Eigen::Matrix< fvar< T >, RB, CB > &  B 
)
inline

Definition at line 39 of file quad_form_sym.hpp.

template<typename TA , int RA, int CA, typename TB , int RB>
boost::enable_if_c< boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, var >::type stan::math::quad_form_sym ( const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, 1 > &  B 
)
inline

Definition at line 43 of file quad_form_sym.hpp.

template<int RA, int CA, int RB, typename T >
fvar<T> stan::math::quad_form_sym ( const Eigen::Matrix< double, RA, CA > &  A,
const Eigen::Matrix< fvar< T >, RB, 1 > &  B 
)
inline

Definition at line 53 of file quad_form_sym.hpp.

template<typename T >
int stan::math::rank ( const std::vector< T > &  v,
int  s 
)
inline

Return the number of components of v less than v[s].

Template Parameters
TType of elements.
Parameters
[in]vInput vector.
[in]sPosition in vector.
Returns
Number of components of v less than v[s].

Definition at line 20 of file rank.hpp.

template<typename T , int R, int C>
int stan::math::rank ( const Eigen::Matrix< T, R, C > &  v,
int  s 
)
inline

Return the number of components of v less than v[s].

Template Parameters
TType of elements of the vector.
Parameters
[in]vInput vector.
sIndex for input vector.
Returns
Number of components of v less than v[s].

Definition at line 42 of file rank.hpp.

template<typename T_y , typename T_scale >
return_type<T_y, T_scale>::type stan::math::rayleigh_ccdf_log ( const T_y &  y,
const T_scale &  sigma 
)

Definition at line 27 of file rayleigh_ccdf_log.hpp.

template<typename T_y , typename T_scale >
return_type<T_y, T_scale>::type stan::math::rayleigh_cdf ( const T_y &  y,
const T_scale &  sigma 
)

Definition at line 28 of file rayleigh_cdf.hpp.

template<typename T_y , typename T_scale >
return_type<T_y, T_scale>::type stan::math::rayleigh_cdf_log ( const T_y &  y,
const T_scale &  sigma 
)

Definition at line 28 of file rayleigh_cdf_log.hpp.

template<bool propto, typename T_y , typename T_scale >
return_type<T_y, T_scale>::type stan::math::rayleigh_log ( const T_y &  y,
const T_scale &  sigma 
)

Definition at line 29 of file rayleigh_log.hpp.

template<typename T_y , typename T_scale >
return_type<T_y, T_scale>::type stan::math::rayleigh_log ( const T_y &  y,
const T_scale &  sigma 
)
inline

Definition at line 109 of file rayleigh_log.hpp.

template<class RNG >
double stan::math::rayleigh_rng ( const double  sigma,
RNG &  rng 
)
inline

Definition at line 27 of file rayleigh_rng.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const size_t  K 
)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

It is generally better to work with the Cholesky factor rather than the correlation matrix itself when the determinant, inverse, etc. of the correlation matrix is needed for some statistical calculation.

See read_corr_matrix(Array, size_t, T) for more information.

Parameters
CPCsThe (K choose 2) canonical partial correlations in (-1, 1).
KDimensionality of correlation matrix.
Returns
Cholesky factor of correlation matrix for specified canonical partial correlations.
Template Parameters
TType of underlying scalar.

Definition at line 41 of file read_corr_L.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_L ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const size_t  K,
T &  log_prob 
)

Return the Cholesky factor of the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

The implementation is Ben Goodrich's Cholesky factor-based approach to the C-vine method of:

  • Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, Generating random correlation matrices based on vines and extended onion method Journal of Multivariate Analysis 100 (2009) 1989–2001
Parameters
CPCsThe (K choose 2) canonical partial correlations in (-1, 1).
KDimensionality of correlation matrix.
log_probReference to variable to increment with the log Jacobian determinant.
Returns
Cholesky factor of correlation matrix for specified partial correlations.
Template Parameters
TType of underlying scalar.

Definition at line 95 of file read_corr_L.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const size_t  K 
)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations.

See read_corr_matrix(Array, size_t, T) for more information.

Parameters
CPCsThe (K choose 2) canonical partial correlations in (-1, 1).
KDimensionality of correlation matrix.
Returns
Cholesky factor of correlation matrix for specified canonical partial correlations.
Template Parameters
TType of underlying scalar.

Definition at line 28 of file read_corr_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_corr_matrix ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const size_t  K,
T &  log_prob 
)

Return the correlation matrix of the specified dimensionality corresponding to the specified canonical partial correlations, incrementing the specified scalar reference with the log absolute determinant of the Jacobian of the transformation.

It is usually preferable to utilize the version that returns the Cholesky factor of the correlation matrix rather than the correlation matrix itself in statistical calculations.

Parameters
CPCsThe (K choose 2) canonical partial correlations in (-1, 1).
KDimensionality of correlation matrix.
log_probReference to variable to increment with the log Jacobian determinant.
Returns
Correlation matrix for specified partial correlations.
Template Parameters
TType of underlying scalar.

Definition at line 56 of file read_corr_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_L ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const Eigen::Array< T, Eigen::Dynamic, 1 > &  sds,
T &  log_prob 
)

This is the function that should be called prior to evaluating the density of any elliptical distribution.

Parameters
CPCson (-1, 1)
sdson (0, inf)
log_probthe log probability value to increment with the Jacobian
Returns
Cholesky factor of covariance matrix for specified partial correlations.

Definition at line 23 of file read_cov_L.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const Eigen::Array< T, Eigen::Dynamic, 1 > &  sds,
T &  log_prob 
)

A generally worse alternative to call prior to evaluating the density of an elliptical distribution.

Parameters
CPCson (-1, 1)
sdson (0, inf)
log_probthe log probability value to increment with the Jacobian
Returns
Covariance matrix for specified partial correlations.

Definition at line 23 of file read_cov_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::read_cov_matrix ( const Eigen::Array< T, Eigen::Dynamic, 1 > &  CPCs,
const Eigen::Array< T, Eigen::Dynamic, 1 > &  sds 
)

Builds a covariance matrix from CPCs and standard deviations.

Parameters
CPCsin (-1, 1)
sdsin (0, inf)

Definition at line 41 of file read_cov_matrix.hpp.

static void stan::math::recover_memory ( )
inlinestatic

Recover memory used for all variables for reuse.

Exceptions
std::logic_errorif empty_nested() returns false

Definition at line 18 of file recover_memory.hpp.

static void stan::math::recover_memory_nested ( )
inlinestatic

Recover only the memory used for the top nested call.

If there is nothing on the nested stack, then a std::logic_error exception is thrown.

Exceptions
std::logic_errorif empty_nested() returns true

Definition at line 20 of file recover_memory_nested.hpp.

template<typename T >
std::vector<T> stan::math::rep_array ( const T &  x,
int  n 
)
inline

Definition at line 13 of file rep_array.hpp.

template<typename T >
std::vector<std::vector<T> > stan::math::rep_array ( const T &  x,
int  m,
int  n 
)
inline

Definition at line 21 of file rep_array.hpp.

template<typename T >
std::vector<std::vector<std::vector<T> > > stan::math::rep_array ( const T &  x,
int  k,
int  m,
int  n 
)
inline

Definition at line 31 of file rep_array.hpp.

template<typename T >
Eigen::Matrix<typename boost::math::tools::promote_args<T>::type, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const T &  x,
int  m,
int  n 
)
inline

Definition at line 16 of file rep_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v,
int  n 
)
inline

Definition at line 26 of file rep_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::rep_matrix ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv,
int  m 
)
inline

Definition at line 36 of file rep_matrix.hpp.

template<typename T >
Eigen::Matrix<typename boost::math::tools::promote_args<T>::type, 1, Eigen::Dynamic> stan::math::rep_row_vector ( const T &  x,
int  m 
)
inline

Definition at line 15 of file rep_row_vector.hpp.

template<typename T >
Eigen::Matrix<typename boost::math::tools::promote_args<T>::type, Eigen::Dynamic, 1> stan::math::rep_vector ( const T &  x,
int  n 
)
inline

Definition at line 16 of file rep_vector.hpp.

template<typename T >
void stan::math::resize ( T &  x,
std::vector< size_t >  dims 
)
inline

Recursively resize the specified vector of vectors, which must bottom out at scalar values, Eigen vectors or Eigen matrices.

Parameters
xArray-like object to resize.
dimsNew dimensions.
Template Parameters
TType of object being resized.

Definition at line 63 of file resize.hpp.

template<typename T >
fvar<T> stan::math::rising_factorial ( const fvar< T > &  x,
const fvar< T > &  n 
)
inline

Definition at line 16 of file rising_factorial.hpp.

template<typename T >
fvar<T> stan::math::rising_factorial ( const fvar< T > &  x,
const double  n 
)
inline

Definition at line 28 of file rising_factorial.hpp.

template<typename T >
fvar<T> stan::math::rising_factorial ( const double  x,
const fvar< T > &  n 
)
inline

Definition at line 41 of file rising_factorial.hpp.

var stan::math::rising_factorial ( const var a,
const double &  b 
)
inline

Definition at line 54 of file rising_factorial.hpp.

template<typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::rising_factorial ( const T1  x,
const T2  n 
)
inline

\[ \mbox{rising\_factorial}(x, n) = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial n} = \begin{cases} \textrm{error} & \mbox{if } x \leq 0\\ \frac{\partial\, x^{(n)}}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN} \end{cases} \]

\[ x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)} \]

\[ \frac{\partial \, x^{(n)}}{\partial x} = x^{(n)}(\Psi(x+n)-\Psi(x)) \]

\[ \frac{\partial \, x^{(n)}}{\partial n} = (x)_n\Psi(x+n) \]

Definition at line 54 of file rising_factorial.hpp.

var stan::math::rising_factorial ( const var a,
const var b 
)
inline

Definition at line 59 of file rising_factorial.hpp.

var stan::math::rising_factorial ( const double &  a,
const var b 
)
inline

Definition at line 64 of file rising_factorial.hpp.

template<typename T >
fvar<T> stan::math::round ( const fvar< T > &  x)
inline

Definition at line 11 of file round.hpp.

var stan::math::round ( const var a)
inline

Returns the rounded form of the specified variable (C99).

See round() for the double-based version.

The derivative is zero everywhere but numbers half way between whole numbers, so for convenience the derivative is defined to be everywhere zero,

$\frac{d}{dx} \mbox{round}(x) = 0$.

\[ \mbox{round}(x) = \begin{cases} \lceil x \rceil & \mbox{if } x-\lfloor x\rfloor \geq 0.5 \\ \lfloor x \rfloor & \mbox{if } x-\lfloor x\rfloor < 0.5 \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{round}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aSpecified variable.
Returns
Rounded variable.

Definition at line 57 of file round.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::row ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m,
size_t  i 
)
inline

Return the specified row of the specified matrix, using start-at-1 indexing.

This is equivalent to calling m.row(i - 1) and assigning the resulting template expression to a row vector.

Template Parameters
TScalar value type for matrix.
Parameters
mMatrix.
iRow index (count from 1).
Returns
Specified row of the matrix.

Definition at line 25 of file row.hpp.

template<typename T , int R, int C>
int stan::math::rows ( const Eigen::Matrix< T, R, C > &  m)
inline

Return the number of rows in the specified matrix, vector, or row vector.

Template Parameters
TType of matrix entries.
RRow type of matrix.
CColumn type of matrix.
Parameters
[in]mInput matrix, vector, or row vector.
Returns
Number of rows.

Definition at line 20 of file rows.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 18 of file rows_dot_product.hpp.

template<int R1, int C1, int R2, int C2>
Eigen::Matrix<double, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Returns the dot product of the specified vectors.

Parameters
v1First vector.
v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorIf the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 22 of file rows_dot_product.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2>
boost::enable_if_c<boost::is_same<T1, var>::value || boost::is_same<T2, var>::value, Eigen::Matrix<var, R1, 1> >::type stan::math::rows_dot_product ( const Eigen::Matrix< T1, R1, C1 > &  v1,
const Eigen::Matrix< T2, R2, C2 > &  v2 
)
inline

Definition at line 25 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< fvar< T >, R2, C2 > &  v2 
)
inline

Definition at line 35 of file rows_dot_product.hpp.

template<typename T , int R1, int C1, int R2, int C2>
Eigen::Matrix<fvar<T>, R1, 1> stan::math::rows_dot_product ( const Eigen::Matrix< fvar< T >, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Definition at line 52 of file rows_dot_product.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, 1> stan::math::rows_dot_self ( const Eigen::Matrix< fvar< T >, R, C > &  x)
inline

Definition at line 15 of file rows_dot_self.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, 1> stan::math::rows_dot_self ( const Eigen::Matrix< T, R, C > &  x)
inline

Returns the dot product of each row of a matrix with itself.

Parameters
xMatrix.
Template Parameters
Tscalar type

Definition at line 16 of file rows_dot_self.hpp.

void stan::math::scaled_add ( std::vector< double > &  x,
const std::vector< double > &  y,
const double  lambda 
)
inline

Definition at line 11 of file scaled_add.hpp.

template<typename T_y , typename T_dof , typename T_scale >
return_type<T_y, T_dof, T_scale>::type stan::math::scaled_inv_chi_square_ccdf_log ( const T_y &  y,
const T_dof &  nu,
const T_scale &  s 
)

Definition at line 31 of file scaled_inv_chi_square_ccdf_log.hpp.

template<typename T_y , typename T_dof , typename T_scale >
return_type<T_y, T_dof, T_scale>::type stan::math::scaled_inv_chi_square_cdf ( const T_y &  y,
const T_dof &  nu,
const T_scale &  s 
)

The CDF of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter.

Parameters
yA scalar variable.
nuDegrees of freedom.
sScale parameter.
Exceptions
std::domain_errorif nu is not greater than 0
std::domain_errorif s is not greater than 0.
std::domain_errorif y is not greater than 0.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.

Definition at line 46 of file scaled_inv_chi_square_cdf.hpp.

template<typename T_y , typename T_dof , typename T_scale >
return_type<T_y, T_dof, T_scale>::type stan::math::scaled_inv_chi_square_cdf_log ( const T_y &  y,
const T_dof &  nu,
const T_scale &  s 
)

Definition at line 32 of file scaled_inv_chi_square_cdf_log.hpp.

template<bool propto, typename T_y , typename T_dof , typename T_scale >
return_type<T_y, T_dof, T_scale>::type stan::math::scaled_inv_chi_square_log ( const T_y &  y,
const T_dof &  nu,
const T_scale &  s 
)

The log of a scaled inverse chi-squared density for y with the specified degrees of freedom parameter and scale parameter.

\begin{eqnarray*} y &\sim& \mbox{\sf{Inv-}}\chi^2(\nu, s^2) \\ \log (p (y \, |\, \nu, s)) &=& \log \left( \frac{(\nu / 2)^{\nu / 2}}{\Gamma (\nu / 2)} s^\nu y^{- (\nu / 2 + 1)} \exp^{-\nu s^2 / (2y)} \right) \\ &=& \frac{\nu}{2} \log(\frac{\nu}{2}) - \log (\Gamma (\nu / 2)) + \nu \log(s) - (\frac{\nu}{2} + 1) \log(y) - \frac{\nu s^2}{2y} \\ & & \mathrm{ where } \; y > 0 \end{eqnarray*}

Parameters
yA scalar variable.
nuDegrees of freedom.
sScale parameter.
Exceptions
std::domain_errorif nu is not greater than 0
std::domain_errorif s is not greater than 0.
std::domain_errorif y is not greater than 0.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.

Definition at line 51 of file scaled_inv_chi_square_log.hpp.

template<typename T_y , typename T_dof , typename T_scale >
return_type<T_y, T_dof, T_scale>::type stan::math::scaled_inv_chi_square_log ( const T_y &  y,
const T_dof &  nu,
const T_scale &  s 
)
inline

Definition at line 173 of file scaled_inv_chi_square_log.hpp.

template<class RNG >
double stan::math::scaled_inv_chi_square_rng ( const double  nu,
const double  s,
RNG &  rng 
)
inline

Definition at line 29 of file scaled_inv_chi_square_rng.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::sd ( const std::vector< T > &  v)
inline

Returns the unbiased sample standard deviation of the coefficients in the specified column vector.

Parameters
vSpecified vector.
Returns
Sample variance of vector.

Definition at line 22 of file sd.hpp.

template<typename T , int R, int C>
boost::math::tools::promote_args<T>::type stan::math::sd ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the unbiased sample standard deviation of the coefficients in the specified vector, row vector, or matrix.

Parameters
mSpecified vector, row vector or matrix.
Returns
Sample variance.

Definition at line 37 of file sd.hpp.

var stan::math::sd ( const std::vector< var > &  v)

Return the sample standard deviation of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters
[in]va vector
Returns
sample standard deviation of specified vector

Definition at line 65 of file sd.hpp.

template<int R, int C>
var stan::math::sd ( const Eigen::Matrix< var, R, C > &  m)

Definition at line 82 of file sd.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::segment ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v,
size_t  i,
size_t  n 
)
inline

Return the specified number of elements as a vector starting from the specified element - 1 of the specified vector.

Definition at line 19 of file segment.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::segment ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  v,
size_t  i,
size_t  n 
)
inline

Definition at line 35 of file segment.hpp.

template<typename T >
std::vector<T> stan::math::segment ( const std::vector< T > &  sv,
size_t  i,
size_t  n 
)

Definition at line 52 of file segment.hpp.

static void stan::math::set_zero_all_adjoints ( )
static

Reset all adjoint values in the stack to zero.

Definition at line 14 of file set_zero_all_adjoints.hpp.

static void stan::math::set_zero_all_adjoints_nested ( )
static

Reset all adjoint values in the top nested portion of the stack to zero.

Definition at line 16 of file set_zero_all_adjoints_nested.hpp.

template<typename T >
int stan::math::sign ( const T &  z)
inline

Definition at line 9 of file sign.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y)

Return the simplex corresponding to the specified free vector.

A simplex is a vector containing values greater than or equal to 0 that sum to 1. A vector with (K-1) unconstrained values will produce a simplex of size K.

The transform is based on a centered stick-breaking process.

Parameters
yFree vector input of dimensionality K - 1.
Returns
Simplex of dimensionality K.
Template Parameters
TType of scalar.

Definition at line 30 of file simplex_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_constrain ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  y,
T &  lp 
)

Return the simplex corresponding to the specified free vector and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The simplex transform is defined through a centered stick-breaking process.

Parameters
yFree vector input of dimensionality K - 1.
lpLog probability reference to increment.
Returns
Simplex of dimensionality K.
Template Parameters
TType of scalar.

Definition at line 69 of file simplex_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::simplex_free ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x)

Return an unconstrained vector that when transformed produces the specified simplex.

It applies to a simplex of dimensionality K and produces an unconstrained vector of dimensionality (K-1).

The simplex transform is defined through a centered stick-breaking process.

Parameters
xSimplex of dimensionality K.
Returns
Free vector of dimensionality (K-1) that transfroms to the simplex.
Template Parameters
TType of scalar.
Exceptions
std::domain_errorif x is not a valid simplex

Definition at line 30 of file simplex_free.hpp.

template<typename T >
fvar<T> stan::math::sin ( const fvar< T > &  x)
inline

Definition at line 14 of file sin.hpp.

var stan::math::sin ( const var a)
inline

Return the sine of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sin x = \cos x$.

\[ \mbox{sin}(x) = \begin{cases} \sin(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sin}(x)}{\partial x} = \begin{cases} \cos(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable for radians of angle.
Returns
Sine of variable.

Definition at line 49 of file sin.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::singular_values ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)

Return the vector of the singular values of the specified matrix in decreasing order of magnitude.

See the documentation for svd() for information on the signular values.

Parameters
mSpecified matrix.
Returns
Singular values of the matrix.

Definition at line 21 of file singular_values.hpp.

template<typename T >
fvar<T> stan::math::sinh ( const fvar< T > &  x)
inline

Definition at line 14 of file sinh.hpp.

var stan::math::sinh ( const var a)
inline

Return the hyperbolic sine of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sinh x = \cosh x$.

\[ \mbox{sinh}(x) = \begin{cases} \sinh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sinh}(x)}{\partial x} = \begin{cases} \cosh(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable.
Returns
Hyperbolic sine of variable.

Definition at line 49 of file sinh.hpp.

template<typename T >
int stan::math::size ( const std::vector< T > &  x)
inline

Return the size of the specified standard vector.

Template Parameters
TType of elements.
Parameters
[in]xInput vector.
Returns
Size of input vector.

Definition at line 17 of file size.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::skew_normal_ccdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  alpha 
)

Definition at line 26 of file skew_normal_ccdf_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::skew_normal_cdf ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  alpha 
)

Definition at line 26 of file skew_normal_cdf.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::skew_normal_cdf_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  alpha 
)

Definition at line 26 of file skew_normal_cdf_log.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::skew_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  alpha 
)

Definition at line 27 of file skew_normal_log.hpp.

template<typename T_y , typename T_loc , typename T_scale , typename T_shape >
return_type<T_y, T_loc, T_scale, T_shape>::type stan::math::skew_normal_log ( const T_y &  y,
const T_loc &  mu,
const T_scale &  sigma,
const T_shape &  alpha 
)
inline

Definition at line 146 of file skew_normal_log.hpp.

template<class RNG >
double stan::math::skew_normal_rng ( const double  mu,
const double  sigma,
const double  alpha,
RNG &  rng 
)
inline

Definition at line 24 of file skew_normal_rng.hpp.

template<typename T >
Eigen::Matrix<fvar<T>, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< fvar< T >, Eigen::Dynamic, 1 > &  alpha)
inline

Definition at line 14 of file softmax.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v)
inline

Return the softmax of the specified vector.

$ \mbox{softmax}(y) = \frac{\exp(y)} {\sum_{k=1}^K \exp(y_k)}, $

The entries in the Jacobian of the softmax function are given by $ \begin{array}{l} \displaystyle \frac{\partial}{\partial y_m} \mbox{softmax}(y)[k] \\[8pt] \displaystyle \mbox{ } \ \ \ = \left\{ \begin{array}{ll} \mbox{softmax}(y)[k] - \mbox{softmax}(y)[k] \times \mbox{softmax}(y)[m] & \mbox{ if } m = k, \mbox{ and} \\[6pt] \mbox{softmax}(y)[k] * \mbox{softmax}(y)[m] & \mbox{ if } m \neq k. \end{array} \right. \end{array} $

Template Parameters
TScalar type of values in vector.
Parameters
[in]vVector to transform.
Returns
Unit simplex result of the softmax transform of the vector.

Definition at line 46 of file softmax.hpp.

Eigen::Matrix<var, Eigen::Dynamic, 1> stan::math::softmax ( const Eigen::Matrix< var, Eigen::Dynamic, 1 > &  alpha)
inline

Return the softmax of the specified Eigen vector.

Softmax is guaranteed to return a simplex.

The gradient calculations are unfolded.

Parameters
alphaUnconstrained input vector.
Returns
Softmax of the input.
Exceptions
std::domain_errorIf the input vector is size 0.

Definition at line 59 of file softmax.hpp.

template<typename T >
std::vector< fvar<T> > stan::math::sort_asc ( std::vector< fvar< T > >  xs)
inline

Definition at line 17 of file sort_asc.hpp.

template<typename T >
std::vector<T> stan::math::sort_asc ( std::vector< T >  xs)
inline

Return the specified standard vector in ascending order.

Parameters
xsStandard vector to order.
Returns
Standard vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 20 of file sort.hpp.

std::vector<var> stan::math::sort_asc ( std::vector< var xs)
inline

Return the specified standard vector in ascending order with gradients kept.

Parameters
xsStandard vector to order.
Returns
Standard vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 21 of file sort_asc.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::sort_asc ( Eigen::Matrix< fvar< T >, R, C >  xs)
inline

Definition at line 25 of file sort_asc.hpp.

template<int R, int C>
Eigen::Matrix<var, R, C> stan::math::sort_asc ( Eigen::Matrix< var, R, C >  xs)
inline

Return the specified eigen vector in ascending order with gradients kept.

Parameters
xsEigen vector to order.
Returns
Eigen vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 35 of file sort_asc.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::sort_asc ( Eigen::Matrix< T, R, C >  xs)
inline

Return the specified eigen vector in ascending order.

Parameters
xsEigen vector to order.
Returns
Eigen vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 46 of file sort.hpp.

template<typename T >
std::vector< fvar<T> > stan::math::sort_desc ( std::vector< fvar< T > >  xs)
inline

Definition at line 17 of file sort_desc.hpp.

std::vector<var> stan::math::sort_desc ( std::vector< var xs)
inline

Return the specified standard vector in descending order with gradients kept.

Parameters
xsStandard vector to order.
Returns
Standard vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 21 of file sort_desc.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::sort_desc ( Eigen::Matrix< fvar< T >, R, C >  xs)
inline

Definition at line 25 of file sort_desc.hpp.

template<typename T >
std::vector<T> stan::math::sort_desc ( std::vector< T >  xs)
inline

Return the specified standard vector in descending order.

Parameters
xsStandard vector to order.
Returns
Standard vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 33 of file sort.hpp.

template<int R, int C>
Eigen::Matrix<var, R, C> stan::math::sort_desc ( Eigen::Matrix< var, R, C >  xs)
inline

Return the specified eigen vector in descending order with gradients kept.

Parameters
xsEigen vector to order.
Returns
Eigen vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 35 of file sort_desc.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::sort_desc ( Eigen::Matrix< T, R, C >  xs)
inline

Return the specified eigen vector in descending order.

Parameters
xsEigen vector to order.
Returns
Eigen vector ordered.
Template Parameters
TType of elements of the vector.

Definition at line 60 of file sort.hpp.

template<typename C >
std::vector<int> stan::math::sort_indices_asc ( const C &  xs)

Return a sorted copy of the argument container in ascending order.

Template Parameters
Ctype of container
Parameters
xsContainer to sort
Returns
sorted version of container

Definition at line 23 of file sort_indices_asc.hpp.

template<typename C >
std::vector<int> stan::math::sort_indices_desc ( const C &  xs)

Return a sorted copy of the argument container in ascending order.

Template Parameters
Ctype of container
Parameters
xsContainer to sort
Returns
sorted version of container

Definition at line 23 of file sort_indices_desc.hpp.

template<typename T >
fvar<T> stan::math::sqrt ( const fvar< T > &  x)
inline

Definition at line 15 of file sqrt.hpp.

var stan::math::sqrt ( const var a)
inline

Return the square root of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \sqrt{x} = \frac{1}{2 \sqrt{x}}$.

\[ \mbox{sqrt}(x) = \begin{cases} \textrm{NaN} & x < 0 \\ \sqrt{x} & \mbox{if } x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{sqrt}(x)}{\partial x} = \begin{cases} \textrm{NaN} & x < 0 \\ \frac{1}{2\sqrt{x}} & x\geq 0\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable whose square root is taken.
Returns
Square root of variable.

Definition at line 50 of file sqrt.hpp.

double stan::math::sqrt2 ( )
inline

Return the square root of two.

Returns
Square root of two.

Definition at line 104 of file constants.hpp.

template<typename T >
fvar<T> stan::math::square ( const fvar< T > &  x)
inline

Definition at line 15 of file square.hpp.

template<typename T >
T stan::math::square ( const T  x)
inline

Return the square of the specified argument.

$\mbox{square}(x) = x^2$.

The implementation of square(x) is just x * x. Given this, this method is mainly useful in cases where x is not a simple primitive type, particularly when it is an auto-dif type.

Parameters
xInput to square.
Returns
Square of input.
Template Parameters
TType of scalar.

Definition at line 22 of file square.hpp.

var stan::math::square ( const var x)
inline

Return the square of the input variable.

Using square(x) is more efficient than using x * x.

\[ \mbox{square}(x) = \begin{cases} x^2 & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{square}(x)}{\partial x} = \begin{cases} 2x & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
xVariable to square.
Returns
Square of variable.

Definition at line 46 of file square.hpp.

template<int R1, int C1, int R2, int C2, typename T1 , typename T2 >
boost::math::tools::promote_args<T1, T2>::type stan::math::squared_distance ( const Eigen::Matrix< T1, R1, C1 > &  v1,
const Eigen::Matrix< T2, R2, C2 > &  v2 
)
inline

Returns the squared distance between the specified vectors.

Parameters
v1First vector.
v2Second vector.
Returns
Dot product of the vectors.
Exceptions
std::domain_errorIf the vectors are not the same size or if they are both not vector dimensioned.

Definition at line 22 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>
var stan::math::squared_distance ( const Eigen::Matrix< var, R1, C1 > &  v1,
const Eigen::Matrix< var, R2, C2 > &  v2 
)
inline

Definition at line 112 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>
var stan::math::squared_distance ( const Eigen::Matrix< var, R1, C1 > &  v1,
const Eigen::Matrix< double, R2, C2 > &  v2 
)
inline

Definition at line 122 of file squared_distance.hpp.

template<int R1, int C1, int R2, int C2>
var stan::math::squared_distance ( const Eigen::Matrix< double, R1, C1 > &  v1,
const Eigen::Matrix< var, R2, C2 > &  v2 
)
inline

Definition at line 132 of file squared_distance.hpp.

void stan::math::stan_print ( std::ostream *  o,
const var x 
)

Definition at line 10 of file stan_print.hpp.

template<typename T >
void stan::math::stan_print ( std::ostream *  o,
const T &  x 
)

Definition at line 12 of file stan_print.hpp.

template<typename T >
void stan::math::stan_print ( std::ostream *  o,
const std::vector< T > &  x 
)

Definition at line 17 of file stan_print.hpp.

template<typename T >
void stan::math::stan_print ( std::ostream *  o,
const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x 
)

Definition at line 27 of file stan_print.hpp.

template<typename T >
void stan::math::stan_print ( std::ostream *  o,
const Eigen::Matrix< T, 1, Eigen::Dynamic > &  x 
)

Definition at line 38 of file stan_print.hpp.

template<typename T >
void stan::math::stan_print ( std::ostream *  o,
const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  x 
)

Definition at line 49 of file stan_print.hpp.

static void stan::math::start_nested ( )
inlinestatic

Record the current position so that recover_memory_nested() can find it.

Definition at line 13 of file start_nested.hpp.

var stan::math::step ( const stan::math::var a)
inline

Return the step, or heaviside, function applied to the specified variable (stan).

See stan::math::step() for the double-based version.

The derivative of the step function is zero everywhere but at 0, so for convenience, it is taken to be everywhere zero,

$\mbox{step}(x) = 0$.

Parameters
aVariable argument.
Returns
The constant variable with value 1.0 if the argument's value is greater than or equal to 0.0, and value 0.0 otherwise.

Definition at line 25 of file step.hpp.

template<typename T >
int stan::math::step ( const T  y)
inline

The step, or Heaviside, function.

The function is defined by

step(y) = (y < 0.0) ? 0 : 1.

\[ \mbox{step}(x) = \begin{cases} 0 & \mbox{if } x \leq 0 \\ 1 & \mbox{if } x > 0 \\[6pt] 0 & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
yScalar argument.
Returns
1 if the specified argument is greater than or equal to 0.0, and 0 otherwise.

Definition at line 29 of file step.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::student_t_ccdf_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 31 of file student_t_ccdf_log.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::student_t_cdf ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 31 of file student_t_cdf.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::student_t_cdf_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  sigma 
)

Definition at line 31 of file student_t_cdf_log.hpp.

template<bool propto, typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::student_t_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  sigma 
)

The log of the Student-t density for the given y, nu, mean, and scale parameter.

The scale parameter must be greater than 0.

\begin{eqnarray*} y &\sim& t_{\nu} (\mu, \sigma^2) \\ \log (p (y \, |\, \nu, \mu, \sigma) ) &=& \log \left( \frac{\Gamma((\nu + 1) /2)} {\Gamma(\nu/2)\sqrt{\nu \pi} \sigma} \left( 1 + \frac{1}{\nu} (\frac{y - \mu}{\sigma})^2 \right)^{-(\nu + 1)/2} \right) \\ &=& \log( \Gamma( (\nu+1)/2 )) - \log (\Gamma (\nu/2) - \frac{1}{2} \log(\nu \pi) - \log(\sigma) -\frac{\nu + 1}{2} \log (1 + \frac{1}{\nu} (\frac{y - \mu}{\sigma})^2) \end{eqnarray*}

Parameters
yA scalar variable.
nuDegrees of freedom.
muThe mean of the Student-t distribution.
sigmaThe scale parameter of the Student-t distribution.
Returns
The log of the Student-t density at y.
Exceptions
std::domain_errorif sigma is not greater than 0.
std::domain_errorif nu is not greater than 0.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.
T_locType of location.
T_scaleType of scale.

Definition at line 56 of file student_t_log.hpp.

template<typename T_y , typename T_dof , typename T_loc , typename T_scale >
return_type<T_y, T_dof, T_loc, T_scale>::type stan::math::student_t_log ( const T_y &  y,
const T_dof &  nu,
const T_loc &  mu,
const T_scale &  sigma 
)
inline

Definition at line 220 of file student_t_log.hpp.

template<class RNG >
double stan::math::student_t_rng ( const double  nu,
const double  mu,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 29 of file student_t_rng.hpp.

void stan::math::sub ( std::vector< double > &  x,
std::vector< double > &  y,
std::vector< double > &  result 
)
inline

Definition at line 10 of file sub.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::sub_col ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m,
size_t  i,
size_t  j,
size_t  nrows 
)
inline

Return a nrows x 1 subcolumn starting at (i-1, j-1).

Parameters
mMatrix
iStarting row + 1
jStarting column + 1
nrowsNumber of rows in block

Definition at line 22 of file sub_col.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::sub_row ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m,
size_t  i,
size_t  j,
size_t  ncols 
)
inline

Return a 1 x nrows subrow starting at (i-1, j-1).

Parameters
mMatrix
iStarting row + 1
jStarting column + 1
ncolsNumber of columns in block

Definition at line 23 of file sub_row.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::subtract ( const Eigen::Matrix< T1, R, C > &  m1,
const Eigen::Matrix< T2, R, C > &  m2 
)
inline

Return the result of subtracting the second specified matrix from the first specified matrix.

The return scalar type is the promotion of the input types.

Template Parameters
T1Scalar type of first matrix.
T2Scalar type of second matrix.
RRow type of matrices.
CColumn type of matrices.
Parameters
m1First matrix.
m2Second matrix.
Returns
Difference between first matrix and second matrix.

Definition at line 27 of file subtract.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::subtract ( const T1 &  c,
const Eigen::Matrix< T2, R, C > &  m 
)
inline

Definition at line 43 of file subtract.hpp.

template<typename T1 , typename T2 , int R, int C>
Eigen::Matrix<typename boost::math::tools::promote_args<T1, T2>::type, R, C> stan::math::subtract ( const Eigen::Matrix< T1, R, C > &  m,
const T2 &  c 
)
inline

Definition at line 56 of file subtract.hpp.

template<typename T >
T stan::math::sum ( const std::vector< T > &  xs)
inline

Return the sum of the values in the specified standard vector.

Template Parameters
TType of elements summed.
Parameters
xsStandard vector to sum.
Returns
Sum of elements.

Definition at line 18 of file sum.hpp.

template<typename T >
fvar<T> stan::math::sum ( const std::vector< fvar< T > > &  m)
inline

Return the sum of the entries of the specified standard vector.

Template Parameters
TType of vector entries.
Parameters
mVector.
Returns
Sum of vector entries.

Definition at line 20 of file sum.hpp.

template<typename T , int R, int C>
fvar<T> stan::math::sum ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Return the sum of the entries of the specified matrix.

Template Parameters
TType of matrix entries.
RRow type of matrix.
CColumn type of matrix.
Parameters
mMatrix.
Returns
Sum of matrix entries.

Definition at line 21 of file sum.hpp.

template<typename T , int R, int C>
double stan::math::sum ( const Eigen::Matrix< T, R, C > &  v)
inline

Returns the sum of the coefficients of the specified column vector.

Template Parameters
TType of elements in matrix.
RRow type of matrix.
CColumn type of matrix.
Parameters
vSpecified vector.
Returns
Sum of coefficients of vector.

Definition at line 22 of file sum.hpp.

template<int R, int C>
var stan::math::sum ( const Eigen::Matrix< var, R, C > &  m)
inline

Returns the sum of the coefficients of the specified matrix, column vector or row vector.

Template Parameters
RRow type for matrix.
CColumn type for matrix.
Parameters
mSpecified matrix or vector.
Returns
Sum of coefficients of matrix.

Definition at line 50 of file sum.hpp.

var stan::math::sum ( const std::vector< var > &  m)
inline

Returns the sum of the entries of the specified vector.

Parameters
mVector.
Returns
Sum of vector entries.

Definition at line 53 of file sum.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::tail ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  v,
size_t  n 
)
inline

Return the specified number of elements as a vector from the back of the specified vector.

Definition at line 23 of file tail.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::tail ( const Eigen::Matrix< T, 1, Eigen::Dynamic > &  rv,
size_t  n 
)
inline

Return the specified number of elements as a row vector from the back of the specified row vector.

Definition at line 38 of file tail.hpp.

template<typename T >
std::vector<T> stan::math::tail ( const std::vector< T > &  sv,
size_t  n 
)

Definition at line 46 of file tail.hpp.

template<typename T >
fvar<T> stan::math::tan ( const fvar< T > &  x)
inline

Definition at line 14 of file tan.hpp.

var stan::math::tan ( const var a)
inline

Return the tangent of a radian-scaled variable (cmath).

The derivative is defined by

$\frac{d}{dx} \tan x = \sec^2 x$.

\[ \mbox{tan}(x) = \begin{cases} \tan(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tan}(x)}{\partial x} = \begin{cases} \sec^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable for radians of angle.
Returns
Tangent of variable.

Definition at line 49 of file tan.hpp.

template<typename T >
fvar<T> stan::math::tanh ( const fvar< T > &  x)
inline

Definition at line 14 of file tanh.hpp.

var stan::math::tanh ( const var a)
inline

Return the hyperbolic tangent of the specified variable (cmath).

The derivative is defined by

$\frac{d}{dx} \tanh x = \frac{1}{\cosh^2 x}$.

\[ \mbox{tanh}(x) = \begin{cases} \tanh(x) & \mbox{if } -\infty\leq x \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tanh}(x)}{\partial x} = \begin{cases} \mbox{sech}^2(x) & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aVariable.
Returns
Hyperbolic tangent of variable.

Definition at line 50 of file tanh.hpp.

matrix_d stan::math::tcrossprod ( const matrix_d M)
inline

Returns the result of post-multiplying a matrix by its own transpose.

Parameters
MMatrix to multiply.
Returns
M times its transpose.

Definition at line 17 of file tcrossprod.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, R> stan::math::tcrossprod ( const Eigen::Matrix< fvar< T >, R, C > &  m)
inline

Definition at line 17 of file tcrossprod.hpp.

matrix_v stan::math::tcrossprod ( const matrix_v M)
inline

Returns the result of post-multiplying a matrix by its own transpose.

Parameters
MMatrix to multiply.
Returns
M times its transpose.

Definition at line 25 of file tcrossprod.hpp.

template<typename T >
fvar<T> stan::math::tgamma ( const fvar< T > &  x)
inline

Definition at line 15 of file tgamma.hpp.

var stan::math::tgamma ( const stan::math::var a)
inline

Return the Gamma function applied to the specified variable (C99).

See boost::math::tgamma() for the double-based version.

The derivative with respect to the argument is

$\frac{d}{dx} \Gamma(x) = \Gamma(x) \Psi^{(0)}(x)$

where $\Psi^{(0)}(x)$ is the digamma function.

See boost::math::digamma() for the double-based version.

\[ \mbox{tgamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Gamma(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{tgamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Gamma(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Gamma(x)=\int_0^{\infty} u^{x - 1} \exp(-u) \, du \]

\[ \frac{\partial \, \Gamma(x)}{\partial x} = \Gamma(x)\Psi(x) \]

Parameters
aArgument to function.
Returns
The Gamma function applied to the specified argument.

Definition at line 65 of file tgamma.hpp.

template<typename T , int R, int C>
std::vector<T> stan::math::to_array_1d ( const Eigen::Matrix< T, R, C > &  matrix)
inline

Definition at line 15 of file to_array_1d.hpp.

template<typename T >
std::vector<T> stan::math::to_array_1d ( const std::vector< T > &  x)
inline

Definition at line 29 of file to_array_1d.hpp.

template<typename T >
std::vector<typename scalar_type<T>::type> stan::math::to_array_1d ( const std::vector< std::vector< T > > &  x)
inline

Definition at line 36 of file to_array_1d.hpp.

template<typename T >
std::vector< std::vector<T> > stan::math::to_array_2d ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  matrix)
inline

Definition at line 13 of file to_array_2d.hpp.

template<typename T >
fvar<T> stan::math::to_fvar ( const T &  x)
inline

Definition at line 16 of file to_fvar.hpp.

template<typename T >
fvar<T> stan::math::to_fvar ( const fvar< T > &  x)
inline

Definition at line 23 of file to_fvar.hpp.

template<int R, int C, typename T >
Eigen::Matrix<T, R, C> stan::math::to_fvar ( const Eigen::Matrix< T, R, C > &  m)
inline

Definition at line 31 of file to_fvar.hpp.

template<int R, int C>
Eigen::Matrix<fvar<double>, R, C> stan::math::to_fvar ( const Eigen::Matrix< double, R, C > &  m)
inline

Definition at line 38 of file to_fvar.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::to_fvar ( const Eigen::Matrix< T, R, C > &  val,
const Eigen::Matrix< T, R, C > &  deriv 
)
inline

Definition at line 48 of file to_fvar.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( Eigen::Matrix< T, R, C >  matrix)
inline

Definition at line 16 of file to_matrix.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< std::vector< T > > &  vec)
inline

Definition at line 23 of file to_matrix.hpp.

Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::to_matrix ( const std::vector< std::vector< int > > &  vec)
inline

Definition at line 40 of file to_matrix.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::to_row_vector ( const Eigen::Matrix< T, R, C > &  matrix)
inline

Definition at line 16 of file to_row_vector.hpp.

template<typename T >
Eigen::Matrix<T, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< T > &  vec)
inline

Definition at line 24 of file to_row_vector.hpp.

Eigen::Matrix<double, 1, Eigen::Dynamic> stan::math::to_row_vector ( const std::vector< int > &  vec)
inline

Definition at line 30 of file to_row_vector.hpp.

var stan::math::to_var ( const double &  x)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]xA scalar value
Returns
An automatic differentiation variable with the input value.

Definition at line 21 of file to_var.hpp.

var stan::math::to_var ( const var x)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]xAn automatic differentiation variable.
Returns
An automatic differentiation variable with the input value.

Definition at line 32 of file to_var.hpp.

matrix_v stan::math::to_var ( const stan::math::matrix_d m)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]mA Matrix with scalars
Returns
A Matrix with automatic differentiation variables

Definition at line 43 of file to_var.hpp.

matrix_v stan::math::to_var ( const matrix_v m)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]mA Matrix with automatic differentiation variables.
Returns
A Matrix with automatic differentiation variables.

Definition at line 58 of file to_var.hpp.

vector_v stan::math::to_var ( const stan::math::vector_d v)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]vA Vector of scalars
Returns
A Vector of automatic differentiation variables with values of v

Definition at line 70 of file to_var.hpp.

vector_v stan::math::to_var ( const vector_v v)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]vA Vector of automatic differentiation variables
Returns
A Vector of automatic differentiation variables with values of v

Definition at line 85 of file to_var.hpp.

row_vector_v stan::math::to_var ( const stan::math::row_vector_d rv)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]rvA row vector of scalars
Returns
A row vector of automatic differentation variables with values of rv.

Definition at line 97 of file to_var.hpp.

row_vector_v stan::math::to_var ( const row_vector_v rv)
inline

Converts argument to an automatic differentiation variable.

Returns a stan::math::var variable with the input value.

Parameters
[in]rvA row vector with automatic differentiation variables
Returns
A row vector with automatic differentiation variables with values of rv.

Definition at line 112 of file to_var.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::to_vector ( const Eigen::Matrix< T, R, C > &  matrix)
inline

Definition at line 16 of file to_vector.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< T > &  vec)
inline

Definition at line 24 of file to_vector.hpp.

Eigen::Matrix<double, Eigen::Dynamic, 1> stan::math::to_vector ( const std::vector< int > &  vec)
inline

Definition at line 30 of file to_vector.hpp.

template<typename T >
T stan::math::trace ( const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &  m)
inline

Returns the trace of the specified matrix.

The trace is defined as the sum of the elements on the diagonal. The matrix is not required to be square. Returns 0 if matrix is empty.

Parameters
[in]mSpecified matrix.
Returns
Trace of the matrix.

Definition at line 20 of file trace.hpp.

template<typename T >
T stan::math::trace ( const T &  m)
inline

Definition at line 26 of file trace.hpp.

template<typename T1 , int R1, int C1, typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c<stan::is_var<T1>::value || stan::is_var<T2>::value || stan::is_var<T3>::value, var>::type stan::math::trace_gen_inv_quad_form_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  D,
const stan::math::LDLT_factor< T2, R2, C2 > &  A,
const Eigen::Matrix< T3, R3, C3 > &  B 
)
inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(D B^T A^-1 B) where D is a square matrix and the LDLT_factor of A is provided.

Definition at line 27 of file trace_gen_inv_quad_form_ldlt.hpp.

template<typename T1 , typename T2 , typename T3 , int R1, int C1, int R2, int C2, int R3, int C3>
boost::enable_if_c<!stan::is_var<T1>::value && !stan::is_var<T2>::value && !stan::is_var<T3>::value, typename boost::math::tools::promote_args<T1, T2, T3>::type>::type stan::math::trace_gen_inv_quad_form_ldlt ( const Eigen::Matrix< T1, R1, C1 > &  D,
const stan::math::LDLT_factor< T2, R2, C2 > &  A,
const Eigen::Matrix< T3, R3, C3 > &  B 
)
inline

Definition at line 30 of file trace_gen_inv_quad_form_ldlt.hpp.

template<int RD, int CD, int RA, int CA, int RB, int CB, typename T >
fvar<T> stan::math::trace_gen_quad_form ( const Eigen::Matrix< fvar< T >, RD, CD > &  D,
const Eigen::Matrix< fvar< T >, RA, CA > &  A,
const Eigen::Matrix< fvar< T >, RB, CB > &  B 
)
inline

Definition at line 15 of file trace_gen_quad_form.hpp.

template<int RD, int CD, int RA, int CA, int RB, int CB>
double stan::math::trace_gen_quad_form ( const Eigen::Matrix< double, RD, CD > &  D,
const Eigen::Matrix< double, RA, CA > &  A,
const Eigen::Matrix< double, RB, CB > &  B 
)
inline

Compute trace(D B^T A B).

Definition at line 17 of file trace_gen_quad_form.hpp.

template<typename TD , int RD, int CD, typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same<TD, var>::value || boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, var >::type stan::math::trace_gen_quad_form ( const Eigen::Matrix< TD, RD, CD > &  D,
const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, CB > &  B 
)
inline

Definition at line 116 of file trace_gen_quad_form.hpp.

template<typename T1 , typename T2 , int R2, int C2, int R3, int C3>
boost::enable_if_c<!stan::is_var<T1>::value && !stan::is_var<T2>::value, typename boost::math::tools::promote_args<T1, T2>::type>::type stan::math::trace_inv_quad_form_ldlt ( const stan::math::LDLT_factor< T1, R2, C2 > &  A,
const Eigen::Matrix< T2, R3, C3 > &  B 
)
inline

Definition at line 27 of file trace_inv_quad_form_ldlt.hpp.

template<typename T2 , int R2, int C2, typename T3 , int R3, int C3>
boost::enable_if_c<stan::is_var<T2>::value || stan::is_var<T3>::value, var>::type stan::math::trace_inv_quad_form_ldlt ( const stan::math::LDLT_factor< T2, R2, C2 > &  A,
const Eigen::Matrix< T3, R3, C3 > &  B 
)
inline

Compute the trace of an inverse quadratic form.

I.E., this computes trace(B^T A^-1 B) where the LDLT_factor of A is provided.

Definition at line 177 of file trace_inv_quad_form_ldlt.hpp.

template<int RA, int CA, int RB, int CB>
double stan::math::trace_quad_form ( const Eigen::Matrix< double, RA, CA > &  A,
const Eigen::Matrix< double, RB, CB > &  B 
)
inline

Compute trace(B^T A B).

Definition at line 17 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB, typename T >
fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< fvar< T >, RA, CA > &  A,
const Eigen::Matrix< fvar< T >, RB, CB > &  B 
)
inline

Definition at line 18 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB, typename T >
fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< fvar< T >, RA, CA > &  A,
const Eigen::Matrix< double, RB, CB > &  B 
)
inline

Definition at line 30 of file trace_quad_form.hpp.

template<int RA, int CA, int RB, int CB, typename T >
fvar<T> stan::math::trace_quad_form ( const Eigen::Matrix< double, RA, CA > &  A,
const Eigen::Matrix< fvar< T >, RB, CB > &  B 
)
inline

Definition at line 42 of file trace_quad_form.hpp.

template<typename TA , int RA, int CA, typename TB , int RB, int CB>
boost::enable_if_c< boost::is_same<TA, var>::value || boost::is_same<TB, var>::value, var >::type stan::math::trace_quad_form ( const Eigen::Matrix< TA, RA, CA > &  A,
const Eigen::Matrix< TB, RB, CB > &  B 
)
inline

Definition at line 98 of file trace_quad_form.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, C, R> stan::math::transpose ( const Eigen::Matrix< T, R, C > &  m)
inline

Definition at line 12 of file transpose.hpp.

template<typename T >
T stan::math::trigamma ( x)
inline

\[ \mbox{trigamma}(x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \Psi_1(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trigamma}(x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \Psi_1(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \Psi_1(x)=\sum_{n=0}^\infty \frac{1}{(x+n)^2} \]

\[ \frac{\partial \, \Psi_1(x)}{\partial x} = -2\sum_{n=0}^\infty \frac{1}{(x+n)^3} \]

Definition at line 50 of file trigamma.hpp.

template<typename T >
fvar<T> stan::math::trunc ( const fvar< T > &  x)
inline

Definition at line 12 of file trunc.hpp.

var stan::math::trunc ( const var a)
inline

Returns the truncatation of the specified variable (C99).

See trunc() for the double-based version.

The derivative is zero everywhere but at integer values, so for convenience the derivative is defined to be everywhere zero,

$\frac{d}{dx} \mbox{trunc}(x) = 0$.

\[ \mbox{trunc}(x) = \begin{cases} \lfloor x \rfloor & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

\[ \frac{\partial\, \mbox{trunc}(x)}{\partial x} = \begin{cases} 0 & \mbox{if } -\infty\leq x\leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases} \]

Parameters
aSpecified variable.
Returns
Truncation of the variable.

Definition at line 60 of file trunc.hpp.

template<typename T , typename TU >
boost::math::tools::promote_args<T, TU>::type stan::math::ub_constrain ( const T  x,
const TU  ub 
)
inline

Return the upper-bounded value for the specified unconstrained scalar and upper bound.

The transform is

$f(x) = U - \exp(x)$

where $U$ is the upper bound.

If the upper bound is positive infinity, this function reduces to identity_constrain(x).

Parameters
xFree scalar.
ubUpper bound.
Returns
Transformed scalar with specified upper bound.
Template Parameters
TType of scalar.
TUType of upper bound.

Definition at line 37 of file ub_constrain.hpp.

template<typename T , typename TU >
boost::math::tools::promote_args<T, TU>::type stan::math::ub_constrain ( const T  x,
const TU  ub,
T &  lp 
)
inline

Return the upper-bounded value for the specified unconstrained scalar and upper bound and increment the specified log probability reference with the log absolute Jacobian determinant of the transform.

The transform is as specified for ub_constrain(T, double). The log absolute Jacobian determinant is

$ \log | \frac{d}{dx} -\mbox{exp}(x) + U | = \log | -\mbox{exp}(x) + 0 | = x$.

If the upper bound is positive infinity, this function reduces to identity_constrain(x, lp).

Parameters
xFree scalar.
ubUpper bound.
lpLog probability reference.
Returns
Transformed scalar with specified upper bound.
Template Parameters
TType of scalar.
TUType of upper bound.

Definition at line 70 of file ub_constrain.hpp.

template<typename T , typename TU >
boost::math::tools::promote_args<T, TU>::type stan::math::ub_free ( const T  y,
const TU  ub 
)
inline

Return the free scalar that corresponds to the specified upper-bounded value with respect to the specified upper bound.

The transform is the reverse of the ub_constrain(T, double) transform,

$f^{-1}(y) = \log -(y - U)$

where $U$ is the upper bound.

If the upper bound is positive infinity, this function reduces to identity_free(y).

Parameters
yUpper-bounded scalar.
ubUpper bound.
Returns
Free scalar corresponding to upper-bounded scalar.
Template Parameters
TType of scalar.
TUType of upper bound.
Exceptions
std::invalid_argumentif y is greater than the upper bound.

Definition at line 39 of file ub_free.hpp.

template<typename T_y , typename T_low , typename T_high >
return_type<T_y, T_low, T_high>::type stan::math::uniform_ccdf_log ( const T_y &  y,
const T_low &  alpha,
const T_high &  beta 
)

Definition at line 23 of file uniform_ccdf_log.hpp.

template<typename T_y , typename T_low , typename T_high >
return_type<T_y, T_low, T_high>::type stan::math::uniform_cdf ( const T_y &  y,
const T_low &  alpha,
const T_high &  beta 
)

Definition at line 22 of file uniform_cdf.hpp.

template<typename T_y , typename T_low , typename T_high >
return_type<T_y, T_low, T_high>::type stan::math::uniform_cdf_log ( const T_y &  y,
const T_low &  alpha,
const T_high &  beta 
)

Definition at line 23 of file uniform_cdf_log.hpp.

template<bool propto, typename T_y , typename T_low , typename T_high >
return_type<T_y, T_low, T_high>::type stan::math::uniform_log ( const T_y &  y,
const T_low &  alpha,
const T_high &  beta 
)

The log of a uniform density for the given y, lower, and upper bound.

\begin{eqnarray*} y &\sim& \mbox{\sf{U}}(\alpha, \beta) \\ \log (p (y \, |\, \alpha, \beta)) &=& \log \left( \frac{1}{\beta-\alpha} \right) \\ &=& \log (1) - \log (\beta - \alpha) \\ &=& -\log (\beta - \alpha) \\ & & \mathrm{ where } \; y \in [\alpha, \beta], \log(0) \; \mathrm{otherwise} \end{eqnarray*}

Parameters
yA scalar variable.
alphaLower bound.
betaUpper bound.
Exceptions
std::invalid_argumentif the lower bound is greater than or equal to the lower bound
Template Parameters
T_yType of scalar.
T_lowType of lower bound.
T_highType of upper bound.

Definition at line 46 of file uniform_log.hpp.

template<typename T_y , typename T_low , typename T_high >
return_type<T_y, T_low, T_high>::type stan::math::uniform_log ( const T_y &  y,
const T_low &  alpha,
const T_high &  beta 
)
inline

Definition at line 124 of file uniform_log.hpp.

template<class RNG >
double stan::math::uniform_rng ( const double  alpha,
const double  beta,
RNG &  rng 
)
inline

Definition at line 22 of file uniform_rng.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< fvar< T >, R, C > &  y)
inline

Definition at line 20 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< T, R, C > &  y)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters
yvector of K unrestricted variables
Returns
Unit length vector of dimension K
Template Parameters
TScalar type.

Definition at line 25 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>
Eigen::Matrix<T, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< T, R, C > &  y,
T &  lp 
)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters
yvector of K unrestricted variables
Returns
Unit length vector of dimension K
Parameters
lpLog probability reference to increment.
Template Parameters
TScalar type.

Definition at line 45 of file unit_vector_constrain.hpp.

template<typename T , int R, int C>
Eigen::Matrix<fvar<T>, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< fvar< T >, R, C > &  y,
fvar< T > &  lp 
)
inline

Definition at line 54 of file unit_vector_constrain.hpp.

template<int R, int C>
Eigen::Matrix<var, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< var, R, C > &  y)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters
yvector of K unrestricted variables
Returns
Unit length vector of dimension K
Template Parameters
TScalar type.

Definition at line 64 of file unit_vector_constrain.hpp.

template<int R, int C>
Eigen::Matrix<var, R, C> stan::math::unit_vector_constrain ( const Eigen::Matrix< var, R, C > &  y,
var lp 
)

Return the unit length vector corresponding to the free vector y.

See https://en.wikipedia.org/wiki/N-sphere#Generating_random_points

Parameters
yvector of K unrestricted variables
Returns
Unit length vector of dimension K
Parameters
lpLog probability reference to increment.
Template Parameters
TScalar type.

Definition at line 112 of file unit_vector_constrain.hpp.

template<typename T >
Eigen::Matrix<T, Eigen::Dynamic, 1> stan::math::unit_vector_free ( const Eigen::Matrix< T, Eigen::Dynamic, 1 > &  x)

Transformation of a unit length vector to a "free" vector However, we are just fixing the unidentified radius to 1.

Thus, the transformation is just the identity

Parameters
xunit vector of dimension K
Returns
Unit vector of dimension K considered "free"
Template Parameters
TScalar type.

Definition at line 24 of file unit_vector_free.hpp.

void stan::math::validate_non_negative_index ( const char *  var_name,
const char *  expr,
int  val 
)
inline

Definition at line 12 of file validate_non_negative_index.hpp.

template<typename T >
T stan::math::value_of ( const fvar< T > &  v)
inline

Return the value of the specified variable.

Parameters
vVariable.
Returns
Value of variable.

Definition at line 16 of file value_of.hpp.

double stan::math::value_of ( const var v)
inline

Return the value of the specified variable.

This function is used internally by auto-dif functions along with stan::math::value_of(T x) to extract the double value of either a scalar or an auto-dif variable. This function will be called when the argument is a stan::math::var even if the function is not referred to by namespace because of argument-dependent lookup.

Parameters
vVariable.
Returns
Value of variable.

Definition at line 22 of file value_of.hpp.

template<typename T >
double stan::math::value_of ( const T  x)
inline

Return the value of the specified scalar argument converted to a double value.

See the stan::math::primitive_value function to extract values without casting to double.

This function is meant to cover the primitive types. For types requiring pass-by-reference, this template function should be specialized.

Template Parameters
TType of scalar.
Parameters
xScalar to convert to double.
Returns
Value of scalar cast to a double.

Definition at line 24 of file value_of.hpp.

template<typename T , int R, int C>
Eigen::Matrix<typename child_type<T>::type, R, C> stan::math::value_of ( const Eigen::Matrix< T, R, C > &  M)
inline

Convert a matrix of type T to a matrix of doubles.

T must implement value_of. See test/math/fwd/mat/fun/value_of.cpp for fvar and var usage.

Template Parameters
TScalar type in matrix
RRows of matrix
CColumns of matrix
Parameters
[in]MMatrix to be converted
Returns
Matrix of values

Definition at line 25 of file value_of.hpp.

template<>
double stan::math::value_of< double > ( const double  x)
inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters
xSpecified value.
Returns
Specified value.

Definition at line 40 of file value_of.hpp.

double stan::math::value_of_rec ( const var v)
inline

Return the value of the specified variable.

Parameters
vVariable.
Returns
Value of variable.

Definition at line 15 of file value_of_rec.hpp.

template<typename T >
double stan::math::value_of_rec ( const fvar< T > &  v)
inline

Return the value of the specified variable.

T must implement value_of_rec.

Template Parameters
TScalar type
Parameters
vVariable.
Returns
Value of variable.

Definition at line 21 of file value_of_rec.hpp.

template<typename T >
double stan::math::value_of_rec ( const T  x)
inline

Return the value of the specified scalar argument converted to a double value.

See the stan::math::primitive_value function to extract values without casting to double.

This function is meant to cover the primitive types. For types requiring pass-by-reference, this template function should be specialized.

Template Parameters
TType of scalar.
Parameters
xScalar to convert to double.
Returns
Value of scalar cast to a double.

Definition at line 24 of file value_of_rec.hpp.

template<typename T , int R, int C>
Eigen::Matrix<double, R, C> stan::math::value_of_rec ( const Eigen::Matrix< T, R, C > &  M)
inline

Convert a matrix of type T to a matrix of doubles.

T must implement value_of_rec. See test/unit/math/fwd/mat/fun/value_of_test.cpp for fvar and var usage.

Template Parameters
TScalar type in matrix
RRows of matrix
CColumns of matrix
Parameters
[in]MMatrix to be converted
Returns
Matrix of values

Definition at line 24 of file value_of_rec.hpp.

template<>
double stan::math::value_of_rec< double > ( const double  x)
inline

Return the specified argument.

See value_of(T) for a polymorphic implementation using static casts.

This inline pass-through no-op should be compiled away.

Parameters
xSpecified value.
Returns
Specified value.

Definition at line 40 of file value_of_rec.hpp.

template<typename T >
boost::math::tools::promote_args<T>::type stan::math::variance ( const std::vector< T > &  v)
inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified standard vector.

Parameters
vSpecified vector.
Returns
Sample variance of vector.
Exceptions
std::domain_errorif the size of the vector is less than 1.

Definition at line 24 of file variance.hpp.

template<typename T , int R, int C>
boost::math::tools::promote_args<T>::type stan::math::variance ( const Eigen::Matrix< T, R, C > &  m)
inline

Returns the sample variance (divide by length - 1) of the coefficients in the specified column vector.

Parameters
mSpecified vector.
Returns
Sample variance of vector.

Definition at line 46 of file variance.hpp.

var stan::math::variance ( const std::vector< var > &  v)

Return the sample variance of the specified standard vector.

Raise domain error if size is not greater than zero.

Parameters
[in]va vector
Returns
sample variance of specified vector

Definition at line 52 of file variance.hpp.

template<int R, int C>
var stan::math::variance ( const Eigen::Matrix< var, R, C > &  m)

Definition at line 69 of file variance.hpp.

template<bool propto, typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::von_mises_log ( T_y const &  y,
T_loc const &  mu,
T_scale const &  kappa 
)

Definition at line 26 of file von_mises_log.hpp.

template<typename T_y , typename T_loc , typename T_scale >
return_type<T_y, T_loc, T_scale>::type stan::math::von_mises_log ( T_y const &  y,
T_loc const &  mu,
T_scale const &  kappa 
)
inline

Definition at line 132 of file von_mises_log.hpp.

template<class RNG >
double stan::math::von_mises_rng ( const double  mu,
const double  kappa,
RNG &  rng 
)
inline

Definition at line 33 of file von_mises_rng.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::weibull_ccdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 27 of file weibull_ccdf_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::weibull_cdf ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 27 of file weibull_cdf.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::weibull_cdf_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 27 of file weibull_cdf_log.hpp.

template<bool propto, typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::weibull_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)

Definition at line 30 of file weibull_log.hpp.

template<typename T_y , typename T_shape , typename T_scale >
return_type<T_y, T_shape, T_scale>::type stan::math::weibull_log ( const T_y &  y,
const T_shape &  alpha,
const T_scale &  sigma 
)
inline

Definition at line 141 of file weibull_log.hpp.

template<class RNG >
double stan::math::weibull_rng ( const double  alpha,
const double  sigma,
RNG &  rng 
)
inline

Definition at line 24 of file weibull_rng.hpp.

template<bool propto, typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type<T_y, T_alpha, T_tau, T_beta, T_delta>::type stan::math::wiener_log ( const T_y &  y,
const T_alpha &  alpha,
const T_tau &  tau,
const T_beta &  beta,
const T_delta &  delta 
)

The log of the first passage time density function for a (Wiener) drift diffusion model for the given $y$, boundary separation $\alpha$, nondecision time $\tau$, relative bias $\beta$, and drift rate $\delta$.

$\alpha$ and $\tau$ must be greater than 0, and $\beta$ must be between 0 and 1. $y$ should contain reaction times in seconds (strictly positive) with upper-boundary responses.

Parameters
yA scalar variate.
alphaThe boundary separation.
tauThe nondecision time.
betaThe relative bias.
deltaThe drift rate.
Returns
The log of the Wiener first passage time density of the specified arguments.

Definition at line 72 of file wiener_log.hpp.

template<typename T_y , typename T_alpha , typename T_tau , typename T_beta , typename T_delta >
return_type<T_y, T_alpha, T_tau, T_beta, T_delta>::type stan::math::wiener_log ( const T_y &  y,
const T_alpha &  alpha,
const T_tau &  tau,
const T_beta &  beta,
const T_delta &  delta 
)
inline

Definition at line 225 of file wiener_log.hpp.

template<bool propto, typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args<T_y, T_dof, T_scale>::type stan::math::wishart_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W,
const T_dof &  nu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S 
)

The log of the Wishart density for the given W, degrees of freedom, and scale matrix.

The scale matrix, S, must be k x k, symmetric, and semi-positive definite. Dimension, k, is implicit. nu must be greater than k-1

\begin{eqnarray*} W &\sim& \mbox{\sf{Wishart}}_{\nu} (S) \\ \log (p (W \, |\, \nu, S) ) &=& \log \left( \left(2^{\nu k/2} \pi^{k (k-1) /4} \prod_{i=1}^k{\Gamma (\frac{\nu + 1 - i}{2})} \right)^{-1} \times \left| S \right|^{-\nu/2} \left| W \right|^{(\nu - k - 1) / 2} \times \exp (-\frac{1}{2} \mbox{tr} (S^{-1} W)) \right) \\ &=& -\frac{\nu k}{2}\log(2) - \frac{k (k-1)}{4} \log(\pi) - \sum_{i=1}^{k}{\log (\Gamma (\frac{\nu+1-i}{2}))} -\frac{\nu}{2} \log(\det(S)) + \frac{\nu-k-1}{2}\log (\det(W)) - \frac{1}{2} \mbox{tr} (S^{-1}W) \end{eqnarray*}

Parameters
WA scalar matrix
nuDegrees of freedom
SThe scale matrix
Returns
The log of the Wishart density at W given nu and S.
Exceptions
std::domain_errorif nu is not greater than k-1
std::domain_errorif S is not square, not symmetric, or not semi-positive definite.
Template Parameters
T_yType of scalar.
T_dofType of degrees of freedom.
T_scaleType of scale.

Definition at line 58 of file wishart_log.hpp.

template<typename T_y , typename T_dof , typename T_scale >
boost::math::tools::promote_args<T_y, T_dof, T_scale>::type stan::math::wishart_log ( const Eigen::Matrix< T_y, Eigen::Dynamic, Eigen::Dynamic > &  W,
const T_dof &  nu,
const Eigen::Matrix< T_scale, Eigen::Dynamic, Eigen::Dynamic > &  S 
)
inline

Definition at line 127 of file wishart_log.hpp.

template<class RNG >
Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> stan::math::wishart_rng ( const double  nu,
const Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > &  S,
RNG &  rng 
)
inline

Definition at line 29 of file wishart_rng.hpp.

Variable Documentation

const double stan::math::CONSTRAINT_TOLERANCE = 1E-8

The tolerance for checking arithmetic bounds In rank and in simplexes.

The default value is 1E-8.

Definition at line 11 of file constraint_tolerance.hpp.

const double stan::math::E = boost::math::constants::e<double>()

The base of the natural logarithm, $ e $.

Definition at line 15 of file constants.hpp.

const double stan::math::EPSILON = std::numeric_limits<double>::epsilon()

Smallest positive value.

Definition at line 61 of file constants.hpp.

const double stan::math::INFTY = std::numeric_limits<double>::infinity()

Positive infinity.

Definition at line 44 of file constants.hpp.

const double stan::math::INV_SQRT_2 = 1.0 / SQRT_2

The value of 1 over the square root of 2, $ 1 / \sqrt{2} $.

Definition at line 27 of file constants.hpp.

const double stan::math::INV_SQRT_TWO_PI = 1.0 / std::sqrt(2.0 * boost::math::constants::pi<double>())

Definition at line 166 of file constants.hpp.

const double stan::math::LOG_10 = std::log(10.0)

The natural logarithm of 10, $ \log 10 $.

Definition at line 39 of file constants.hpp.

const double stan::math::LOG_2 = std::log(2.0)

The natural logarithm of 2, $ \log 2 $.

Definition at line 33 of file constants.hpp.

const double stan::math::LOG_HALF = std::log(0.5)

Definition at line 179 of file constants.hpp.

const double stan::math::LOG_PI = std::log(boost::math::constants::pi<double>())

Definition at line 170 of file constants.hpp.

const double stan::math::LOG_PI_OVER_FOUR = std::log(boost::math::constants::pi<double>()) / 4.0

Log pi divided by 4 $ \log \pi / 4 $.

Definition at line 79 of file constants.hpp.

const double stan::math::LOG_SQRT_PI = std::log(SQRT_PI)

Definition at line 173 of file constants.hpp.

const double stan::math::LOG_TWO = std::log(2.0)

Definition at line 177 of file constants.hpp.

const double stan::math::LOG_TWO_PI = LOG_TWO + LOG_PI

Definition at line 193 of file constants.hpp.

const double stan::math::LOG_ZERO = std::log(0.0)

Definition at line 175 of file constants.hpp.

const std::string stan::math::MAJOR_VERSION = STAN_STRING(STAN_MATH_MAJOR)

Major version number for Stan math library.

Definition at line 22 of file version.hpp.

const std::string stan::math::MINOR_VERSION = STAN_STRING(STAN_MATH_MINOR)

Minor version number for Stan math library.

Definition at line 25 of file version.hpp.

const double stan::math::NEG_LOG_PI = - LOG_PI

Definition at line 186 of file constants.hpp.

const double stan::math::NEG_LOG_SQRT_PI = -std::log(std::sqrt(boost::math::constants::pi<double>()))

Definition at line 189 of file constants.hpp.

const double stan::math::NEG_LOG_SQRT_TWO_PI = - std::log(std::sqrt(2.0 * boost::math::constants::pi<double>()))

Definition at line 184 of file constants.hpp.

const double stan::math::NEG_LOG_TWO = - LOG_TWO

Definition at line 181 of file constants.hpp.

const double stan::math::NEG_LOG_TWO_OVER_TWO = - LOG_TWO / 2.0

Definition at line 191 of file constants.hpp.

const double stan::math::NEG_LOG_TWO_PI = - LOG_TWO_PI

Definition at line 195 of file constants.hpp.

const double stan::math::NEG_TWO_OVER_SQRT_PI = -TWO_OVER_SQRT_PI

Definition at line 163 of file constants.hpp.

const double stan::math::NEGATIVE_EPSILON = - std::numeric_limits<double>::epsilon()

Largest negative value (i.e., smallest absolute value).

Definition at line 67 of file constants.hpp.

const double stan::math::NEGATIVE_INFTY = - std::numeric_limits<double>::infinity()

Negative infinity.

Definition at line 50 of file constants.hpp.

const double stan::math::NOT_A_NUMBER = std::numeric_limits<double>::quiet_NaN()

(Quiet) not-a-number value.

Definition at line 56 of file constants.hpp.

const std::string stan::math::PATCH_VERSION = STAN_STRING(STAN_MATH_PATCH)

Patch version for Stan math library.

Definition at line 28 of file version.hpp.

const double stan::math::POISSON_MAX_RATE = std::pow(2.0, 30)

Largest rate parameter allowed in Poisson RNG.

Definition at line 72 of file constants.hpp.

const double stan::math::SQRT_2 = std::sqrt(2.0)

The value of the square root of 2, $ \sqrt{2} $.

Definition at line 21 of file constants.hpp.

const double stan::math::SQRT_2_TIMES_SQRT_PI = SQRT_2 * SQRT_PI

Definition at line 158 of file constants.hpp.

const double stan::math::SQRT_PI = std::sqrt(boost::math::constants::pi<double>())

Definition at line 156 of file constants.hpp.

const double stan::math::TWO_OVER_SQRT_PI = 2.0 / SQRT_PI

Definition at line 161 of file constants.hpp.


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