# !/usr/bin/env python
"""numerical differentiation functions:
Derivative, Gradient, Jacobian, and Hessian
Author: Per A. Brodtkorb
Created: 01.08.2008
Copyright: (c) pab 2008
Licence: New BSD
Based on matlab functions derivest.m gradest.m hessdiag.m, hessian.m
and jacobianest.m version 1.0 released 12/27/2006 by John D'Errico
(e-mail: woodchips@rochester.rr.com)
Also based on the python functions approx_fprime, approx_fprime_cs,
approx_hess_cs, approx_hess1, approx_hess2 and approx_hess3 in the
statsmodels.tools.numdiff module released in 2014 written by Josef Perktold.
"""
from __future__ import division, print_function
import numpy as np
from collections import namedtuple
from numdifftools.multicomplex import bicomplex
from numdifftools.extrapolation import Richardson, dea3, convolve
from numdifftools.test_functions import get_function # , function_names
from numpy import linalg
from scipy import misc
from scipy.ndimage.filters import convolve1d
import warnings
__all__ = ('dea3', 'Derivative', 'Jacobian', 'Gradient', 'Hessian', 'Hessdiag',
'MinStepGenerator', 'MaxStepGenerator', 'Richardson',
'directionaldiff')
# NOTE: we only do double precision internally so far
_TINY = np.finfo(float).tiny
_EPS = np.finfo(float).eps
EPS = np.MachAr().eps
_SQRT_J = (1j + 1.0) / np.sqrt(2.0) # = 1j**0.5
_CENTRAL_WEIGHTS_AND_POINTS = {
(1, 3): (np.array([-1, 0, 1]) / 2.0, np.arange(-1, 2)),
(1, 5): (np.array([1, -8, 0, 8, -1]) / 12.0, np.arange(-2, 3)),
(1, 7): (np.array([-1, 9, -45, 0, 45, -9, 1]) / 60.0, np.arange(-3, 4)),
(1, 9): (np.array([3, -32, 168, -672, 0, 672, -168, 32, -3]) / 840.0,
np.arange(-4, 5)),
(2, 3): (np.array([1, -2.0, 1]), np.arange(-1, 2)),
(2, 5): (np.array([-1, 16, -30, 16, -1]) / 12.0, np.arange(-2, 3)),
(2, 7): (np.array([2, -27, 270, -490, 270, -27, 2]) / 180.0,
np.arange(-3, 4)),
(2, 9): (np.array([-9, 128, -1008, 8064, -14350,
8064, -1008, 128, -9]) / 5040.0,
np.arange(-4, 5))}
[docs]def fornberg_weights_all(x, x0, M=1):
"""
Return finite difference weights_and_points for derivatives of all orders.
Parameters
----------
x : vector, length n
x-coordinates for grid points
x0 : scalar
location where approximations are to be accurate
m : scalar integer
highest derivative that we want to find weights_and_points for
Returns
-------
C : array, shape n x m+1
contains coefficients for the j'th derivative in column j (0 <= j <= m)
See also:
---------
fornberg_weights
Reference
---------
B. Fornberg (1998)
"Calculation of weights_and_points in finite difference formulas",
SIAM Review 40, pp. 685-691.
http://www.scholarpedia.org/article/Finite_difference_method
"""
N = len(x)
if M >= N:
raise ValueError('length(x) must be larger than m')
c1, c4 = 1, x[0] - x0
C = np.zeros((N, M + 1))
C[0, 0] = 1
for n in range(1, N):
m = np.arange(0, min(n, M) + 1)
c2, c5, c4 = 1, c4, x[n] - x0
for v in range(n):
c3 = x[n] - x[v]
c2, c6, c7 = c2 * c3, m * C[v, m-1], C[v, m]
C[v, m] = (c4 * c7 - c6) / c3
C[n, m] = c1 * (c6 - c5 * c7) / c2
c1 = c2
return C
[docs]def fornberg_weights(x, x0, m=1):
"""
Return weights for finite difference approximation of the m'th derivative
U^m(x0), evaluated at x0, based on n values of U at x[0], x[1],... x[n-1]:
U^m(x0) = sum weights[i] * U(x[i])
Parameters
----------
x : vector
abscissas used for the evaluation for the derivative at x0.
x0 : scalar
location where approximations are to be accurate
m : integer
order of derivative. Note for m=0 this can be used to evaluate the
interpolating polynomial itself.
Notes
-----
The x values can be arbitrarily spaced but must be distinct and len(x) > m.
The Fornberg algorithm is much more stable numerically than regular
vandermonde systems for large values of n.
See also
--------
fornberg_weights_all
"""
return fornberg_weights_all(x, x0, m)[:, -1]
def _make_exact(h):
"""Make sure h is an exact representable number
This is important when calculating numerical derivatives and is
accomplished by adding 1 and then subtracting 1..
"""
return (h + 1.0) - 1.0
[docs]def default_scale(method='forward', n=1, order=2):
# is_odd = (n % 2) == 1
high_order = int(n > 1 or order >= 4)
order2 = max(order // 2-1, 0)
n4 = n // 4
return (dict(multicomplex=1.35, complex=1.35).get(method, 2.5) +
int((n - 1)) * dict(multicomplex=0, complex=0.0).get(method, 1.3) +
order2 * dict(central=3, forward=2, backward=2).get(method, 0) +
# is_odd * dict(complex=2.65*int(n//2)).get(method, 0) +
(n % 4 == 1) * high_order * dict(complex=3.65 + n4 * (5 + 1.5**n4)
).get(method, 0) +
(n % 4 == 3) * dict(complex=3.65*2 + n4 * (5 + 2.1**n4)
).get(method, 0) +
(n % 4 == 2) * dict(complex=3.65 + n4 * (5 + 1.7**n4)
).get(method, 0) +
(n % 4 == 0) * dict(complex=(n//4) * (10 + 1.5*int(n > 10))
).get(method, 0))
[docs]def valarray(shape, value=np.NaN, typecode=None):
"""Return an array of all value."""
if typecode is None:
typecode = bool
out = np.ones(shape, dtype=typecode) * value
if not isinstance(out, np.ndarray):
out = np.asarray(out)
return out
def nom_step(x=None):
"""Return nominal step."""
if x is None:
return 1.0
return np.maximum(np.log1p(np.abs(x)), 1.0)
def _default_base_step(x, scale, epsilon=None):
if epsilon is None:
h = EPS ** (1. / scale) * nom_step(x)
else:
h = valarray(x.shape, value=epsilon)
return h
[docs]class MinStepGenerator(object):
"""
Generates a sequence of steps
where steps = base_step * step_ratio ** (np.arange(num_steps) + offset)
Parameters
----------
base_step : float, array-like, optional
Defines the base step, if None, then base_step is set to
EPS**(1/scale)*max(log(1+|x|), 1) where x is supplied at runtime
through the __call__ method.
step_ratio : real scalar, optional, default 2
Ratio between sequential steps generated.
Note: Ratio > 1
If None then step_ratio is 2 for n=1 otherwise step_ratio is 1.6
num_steps : scalar integer, optional, default n + order - 1 + num_extrap
defines number of steps generated. It should be larger than
n + order - 1
offset : real scalar, optional, default 0
offset to the base step
scale : real scalar, optional
scale used in base step. If not None it will override the default
computed with the default_scale function.
"""
[docs] def __init__(self, base_step=None, step_ratio=2, num_steps=None,
offset=0, scale=None, num_extrap=0, use_exact_steps=True,
check_num_steps=True):
self.base_step = base_step
self.num_steps = num_steps
self.step_ratio = step_ratio
self.offset = offset
self.scale = scale
self.check_num_steps = check_num_steps
self.use_exact_steps = use_exact_steps
self.num_extrap = num_extrap
def __repr__(self):
class_name = self.__class__.__name__
kwds = ['%s=%s' % (name, str(getattr(self, name)))
for name in self.__dict__.keys()]
return """%s(%s)""" % (class_name, ','.join(kwds))
def _default_scale(self, method, n, order):
scale = self.scale
if scale is None:
scale = default_scale(method, n, order)
return scale
def _default_base_step(self, xi, method, n, order=2):
scale = self._default_scale(method, n, order)
base_step = _default_base_step(xi, scale, self.base_step)
if self.use_exact_steps:
base_step = _make_exact(base_step)
return base_step
def _min_num_steps(self, method, n, order):
num_steps = n + order - 1
if method in ['central', 'central2', 'complex', 'multicomplex']:
step = 2
if method == 'complex':
step = 4 if n > 2 or order >= 4 else 2
num_steps = (n + order-1) // step
return max(int(num_steps), 1)
def _default_num_steps(self, method, n, order):
min_num_steps = self._min_num_steps(method, n, order)
if self.num_steps is not None:
num_steps = int(self.num_steps)
if self.check_num_steps:
num_steps = max(num_steps, min_num_steps)
return num_steps
return min_num_steps + int(self.num_extrap)
def _default_step_ratio(self, n):
if self.step_ratio is None:
step_ratio = {1: 2.0}.get(n, 1.6)
else:
step_ratio = float(self.step_ratio)
if self.use_exact_steps:
step_ratio = _make_exact(step_ratio)
return step_ratio
def __call__(self, x, method='central', n=1, order=2):
xi = np.asarray(x)
base_step = self._default_base_step(xi, method, n, order)
step_ratio = self._default_step_ratio(n)
num_steps = self._default_num_steps(method, n, order)
offset = self.offset
for i in range(num_steps-1, -1, -1):
h = (base_step * step_ratio**(i + offset))
if (np.abs(h) > 0).all():
yield h
[docs]class MinMaxStepGenerator(object):
"""
Generates a sequence of steps
where
steps = logspace(log10(step_min), log10(step_max), num_steps)
Parameters
----------
step_min : float, array-like, optional
Defines the minimim step. Default value is:
EPS**(1/scale)*max(log(1+|x|), 1)
where x and scale are supplied at runtime through the __call__ method.
step_max : real scalar, optional
maximum step generated. Default value is:
exp(log(step_min) * scale / (scale + 1.5))
num_steps : scalar integer, optional
defines number of steps generated.
scale : real scalar, optional
scale used in base step. If set to a value it will override the scale
supplied at runtime.
"""
[docs] def __init__(self, step_min=None, step_max=None, num_steps=10, scale=None,
num_extrap=0):
self.step_min = step_min
self.num_steps = num_steps
self.step_max = step_max
self.scale = scale
self.num_extrap = num_extrap
def __repr__(self):
class_name = self.__class__.__name__
kwds = ['%s=%s' % (name, str(getattr(self, name)))
for name in self.__dict__.keys()]
return """%s(%s)""" % (class_name, ','.join(kwds))
def __call__(self, x, method='forward', n=1, order=None):
if self.scale is not None:
scale = self.scale
xi = np.asarray(x)
step_min, step_max = self.step_min, self.step_max
delta = _default_base_step(xi, scale, step_min)
if step_min is None:
step_min = (10 * EPS)**(1. / scale)
if step_max is None:
step_max = np.exp(np.log(step_min) * scale / (scale + 1.5))
steps = np.logspace(0, np.log10(step_max) - np.log10(step_min),
self.num_steps)[::-1]
for step in steps:
h = _make_exact(delta * step)
if (np.abs(h) > 0).all():
yield h
[docs]class MaxStepGenerator(MinStepGenerator):
"""
Generates a sequence of steps
where
steps = base_step * step_ratio ** (-np.arange(num_steps) + offset)
base_step = step_max * step_nom
Parameters
----------
max_step : float, array-like, optional default 2
Defines the maximum step
step_ratio : real scalar, optional, default 2
Ratio between sequential steps generated.
Note: Ratio > 1
num_steps : scalar integer, optional, default n + order - 1 + num_extrap
defines number of steps generated. It should be larger than
n + order - 1
step_nom : default maximum(log1p(abs(x)), 1)
Nominal step.
offset : real scalar, optional, default 0
offset to the base step: max_step * nom_step
"""
[docs] def __init__(self, step_max=2.0, step_ratio=2.0, num_steps=15,
step_nom=None, offset=0, num_extrap=0,
use_exact_steps=False, check_num_steps=True):
self.base_step = None
self.step_max = step_max
self.step_ratio = step_ratio
self.num_steps = num_steps
self.step_nom = step_nom
self.offset = offset
self.num_extrap = num_extrap
self.check_num_steps = check_num_steps
self.use_exact_steps = use_exact_steps
def _default_step_nom(self, x):
if self.step_nom is None:
return nom_step(x)
return valarray(x.shape, value=self.step_nom)
def _default_base_step(self, xi, method, n, order=1):
base_step = self.base_step
if base_step is None:
base_step = self.step_max * self._default_step_nom(xi)
if self.use_exact_steps:
base_step = _make_exact(base_step)
return base_step
def __call__(self, x, method='forward', n=1, order=None):
xi = np.asarray(x)
offset = self.offset
base_step = self._default_base_step(xi, method, n)
step_ratio = self._default_step_ratio(n)
num_steps = self._default_num_steps(method, n, order)
for i in range(num_steps):
h = base_step * step_ratio**(-i + offset)
if (np.abs(h) > 0).all():
yield h
_cmn_doc = """
Calculate %(derivative)s with finite difference approximation
Parameters
----------
f : function
function of one array f(x, `*args`, `**kwds`)
step : float, array-like or StepGenerator object, optional
Defines the spacing used in the approximation.
Default is MinStepGenerator(base_step=step, step_ratio=None)
if step or method in in ['complex', 'multicomplex'], otherwise
MaxStepGenerator(step_ratio=None, num_extrap=14)
The results are extrapolated if the StepGenerator generate more than 3
steps.
method : {'central', 'complex', 'multicomplex', 'forward', 'backward'}
defines the method used in the approximation%(extra_parameter)s
full_output : bool, optional
If `full_output` is False, only the derivative is returned.
If `full_output` is True, then (der, r) is returned `der` is the
derivative, and `r` is a Results object.
Call Parameters
---------------
x : array_like
value at which function derivative is evaluated
args : tuple
Arguments for function `f`.
kwds : dict
Keyword arguments for function `f`.
%(returns)s
Notes
-----
Complex methods are usually the most accurate provided the function to
differentiate is analytic. The complex-step methods also requires fewer
steps than the other methods and can work very close to the support of
a function.
The complex-step derivative has truncation error O(steps**2) for `n=1` and
O(steps**4) for `n` larger, so truncation error can be eliminated by
choosing steps to be very small.
Especially the first order complex-step derivative avoids the problem of
round-off error with small steps because there is no subtraction. However,
this method fails if f(x) does not support complex numbers or involves
non-analytic functions such as e.g.: abs, max, min.
Central difference methods are almost as accurate and has no restriction on
type of function. For this reason the 'central' method is the default
method, but sometimes one can only allow evaluation in forward or backward
direction.
For all methods one should be careful in decreasing the step size too much
due to round-off errors.
%(extra_note)s
Reference
---------
Ridout, M.S. (2009) Statistical applications of the complex-step method
of numerical differentiation. The American Statistician, 63, 66-74
K.-L. Lai, J.L. Crassidis, Y. Cheng, J. Kim (2005), New complex step
derivative approximations with application to second-order
kalman filtering, AIAA Guidance, Navigation and Control Conference,
San Francisco, California, August 2005, AIAA-2005-5944.
Lyness, J. M., Moler, C. B. (1966). Vandermonde Systems and Numerical
Differentiation. *Numerische Mathematik*.
Lyness, J. M., Moler, C. B. (1969). Generalized Romberg Methods for
Integrals of Derivatives. *Numerische Mathematik*.
%(example)s
%(see_also)s
"""
class _Derivative(object):
info = namedtuple('info', ['error_estimate', 'final_step', 'index'])
def __init__(self, f, step=None, method='central', order=2, n=1,
full_output=False):
self.f = f
self.n = n
self.order = order
self.method = method
self.full_output = full_output
self.richardson_terms = 2
self.step = self._make_generator(step)
def _make_generator(self, step):
if hasattr(step, '__call__'):
return step
if step is None and self.method not in ['complex', 'multicomplex']:
return MaxStepGenerator(step_ratio=None, num_extrap=14)
return MinStepGenerator(base_step=step, step_ratio=None, num_extrap=0)
def _get_arg_min(self, errors):
shape = errors.shape
try:
arg_mins = np.nanargmin(errors, axis=0)
min_errors = np.nanmin(errors, axis=0)
except ValueError as msg:
warnings.warn(str(msg))
ix = np.arange(shape[1])
return ix
for i, min_error in enumerate(min_errors):
idx = np.flatnonzero(errors[:, i] == min_error)
arg_mins[i] = idx[idx.size // 2]
ix = np.ravel_multi_index((arg_mins, np.arange(shape[1])), shape)
return ix
def _add_error_to_outliers(self, der, trim_fact=10):
try:
median = np.nanmedian(der, axis=0)
p75 = np.nanpercentile(der, 75, axis=0)
p25 = np.nanpercentile(der, 25, axis=0)
iqr = np.abs(p75-p25)
except ValueError as msg:
warnings.warn(str(msg))
return 0 * der
a_median = np.abs(median)
outliers = (((abs(der) < (a_median / trim_fact)) +
(abs(der) > (a_median * trim_fact))) * (a_median > 1e-8) +
((der < p25-1.5*iqr) + (p75+1.5*iqr < der)))
errors = outliers * np.abs(der - median)
return errors
def _get_best_estimate(self, der, errors, steps, shape):
errors += self._add_error_to_outliers(der)
ix = self._get_arg_min(errors)
final_step = steps.flat[ix].reshape(shape)
err = errors.flat[ix].reshape(shape)
return der.flat[ix].reshape(shape), self.info(err, final_step, ix)
@property
def _method_order(self):
step = self._richardson_step()
# Make sure it is even and at least 2 or 4
order = max((self.order // step) * step, step)
return order
def _complex_high_order(self):
return self.method == 'complex' and (self.n > 1 or self.order >= 4)
def _richardson_step(self):
# complex_step = 4 if self.n % 2 == 0 else 2
complex_step = 4 if self._complex_high_order() else 2
return dict(central=2, central2=2, complex=complex_step,
multicomplex=2).get(self.method, 1)
def _set_richardson_rule(self, step_ratio, num_terms=2):
order = self._method_order
step = self._richardson_step()
self._richardson_extrapolate = Richardson(step_ratio=step_ratio,
step=step, order=order,
num_terms=num_terms)
def _wynn_extrapolate(self, der, steps):
der, errors = dea3(der[0:-2], der[1:-1], der[2:], symmetric=False)
return der, errors, steps[2:]
def _extrapolate(self, results, steps, shape):
der, errors, steps = self._richardson_extrapolate(results, steps)
if len(der) > 2:
# der, errors, steps = self._richardson_extrapolate(results, steps)
der, errors, steps = self._wynn_extrapolate(der, steps)
der, info = self._get_best_estimate(der, errors, steps, shape)
return der, info
def _get_function_name(self):
name = '_%s' % self.method
even_derivative_order = self._is_even_derivative()
if even_derivative_order and self.method in ('central', 'complex'):
name = name + '_even'
if self.method in ('complex') and self._is_fourth_derivative():
name = name + '_higher'
else:
if self._complex_high_order() and self._is_odd_derivative():
name = name + '_odd'
if self._is_third_derivative():
name = name + '_higher'
elif self.method == 'multicomplex' and self.n > 1:
if self.n == 2:
name = name + '2'
else:
raise ValueError('Multicomplex method only support first '
'and second order derivatives.')
return name
def _get_functions(self):
name = self._get_function_name()
return getattr(self, name), self.f
def _get_steps(self, xi):
method, n, order = self.method, self.n, self._method_order
return [step for step in self.step(xi, method, n, order)]
def _is_odd_derivative(self):
return self.n % 2 == 1
def _is_even_derivative(self):
return self.n % 2 == 0
def _is_third_derivative(self):
return self.n % 4 == 3
def _is_fourth_derivative(self):
return self.n % 4 == 0
def _eval_first_condition(self):
even_derivative = self._is_even_derivative()
return ((even_derivative and self.method in ('central', 'central2')) or
self.method in ['forward', 'backward'] or
self.method == 'complex' and self._is_fourth_derivative())
def _eval_first(self, f, x, *args, **kwds):
if self._eval_first_condition():
return f(x, *args, **kwds)
return 0.0
def _vstack(self, sequence, steps):
# sequence = np.atleast_2d(sequence)
original_shape = np.shape(sequence[0])
f_del = np.vstack(list(np.ravel(r)) for r in sequence)
h = np.vstack(list(np.ravel(np.ones(original_shape)*step))
for step in steps)
if f_del.size != h.size:
raise ValueError('fun did not return data of correct size ' +
'(it must be vectorized)')
return f_del, h, original_shape
def _compute_step_ratio(self, steps):
if len(steps) < 2:
return 1
return np.unique(steps[0]/steps[1]).mean()
def __call__(self, x, *args, **kwds):
xi = np.asarray(x)
results = self._derivative(xi, args, kwds)
derivative, info = self._extrapolate(*results)
if self.full_output:
return derivative, info
return derivative
[docs]class Derivative(_Derivative):
__doc__ = _cmn_doc % dict(
derivative='n-th derivative',
extra_parameter="""
order : int, optional
defines the order of the error term in the Taylor approximation used.
For 'central' and 'complex' methods, it must be an even number.
n : int, optional
Order of the derivative.""",
extra_note="""
Higher order approximation methods will generally be more accurate, but may
also suffer more from numerical problems. First order methods is usually
not recommended.
""", returns="""
Returns
-------
der : ndarray
array of derivatives
""", example="""
Example
-------
>>> import numpy as np
>>> import numdifftools as nd
# 1'st derivative of exp(x), at x == 1
>>> fd = nd.Derivative(np.exp)
>>> np.allclose(fd(1), 2.71828183)
True
>>> d2 = fd([1, 2])
>>> np.allclose(d2, [ 2.71828183, 7.3890561 ])
True
>>> def f(x):
... return x**3 + x**2
>>> df = nd.Derivative(f)
>>> np.allclose(df(1), 5)
True
>>> ddf = nd.Derivative(f, n=2)
>>> np.allclose(ddf(1), 8)
True
""", see_also="""
See also
--------
Gradient,
Hessian
""")
"""
Find the n-th derivative of a function at a point.
Given a function, use a difference formula with spacing `dx` to
compute the `n`-th derivative at `x0`.
Parameters
----------
f : function
Input function.
x0 : float
The point at which `n`-th derivative is found.
dx : float, optional
Spacing.
method : Method of estimation. Valid options are:
'central', 'forward' or 'backward'. (Default 'central')
n : int, optional (Default 1)
Order of the derivative.
order : int, optional (Default 2)
defining order of basic method used.
For 'central' methods, it must be an even number eg. [2,4].
Notes
-----
Decreasing the step size too small can result in round-off error.
Note on order: higher order methods will generally be more accurate,
but may also suffer more from numerical problems. First order
methods would usually not be recommended.
Complex methods are usually the most accurate provided the function to
differentiate is analytic. The complex-step methods also requires fewer
steps than the other methods and can work very close to the support of
a function. Central difference methods are almost as accurate and has
no restriction on type of function, but sometimes one can only allow
evaluation in forward or backward direction.
"""
@property
def n(self):
return self._n
@n.setter
def n(self, n):
self._n = n
if n == 0:
self._derivative = self._derivative_zero_order
else:
self._derivative = self._derivative_nonzero_order
@staticmethod
def _fd_matrix(step_ratio, parity, nterms):
"""
Return matrix for finite difference and complex step derivation.
Parameters
----------
step_ratio : real scalar
ratio between steps in unequally spaced difference rule.
parity : scalar, integer
0 (one sided, all terms included but zeroth order)
1 (only odd terms included)
2 (only even terms included)
3 (only every 4'th order terms included starting from order 2)
4 (only every 4'th order terms included starting from order 4)
nterms : scalar, integer
number of terms
"""
try:
step = [1, 2, 2, 4, 4, 4, 4][parity]
except Exception as e:
msg = '%s. Parity must be 0, 1, 2, 3, 4, 5 or 6! (%d)' % (str(e),
parity)
raise ValueError(msg)
inv_sr = 1.0 / step_ratio
offset = [1, 1, 2, 2, 4, 1, 3][parity]
c0 = [1.0, 1.0, 1.0, 2.0, 24.0, 1.0, 6.0][parity]
c = c0/misc.factorial(np.arange(offset, step * nterms + offset, step))
[i, j] = np.ogrid[0:nterms, 0:nterms]
return np.atleast_2d(c[j] * inv_sr ** (i * (step * j + offset)))
def _flip_fd_rule(self):
n = self.n
return ((self._is_even_derivative() and (self.method == 'backward')) or
(self.method == 'complex' and (n % 8 in [3, 4, 5, 6])))
def _get_finite_difference_rule(self, step_ratio):
"""
Generate finite differencing rule in advance.
The rule is for a nominal unit step size, and will
be scaled later to reflect the local step size.
Member methods used
-------------------
_fd_matrix
Member variables used
---------------------
n
order
method
"""
method = self.method
if method in ('multicomplex', ) or self.n == 0:
return np.ones((1,))
order, method_order = self.n - 1, self._method_order
parity = 0
if (method.startswith('central') or
(method.startswith('complex') and self.n == 1 and
method_order < 4)):
parity = (order % 2) + 1
elif self.method == 'complex':
if self._is_odd_derivative():
parity = 6 if self._is_third_derivative() else 5
else:
parity = 4 if self._is_fourth_derivative() else 3
step = self._richardson_step()
num_terms, ix = (order + method_order) // step, order // step
fd_mat = self._fd_matrix(step_ratio, parity, num_terms)
fd_rule = linalg.pinv(fd_mat)[ix]
if self._flip_fd_rule():
fd_rule *= -1
return fd_rule
def _apply_fd_rule(self, fd_rule, sequence, steps):
"""
Return derivative estimates of f at x0 for a sequence of stepsizes h
Member variables used
---------------------
n
"""
f_del, h, original_shape = self._vstack(sequence, steps)
ne = h.shape[0]
if ne < fd_rule.size:
raise ValueError('num_steps (%d) must be larger than '
'(%d) n + order - 1 = %d + %d -1'
' (%s)' % (ne, fd_rule.size, self.n, self.order,
self.method)
)
nr = (fd_rule.size-1)
f_diff = convolve(f_del, fd_rule[::-1], axis=0, origin=nr//2)
der_init = f_diff / (h ** self.n)
ne = max(ne - nr, 1)
return der_init[:ne], h[:ne], original_shape
def _derivative_zero_order(self, xi, args, kwds):
steps = [np.zeros_like(xi)]
results = [self.f(xi, *args, **kwds)]
self._set_richardson_rule(2, 0)
return self._vstack(results, steps)
def _derivative_nonzero_order(self, xi, args, kwds):
diff, f = self._get_functions()
steps = self._get_steps(xi)
fxi = self._eval_first(f, xi, *args, **kwds)
results = [diff(f, fxi, xi, h, *args, **kwds) for h in steps]
step_ratio = self._compute_step_ratio(steps)
self._set_richardson_rule(step_ratio, self.richardson_terms)
fd_rule = self._get_finite_difference_rule(step_ratio)
return self._apply_fd_rule(fd_rule, results, steps)
@staticmethod
def _central_even(f, f_x0i, x0i, h, *args, **kwds):
return (f(x0i + h, *args, **kwds) +
f(x0i - h, *args, **kwds)) / 2.0 - f_x0i
@staticmethod
def _central(f, f_x0i, x0i, h, *args, **kwds):
return (f(x0i + h, *args, **kwds) -
f(x0i - h, *args, **kwds)) / 2.0
@staticmethod
def _forward(f, f_x0i, x0i, h, *args, **kwds):
return (f(x0i + h, *args, **kwds) - f_x0i)
@staticmethod
def _backward(f, f_x0i, x0i, h, *args, **kwds):
return (f_x0i - f(x0i - h, *args, **kwds))
@staticmethod
def _complex(f, fx, x, h, *args, **kwds):
return f(x + 1j * h, *args, **kwds).imag
@staticmethod
def _complex_odd(f, fx, x, h, *args, **kwds):
ih = h * _SQRT_J
return ((_SQRT_J/2.) * (f(x + ih, *args, **kwds) -
f(x - ih, *args, **kwds))).imag
@staticmethod
def _complex_odd_higher(f, fx, x, h, *args, **kwds):
ih = h * _SQRT_J
return ((3 * _SQRT_J) * (f(x + ih, *args, **kwds) -
f(x - ih, *args, **kwds))).real
@staticmethod
def _complex_even(f, fx, x, h, *args, **kwds):
ih = h * _SQRT_J
return (f(x + ih, *args, **kwds) +
f(x - ih, *args, **kwds)).imag
@staticmethod
def _complex_even_higher(f, fx, x, h, *args, **kwds):
ih = h * _SQRT_J
return 12.0 * (f(x + ih, *args, **kwds) +
f(x - ih, *args, **kwds) - 2 * fx).real
@staticmethod
def _multicomplex(f, fx, x, h, *args, **kwds):
z = bicomplex(x + 1j * h, 0)
return f(z, *args, **kwds).imag
@staticmethod
def _multicomplex2(f, fx, x, h, *args, **kwds):
z = bicomplex(x + 1j * h, h)
return f(z, *args, **kwds).imag12
[docs]def directionaldiff(f, x0, vec, **options):
"""
Return directional derivative of a function of n variables
Parameters
----------
f: function
analytical function to differentiate.
x0: array
vector location at which to differentiate f. If x0 is an nxm array,
then fun is assumed to be a function of n*m variables.
vec: array
vector defining the line along which to take the derivative. It should
be the same size as x0, but need not be a vector of unit length.
**options:
optional arguments to pass on to Derivative.
Returns
-------
dder: scalar
estimate of the first derivative of f in the specified direction.
Example
-------
At the global minimizer (1,1) of the Rosenbrock function,
compute the directional derivative in the direction [1 2]
>>> import numpy as np
>>> import numdifftools as nd
>>> vec = np.r_[1, 2]
>>> rosen = lambda x: (1-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> dd, info = nd.directionaldiff(rosen, [1, 1], vec, full_output=True)
>>> np.allclose(dd, 0)
True
>>> np.abs(info.error_estimate)<1e-14
True
See also
--------
Derivative,
Gradient
"""
x0 = np.asarray(x0)
vec = np.asarray(vec)
if x0.size != vec.size:
raise ValueError('vec and x0 must be the same shapes')
vec = np.reshape(vec/np.linalg.norm(vec.ravel()), x0.shape)
return Derivative(lambda t: f(x0+t*vec), **options)(0)
[docs]class Gradient(Derivative):
[docs] def __init__(self, f, step=None, method='central', order=2,
full_output=False):
super(Gradient, self).__init__(f, step=step, method=method, n=1,
order=order, full_output=full_output)
__doc__ = _cmn_doc % dict(
derivative='Gradient',
extra_parameter="""
order : int, optional
defines the order of the error term in the Taylor approximation used.
For 'central' and 'complex' methods, it must be an even number.""",
returns="""
Returns
-------
grad : array
gradient
""", extra_note="""
Higher order approximation methods will generally be more accurate, but may
also suffer more from numerical problems. First order methods is usually
not recommended.
""", example="""
Example
-------
>>> import numpy as np
>>> import numdifftools as nd
>>> fun = lambda x: np.sum(x**2)
>>> dfun = nd.Gradient(fun)
>>> dfun([1,2,3])
array([ 2., 4., 6.])
# At [x,y] = [1,1], compute the numerical gradient
# of the function sin(x-y) + y*exp(x)
>>> sin = np.sin; exp = np.exp
>>> z = lambda xy: sin(xy[0]-xy[1]) + xy[1]*exp(xy[0])
>>> dz = nd.Gradient(z)
>>> grad2 = dz([1, 1])
>>> grad2
array([ 3.71828183, 1.71828183])
# At the global minimizer (1,1) of the Rosenbrock function,
# compute the gradient. It should be essentially zero.
>>> rosen = lambda x : (1-x[0])**2 + 105.*(x[1]-x[0]**2)**2
>>> rd = nd.Gradient(rosen)
>>> grad3 = rd([1,1])
>>> np.allclose(grad3,[0, 0])
True""", see_also="""
See also
--------
Derivative, Hessian, Jacobian
""")
@staticmethod
def _central(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [(f(x + hi, *args, **kwds) - f(x - hi, *args, **kwds)) / 2.0
for hi in increments]
return np.array(partials).T
@staticmethod
def _backward(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [(fx - f(x - hi, *args, **kwds)) for hi in increments]
return np.array(partials).T
@staticmethod
def _forward(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [(f(x + hi, *args, **kwds) - fx) for hi in increments]
return np.array(partials).T
@staticmethod
def _complex(f, fx, x, h, *args, **kwds):
# From Guilherme P. de Freitas, numpy mailing list
# http://mail.scipy.org/pipermail/numpy-discussion/2010-May/050250.html
n = len(x)
increments = np.identity(n) * 1j * h
partials = [f(x + ih, *args, **kwds).imag for ih in increments]
return np.array(partials).T
@staticmethod
def _complex_odd(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * _SQRT_J * h
partials = [((_SQRT_J/2.) * (f(x + ih, *args, **kwds) -
f(x - ih, *args, **kwds))).imag
for ih in increments]
return np.array(partials).T
@staticmethod
def _multicomplex(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * 1j * h
partials = [f(bicomplex(x + hi, 0), *args, **kwds).imag
for hi in increments]
return np.array(partials).T
def __call__(self, x, *args, **kwds):
return super(Gradient, self).__call__(np.atleast_1d(x), *args, **kwds)
[docs]class Jacobian(Gradient):
__doc__ = _cmn_doc % dict(
derivative='Jacobian',
extra_parameter="""
order : int, optional
defines the order of the error term in the Taylor approximation used.
For 'central' and 'complex' methods, it must be an even number.""",
returns="""
Returns
-------
jacob : array
Jacobian
""", extra_note="""
Higher order approximation methods will generally be more accurate, but may
also suffer more from numerical problems. First order methods is usually
not recommended.
If f returns a 1d array, it returns a Jacobian. If a 2d array is returned
by f (e.g., with a value for each observation), it returns a 3d array
with the Jacobian of each observation with shape xk x nobs x xk. I.e.,
the Jacobian of the first observation would be [:, 0, :]
""", example="""
Example
-------
>>> import numdifftools as nd
#(nonlinear least squares)
>>> xdata = np.reshape(np.arange(0,1,0.1),(-1,1))
>>> ydata = 1+2*np.exp(0.75*xdata)
>>> fun = lambda c: (c[0]+c[1]*np.exp(c[2]*xdata) - ydata)**2
>>> Jfun = nd.Jacobian(fun)
>>> val = Jfun([1,2,0.75])
>>> np.allclose(val, np.zeros((10,3)))
True
>>> fun2 = lambda x : x[0]*x[1]*x[2] + np.exp(x[0])*x[1]
>>> Jfun3 = nd.Jacobian(fun2)
>>> Jfun3([3.,5.,7.])
array([ 135.42768462, 41.08553692, 15. ])
""", see_also="""
See also
--------
Derivative, Hessian, Gradient
""")
[docs]class Hessdiag(Derivative):
[docs] def __init__(self, f, step=None, method='central', order=2,
full_output=False):
super(Hessdiag, self).__init__(f, step=step, method=method, n=2,
order=order, full_output=full_output)
__doc__ = _cmn_doc % dict(
derivative='Hessian diagonal',
extra_parameter="""order : int, optional
defines the order of the error term in the Taylor approximation used.
For 'central' and 'complex' methods, it must be an even number.""",
returns="""
Returns
-------
hessdiag : array
hessian diagonal
""", extra_note="""
Higher order approximation methods will generally be more accurate, but may
also suffer more from numerical problems. First order methods is usually
not recommended.
""", example="""
Example
-------
>>> import numpy as np
>>> import numdifftools as nd
>>> fun = lambda x : x[0] + x[1]**2 + x[2]**3
>>> Hfun = nd.Hessdiag(fun, full_output=True)
>>> hd, info = Hfun([1,2,3])
>>> np.allclose(hd, [ 0., 2., 18.])
True
>>> info.error_estimate < 1e-11
array([ True, True, True], dtype=bool)
""", see_also="""
See also
--------
Derivative, Hessian, Jacobian, Gradient
""")
@staticmethod
def _central2(f, fx, x, h, *args, **kwds):
"""Eq. 8"""
n = len(x)
increments = np.identity(n) * h
partials = [(f(x + 2*hi, *args, **kwds) +
f(x - 2*hi, *args, **kwds) + 2*fx -
2*f(x + hi, *args, **kwds) -
2*f(x - hi, *args, **kwds)) / 4.0
for hi in increments]
return np.array(partials)
@staticmethod
def _central_even(f, fx, x, h, *args, **kwds):
"""Eq. 9"""
n = len(x)
increments = np.identity(n) * h
partials = [(f(x + hi, *args, **kwds) +
f(x - hi, *args, **kwds)) / 2.0 - fx
for hi in increments]
return np.array(partials)
@staticmethod
def _backward(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [(fx - f(x - hi, *args, **kwds)) for hi in increments]
return np.array(partials)
@staticmethod
def _forward(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [(f(x + hi, *args, **kwds) - fx) for hi in increments]
return np.array(partials)
@staticmethod
def _multicomplex2(f, fx, x, h, *args, **kwds):
n = len(x)
increments = np.identity(n) * h
partials = [f(bicomplex(x + 1j * hi, hi), *args, **kwds).imag12
for hi in increments]
return np.array(partials)
@staticmethod
def _complex_even(f, fx, x, h, *args, **kwargs):
n = len(x)
increments = np.identity(n) * h * (1j+1) / np.sqrt(2)
partials = [(f(x + hi, *args, **kwargs) +
f(x - hi, *args, **kwargs)).imag
for hi in increments]
return np.array(partials)
def __call__(self, x, *args, **kwds):
return super(Hessdiag, self).__call__(np.atleast_1d(x), *args, **kwds)
[docs]class Hessian(_Derivative):
[docs] def __init__(self, f, step=None, method='central', full_output=False):
order = dict(backward=1, forward=1, complex=2).get(method, 2)
super(Hessian, self).__init__(f, n=2, step=step, method=method,
order=order, full_output=full_output)
__doc__ = _cmn_doc % dict(
derivative='Hessian',
extra_parameter="",
returns="""
Returns
-------
hess : ndarray
array of partial second derivatives, Hessian
""", extra_note="""
Computes the Hessian according to method as:
'forward' :eq:`7`, 'central' :eq:`9` and 'complex' :eq:`10`:
.. math::
\quad ((f(x + d_j e_j + d_k e_k) - f(x + d_j e_j))) / (d_j d_k)
:label: 7
.. math::
\quad ((f(x + d_j e_j + d_k e_k) - f(x + d_j e_j - d_k e_k)) - (f(x - d_j e_j + d_k e_k) - f(x - d_j e_j - d_k e_k)) / (4 d_j d_k)
:label: 9
.. math::
imag(f(x + i d_j e_j + d_k e_k) - f(x + i d_j e_j - d_k e_k)) /(2 d_j d_k)
:label: 10
where :math:`e_j` is a vector with element :math:`j` is one and the rest
are zero and :math:`d_j` is a scalar spacing :math:`steps_j`.
""", example="""
Example
-------
>>> import numpy as np
>>> import numdifftools as nd
# Rosenbrock function, minimized at [1,1]
>>> rosen = lambda x : (1.-x[0])**2 + 105*(x[1]-x[0]**2)**2
>>> Hfun = nd.Hessian(rosen)
>>> h = Hfun([1, 1])
>>> h
array([[ 842., -420.],
[-420., 210.]])
# cos(x-y), at (0,0)
>>> cos = np.cos
>>> fun = lambda xy : cos(xy[0]-xy[1])
>>> Hfun2 = nd.Hessian(fun)
>>> h2 = Hfun2([0, 0])
>>> h2
array([[-1., 1.],
[ 1., -1.]])""", see_also="""
See also
--------
Derivative, Hessian
""")
def _complex_high_order(self):
return False
def _derivative(self, xi, args, kwds):
xi = np.atleast_1d(xi)
diff, f = self._get_functions()
steps = self._get_steps(xi)
fxi = self._eval_first(f, xi, *args, **kwds)
results = [diff(f, fxi, xi, h, *args, **kwds) for h in steps]
step_ratio = self._compute_step_ratio(steps)
self._set_richardson_rule(step_ratio, self.richardson_terms)
return self._vstack(results, steps)
@staticmethod
def _complex_even(f, fx, x, h, *args, **kwargs):
"""
Calculate Hessian with complex-step derivative approximation
The stepsize is the same for the complex and the finite difference part
"""
n = len(x)
# h = _default_base_step(x, 3, base_step, n)
ee = np.diag(h)
hes = 2. * np.outer(h, h)
for i in range(n):
for j in range(i, n):
hes[i, j] = (f(x + 1j * ee[i] + ee[j], *args, **kwargs) -
f(x + 1j * ee[i] - ee[j], *args, **kwargs)
).imag / hes[j, i]
hes[j, i] = hes[i, j]
return hes
@staticmethod
def _multicomplex2(f, fx, x, h, *args, **kwargs):
"""Calculate Hessian with bicomplex-step derivative approximation"""
n = len(x)
ee = np.diag(h)
hess = np.outer(h, h)
for i in range(n):
for j in range(i, n):
zph = bicomplex(x + 1j * ee[i, :], ee[j, :])
hess[i, j] = (f(zph, *args, **kwargs)).imag12 / hess[j, i]
hess[j, i] = hess[i, j]
return hess
@staticmethod
def _central_even(f, fx, x, h, *args, **kwargs):
"""Eq 9."""
n = len(x)
# h = _default_base_step(x, 4, base_step, n)
ee = np.diag(h)
hess = np.outer(h, h)
for i in range(n):
hess[i, i] = (f(x + 2*ee[i, :], *args, **kwargs) - 2*fx +
f(x - 2*ee[i, :], *args, **kwargs)
) / (4. * hess[i, i])
for j in range(i+1, n):
hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) -
f(x + ee[i, :] - ee[j, :], *args, **kwargs) -
f(x - ee[i, :] + ee[j, :], *args, **kwargs) +
f(x - ee[i, :] - ee[j, :], *args, **kwargs)
) / (4. * hess[j, i])
hess[j, i] = hess[i, j]
return hess
@staticmethod
def _central2(f, fx, x, h, *args, **kwargs):
"""Eq. 8"""
n = len(x)
# NOTE: ridout suggesting using eps**(1/4)*theta
# h = _default_base_step(x, 3, base_step, n)
ee = np.diag(h)
dtype = np.result_type(fx)
g = np.empty(n, dtype=dtype)
gg = np.empty(n, dtype=dtype)
for i in range(n):
g[i] = f(x + ee[i], *args, **kwargs)
gg[i] = f(x - ee[i], *args, **kwargs)
hess = np.empty((n, n), dtype=dtype)
np.outer(h, h, out=hess)
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) -
g[i] - g[j] + fx +
f(x - ee[i, :] - ee[j, :], *args, **kwargs) -
gg[i] - gg[j] + fx) / (2 * hess[j, i])
hess[j, i] = hess[i, j]
return hess
@staticmethod
def _forward(f, fx, x, h, *args, **kwargs):
"""Eq. 7"""
n = len(x)
ee = np.diag(h)
dtype = np.result_type(fx)
g = np.empty(n, dtype=dtype)
for i in range(n):
g[i] = f(x + ee[i, :], *args, **kwargs)
hess = np.empty((n, n), dtype=dtype)
np.outer(h, h, out=hess)
for i in range(n):
for j in range(i, n):
hess[i, j] = (f(x + ee[i, :] + ee[j, :], *args, **kwargs) -
g[i] - g[j] + fx) / hess[j, i]
hess[j, i] = hess[i, j]
return hess
def _backward(self, f, fx, x, h, *args, **kwargs):
return self._forward(f, fx, x, -h, *args, **kwargs)
def test_docstrings():
import doctest
doctest.testmod(optionflags=doctest.NORMALIZE_WHITESPACE)
if __name__ == '__main__': # pragma : no cover
test_docstrings()