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# -*- coding: utf-8 -*- 

 

u'''I{Karney}'s elliptic functions and integrals. 

 

Class L{Elliptic} transcoded from I{Charles Karney}'s C++ class U{EllipticFunction 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1EllipticFunction.html>} 

to pure Python, including symmetric integrals L{Elliptic.fRC}, L{Elliptic.fRD}, 

L{Elliptic.fRF}, L{Elliptic.fRG} and L{Elliptic.fRJ} as C{static methods}. 

 

Python method names follow the C++ member functions, I{except}: 

 

- member functions I{without arguments} are mapped to Python properties 

prefixed with C{"c"}, for example C{E()} is property C{cE}, 

 

- member functions with 1 or 3 arguments are renamed to Python methods 

starting with an C{"f"}, example C{E(psi)} to C{fE(psi)} and C{E(sn, 

cn, dn)} to C{fE(sn, cn, dn)}, 

 

- other Python method names conventionally start with a lower-case 

letter or an underscore if private. 

 

Following is a copy of I{Karney}'s U{EllipticFunction.hpp 

<https://GeographicLib.SourceForge.io/html/EllipticFunction_8hpp_source.html>} 

file C{Header}. 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2021) 

and licensed under the MIT/X11 License. For more information, see the 

U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. 

 

B{Elliptic integrals and functions.} 

 

This provides the elliptic functions and integrals needed for 

C{Ellipsoid}, C{GeodesicExact}, and C{TransverseMercatorExact}. Two 

categories of function are provided: 

 

- functions to compute U{symmetric elliptic integrals 

<https://DLMF.NIST.gov/19.16.i>} 

 

- methods to compute U{Legrendre's elliptic integrals 

<https://DLMF.NIST.gov/19.2.ii>} and U{Jacobi elliptic 

functions<https://DLMF.NIST.gov/22.2>}. 

 

In the latter case, an object is constructed giving the modulus 

C{k} (and optionally the parameter C{alpha}). The modulus (and 

parameter) are always passed as squares which allows C{k} to be 

pure imaginary. (Confusingly, Abramowitz and Stegun call C{m = k**2} 

the "parameter" and C{n = alpha**2} the "characteristic".) 

 

In geodesic applications, it is convenient to separate the incomplete 

integrals into secular and periodic components, e.g. 

 

I{C{E(phi, k) = (2 E(k) / pi) [ phi + delta E(phi, k) ]}} 

 

where I{C{delta E(phi, k)}} is an odd periodic function with 

period I{C{pi}}. 

 

The computation of the elliptic integrals uses the algorithms given 

in U{B. C. Carlson, Computation of real or complex elliptic integrals 

<https://DOI.org/10.1007/BF02198293>} (also available U{here 

<https://ArXiv.org/pdf/math/9409227.pdf>}), Numerical Algorithms 10, 

13--26 (1995) with the additional optimizations given U{here 

<https://DLMF.NIST.gov/19.36.i>}. 

 

The computation of the Jacobi elliptic functions uses the algorithm 

given in U{R. Bulirsch, Numerical Calculation of Elliptic Integrals 

and Elliptic Functions<https://DOI.org/10.1007/BF01397975>}, 

Numerische Mathematik 7, 78--90 (1965). 

 

The notation follows U{NIST Digital Library of Mathematical Functions 

<https://DLMF.NIST.gov>} chapters U{19<https://DLMF.NIST.gov/19>} and 

U{22<https://DLMF.NIST.gov/22>}. 

''' 

# make sure int/int division yields float quotient, see .basics 

from __future__ import division as _; del _ # PYCHOK semicolon 

 

from pygeodesy.basics import copysign0, map2, neg 

# from pygeodesy.errors import _ValueError # from .fmath 

from pygeodesy.fmath import fdot, Fsum, hypot1, _ValueError 

# from pygeodesy.fsums import Fsum # from .fmath 

from pygeodesy.interns import EPS, INF, NN, PI, PI_2, PI_4, \ 

_DOT_, _EPStol as _TolJAC, _convergence_, \ 

_f_, _no_, _SPACE_, _0_0, _0_125, _0_25, \ 

_0_5, _1_0, _2_0, _N_2_0, _3_0, _4_0, \ 

_5_0, _6_0, _8_0, _180_0, _360_0 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import _Named, _NamedTuple, unstr 

from pygeodesy.props import Property_RO, property_RO, _update_all 

# from pygeodesy.streprs import unstr # from .named 

from pygeodesy.units import Scalar, Scalar_ 

from pygeodesy.utily import sincos2, sincos2d 

 

from math import asinh, atan, atan2, ceil, cosh, floor, sin, \ 

sqrt, tanh 

 

__all__ = _ALL_LAZY.elliptic 

__version__ = '22.05.04' 

 

_delta_ = 'delta' 

_invokation_ = 'invokation' 

_1_64th = _1_0 / 64 # pow(2.0, -6) 

_TolRD = pow(EPS * 0.002, _0_125) 

_TolRF = pow(EPS * 0.030, _0_125) 

_TolRG0 = _TolJAC * 2.7 

_TRIPS = 31 # Max depth, 7 might be sufficient 

 

 

class _Complete(object): 

'''(INTERAL) Hold complete integrals. 

''' 

def __init__(self, **kwds): 

self.__dict__ = kwds 

 

 

class Elliptic(_Named): 

'''Elliptic integrals and functions. 

 

@see: I{Karney}'s U{Detailed Description<https://GeographicLib.SourceForge.io/ 

html/classGeographicLib_1_1EllipticFunction.html#details>}. 

''' 

_alpha2 = 0 

_alphap2 = 0 

_eps = EPS 

_iteration = None # Iteration number (C{int}) or C{None} 

_k2 = 0 

_kp2 = 0 

 

def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None): 

'''Constructor, specifying the C{modulus} and C{parameter}. 

 

@see: Method L{Elliptic.reset} for further details. 

 

@note: If only elliptic integrals of the first and second kinds 

are needed, use C{B{alpha2}=0}, the default value. In 

that case, we have C{Π(φ, 0, k) = F(φ, k), G(φ, 0, k) = 

E(φ, k)} and C{H(φ, 0, k) = F(φ, k) - D(φ, k)}. 

''' 

self.reset(k2=k2, alpha2=alpha2, kp2=kp2, alphap2=alphap2) 

 

@Property_RO 

def alpha2(self): 

'''Get α^2, the square of the parameter (C{float}). 

''' 

return self._alpha2 

 

@Property_RO 

def alphap2(self): 

'''Get α'^2, the square of the complementary parameter (C{float}). 

''' 

return self._alphap2 

 

@Property_RO 

def cD(self): 

'''Get Jahnke's complete integral C{D(k)} (C{float}), 

U{defined<https://DLMF.NIST.gov/19.2.E6>}. 

''' 

return self._reset4integrals.cD 

 

@Property_RO 

def cE(self): 

'''Get the complete integral of the second kind C{E(k)} 

(C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}. 

''' 

return self._reset4integrals.cE 

 

@Property_RO 

def cG(self): 

'''Get Legendre's complete geodesic longitude integral 

C{G(α^2, k)} (C{float}). 

''' 

return self._reset3integrals.cG 

 

@Property_RO 

def cH(self): 

'''Get Cayley's complete geodesic longitude difference integral 

C{H(α^2, k)} (C{float}). 

''' 

return self._reset3integrals.cH 

 

@Property_RO 

def cK(self): 

'''Get the complete integral of the first kind C{K(k)} 

(C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}. 

''' 

return self._reset4integrals.cK 

 

@Property_RO 

def cKE(self): 

'''Get the difference between the complete integrals of the 

first and second kinds, C{K(k) − E(k)} (C{float}). 

''' 

return self._reset4integrals.cKE 

 

@Property_RO 

def cPi(self): 

'''Get the complete integral of the third kind C{Pi(α^2, k)} 

(C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}. 

''' 

return self._reset3integrals.cPi 

 

def deltaD(self, sn, cn, dn): 

'''The periodic Jahnke's incomplete elliptic integral. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cD, self.fD) 

 

def deltaE(self, sn, cn, dn): 

'''The periodic incomplete integral of the second kind. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cE, self.fE) 

 

def deltaEinv(self, stau, ctau): 

'''The periodic inverse of the incomplete integral of the second kind. 

 

@arg stau: sin(τ) 

@arg ctau: cos(τ) 

 

@return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}). 

 

@raise EllipticError: No convergence. 

''' 

# Function is periodic with period pi 

t = atan2(-stau, -ctau) if ctau < 0 else atan2(stau, ctau) 

return self.fEinv(t * self.cE / PI_2) - t 

 

def deltaF(self, sn, cn, dn): 

'''The periodic incomplete integral of the first kind. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cK, self.fF) 

 

def deltaG(self, sn, cn, dn): 

'''Legendre's periodic geodesic longitude integral. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cG, self.fG) 

 

def deltaH(self, sn, cn, dn): 

'''Cayley's periodic geodesic longitude difference integral. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cH, self.fH) 

 

def deltaPi(self, sn, cn, dn): 

'''The periodic incomplete integral of the third kind. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

@arg dn: sqrt(1 − k2 sin(2φ)). 

 

@return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ 

(C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._deltaX(sn, cn, dn, self.cPi, self.fPi) 

 

def _deltaX(self, sn, cn, dn, cX, fX): 

'''(INTERNAL) Helper for C{.deltaD} thru C{.deltaPi}. 

''' 

if cn is None or dn is None: 

n = NN(_delta_, fX.__name__[1:]) 

raise _callError(n, sn, cn, dn) 

 

if cn < 0: 

cn, sn = -cn, -sn 

return fX(sn, cn, dn) * PI_2 / cX - atan2(sn, cn) 

 

@Property_RO 

def eps(self): 

'''Get epsilon (C{float}). 

''' 

return self._reset4integrals.eps 

 

def fD(self, phi_or_sn, cn=None, dn=None): 

'''Jahnke's incomplete elliptic integral in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: D(φ, k) as though φ ∈ (−π, π] (C{float}), 

U{defined<https://DLMF.NIST.gov/19.2.E4>}. 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

def _fD(sn, cn, dn): 

r = abs(sn)**3 

if r: 

r = _RD(self, cn**2, dn**2, _1_0, _3_0 / r) 

return r 

 

return self._fXf(phi_or_sn, cn, dn, self.cD, 

self.deltaD, _fD) 

 

def fDelta(self, sn, cn): 

'''The C{Delta} amplitude function. 

 

@arg sn: sin(φ). 

@arg cn: cos(φ). 

 

@return: sqrt(1 − k2 sin(2φ)) (C{float}). 

''' 

k2, kp2 = self.k2, self.kp2 

s = (_1_0 - k2 * sn**2) if k2 < 0 else ( 

(kp2 + k2 * cn**2) if k2 > 0 else kp2) 

return sqrt(s) if s else _0_0 

 

def fE(self, phi_or_sn, cn=None, dn=None): 

'''The incomplete integral of the second kind in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: E(φ, k) as though φ ∈ (−π, π] (C{float}), 

U{defined<https://DLMF.NIST.gov/19.2.E5>}. 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

def _fE(sn, cn, dn): 

'''(INTERNAL) Core of C{.fE}. 

''' 

if sn: 

sn2, cn2, dn2 = sn**2, cn**2, dn**2 

kp2, k2 = self.kp2, self.k2 

if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9> 

ei = _RF3(self, cn2, dn2, _1_0) 

if k2: 

ei -= _RD(self, cn2, dn2, _1_0, _3rd(k2, sn2)) 

elif kp2 >= 0: # <https://DLMF.NIST.gov/19.25.E10> 

ei = k2 * abs(cn) / dn 

if kp2: 

ei += (_RD(self, cn2, _1_0, dn2, _3rd(k2, sn2)) + 

_RF3(self, cn2, dn2, _1_0)) * kp2 

else: # <https://DLMF.NIST.gov/19.25.E11> 

ei = dn / abs(cn) - _RD(self, dn2, _1_0, cn2, _3rd(kp2, sn2)) 

ei *= abs(sn) 

else: # PYCHOK no cover 

ei = _0_0 

return ei 

 

return self._fXf(phi_or_sn, cn, dn, self.cE, 

self.deltaE, _fE) 

 

def fEd(self, deg): 

'''The incomplete integral of the second kind with 

the argument given in degrees. 

 

@arg deg: Angle (C{degrees}). 

 

@return: E(π B{C{deg}}/180, k) (C{float}). 

 

@raise EllipticError: No convergence. 

''' 

if abs(deg) < _180_0: 

e = _0_0 

else: # PYCHOK no cover 

e = ceil(deg / _360_0 - _0_5) 

deg -= _360_0 * e 

e *= _4_0 * self.cE 

sn, cn = sincos2d(deg) 

return self.fE(sn, cn, self.fDelta(sn, cn)) + e 

 

def fEinv(self, x): 

'''The inverse of the incomplete integral of the second kind. 

 

@arg x: Argument (C{float}). 

 

@return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}} 

(C{float}). 

 

@raise EllipticError: No convergence. 

''' 

E2 = self.cE * _2_0 

n = floor(x / E2 + _0_5) 

y = x - E2 * n # y now in [-ec, ec) 

# linear approximation 

phi = PI * y / E2 # phi in [-pi/2, pi/2) 

Phi = Fsum(phi) 

# first order correction 

phi = Phi.fsum_(self.eps * sin(_2_0 * phi) / _N_2_0) 

# For kp2 close to zero use asin(x/.cE) or J. P. Boyd, 

# Applied Math. and Computation 218, 7005-7013 (2012) 

# <https://DOI.org/10.1016/j.amc.2011.12.021> 

Phi_fsum2_ = Phi.fsum2_ 

fE = self.fE 

_sncndnPhi = self._sncndnPhi 

self._iteration = 0 # aggregate 

for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC 

sn, cn, dn = _sncndnPhi(phi) 

phi, e = Phi_fsum2_((y - fE(sn, cn, dn)) / dn) 

if abs(e) < _TolJAC: 

self._iteration += i 

break 

else: # PYCHOK no cover 

raise _convergenceError(_TolJAC, self.fEinv, x) 

return Phi.fsum_(n * PI) if n else phi 

 

def fF(self, phi_or_sn, cn=None, dn=None): 

'''The incomplete integral of the first kind in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: F(φ, k) as though φ ∈ (−π, π] (C{float}), 

U{defined<https://DLMF.NIST.gov/19.2.E4>}. 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

def _fF(sn, cn, dn): 

r = abs(sn) 

if r: 

r *= _RF3(self, cn**2, dn**2, _1_0) 

return r 

 

return self._fXf(phi_or_sn, cn, dn, self.cK, 

self.deltaF, _fF) 

 

def fG(self, phi_or_sn, cn=None, dn=None): 

'''Legendre's geodesic longitude integral in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: G(φ, k) as though φ ∈ (−π, π] (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

 

@note: Legendre expresses the longitude of a point on the 

geodesic in terms of this combination of elliptic 

integrals in U{Exercices de Calcul Intégral, Vol 1 

(1811), p 181<https://Books.Google.com/books?id= 

riIOAAAAQAAJ&pg=PA181>}. 

 

@see: U{Geodesics in terms of elliptic integrals<https:// 

GeographicLib.SourceForge.io/html/geodesic.html#geodellip>} 

for the expression for the longitude in terms of this function. 

''' 

return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2, 

self.cG, self.deltaG) 

 

def fH(self, phi_or_sn, cn=None, dn=None): 

'''Cayley's geodesic longitude difference integral in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: H(φ, k) as though φ ∈ (−π, π] (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

 

@note: Cayley expresses the longitude difference of a point 

on the geodesic in terms of this combination of 

elliptic integrals in U{Phil. Mag. B{40} (1870), p 333 

<https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}. 

 

@see: U{Geodesics in terms of elliptic integrals<https:// 

GeographicLib.SourceForge.io/html/geodesic.html#geodellip>} 

for the expression for the longitude in terms of this function. 

''' 

return self._fXa(phi_or_sn, cn, dn, -self.alphap2, 

self.cH, self.deltaH) 

 

def fPi(self, phi_or_sn, cn=None, dn=None): 

'''The incomplete integral of the third kind in terms of 

Jacobi elliptic functions. 

 

@arg phi_or_sn: φ or sin(φ). 

@kwarg cn: C{None} or cos(φ). 

@kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). 

 

@return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}). 

 

@raise EllipticError: Invalid invokation or no convergence. 

''' 

return self._fXa(phi_or_sn, cn, dn, self.alpha2, 

self.cPi, self.deltaPi) 

 

def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX): 

'''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}. 

''' 

def _fX(sn, cn, dn): 

if sn: 

cn2, sn2, dn2 = cn**2, sn**2, dn**2 

r = _RF3(self, cn2, dn2, _1_0) 

if aX: 

z = cn2 + sn2 * self.alphap2 

r += _RJ(self, cn2, dn2, _1_0, z, _3rd(aX, sn2)) 

r *= abs(sn) 

else: # PYCHOK no cover 

r = _0_0 

return r 

 

return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX) 

 

def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX): 

'''(INTERNAL) Helper for C{f.D}, C{.fE}, C{.fF} and C{._fXa}. 

''' 

self._iteration = 0 # aggregate 

phi = sn = phi_or_sn 

if cn is dn is None: # fX(phi) call 

sn, cn, dn = self._sncndnPhi(phi) 

if abs(phi) >= PI: # PYCHOK no cover 

return (deltaX(sn, cn, dn) + phi) * cX / PI_2 

# fall through 

elif cn is None or dn is None: 

n = NN(_f_, deltaX.__name__[5:]) 

raise _callError(n, sn, cn, dn) 

 

if cn < 0: # enforce usual trig-like symmetries 

xi = _2_0 * cX - fX(sn, cn, dn) 

elif cn > 0: 

xi = fX(sn, cn, dn) 

else: 

xi = cX 

return copysign0(xi, sn) 

 

@property_RO 

def iteration(self): 

'''Get the most recent C{Elliptic.fEinv} or C{Elliptic.sncndn} 

iteration number (C{int}) or C{None} if not available/applicable. 

 

@note: The interation number may be an aggregate number over 

several, nested functions. 

''' 

return self._iteration 

 

@Property_RO 

def k2(self): 

'''Get k^2, the square of the modulus (C{float}). 

''' 

return self._k2 

 

@Property_RO 

def kp2(self): 

'''Get k'^2, the square of the complementary modulus (C{float}). 

''' 

return self._kp2 

 

def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13 

'''Reset the modulus, parameter and the complementaries. 

 

@kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1). 

@kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1). 

@kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0). 

@kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0). 

 

@raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}} 

or B{C{alphap2}}. 

 

@note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and 

C{B{alpha2} + B{alphap2} = 1}. No checking is done 

that these conditions are met to enable accuracy to be 

maintained, e.g., when C{k} is very close to unity. 

''' 

_update_all(self, Elliptic.iteration._uname) 

 

self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0) 

self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError) 

 

self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0) 

self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2), 

Error=EllipticError) 

 

# Values of complete elliptic integrals for k = 0,1 and alpha = 0,1 

# K E D 

# k = 0: pi/2 pi/2 pi/4 

# k = 1: inf 1 inf 

# Pi G H 

# k = 0, alpha = 0: pi/2 pi/2 pi/4 

# k = 1, alpha = 0: inf 1 1 

# k = 0, alpha = 1: inf inf pi/2 

# k = 1, alpha = 1: inf inf inf 

# 

# G(0, k) = Pi(0, k) = H(1, k) = E(k) 

# H(0, k) = K(k) - D(k) 

# Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2)) 

# H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1)) 

# Pi(alpha2, 1) = inf 

# G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2) 

 

@Property_RO 

def _reset3integrals(self): 

'''(INTERNAL) Get the complete integrals G, H and Pi. 

''' 

self._iteration = 0 

alpha2 = self.alpha2 

if alpha2: 

alphap2 = self.alphap2 

if alphap2: 

kp2 = self.kp2 

if kp2: # <https://DLMF.NIST.gov/19.25.E2> 

rj = _RJ(self, _0_0, kp2, _1_0, alphap2, _3_0) 

cPi = cH = cG = self.cK 

cG += (alpha2 - self.k2) * rj # G(alpha2, k) 

cH -= alphap2 * rj # H(alpha2, k) 

cPi += alpha2 * rj # Pi(alpha2, k) 

else: # PYCHOK no cover 

cG = cH = _RC(self, _1_0, alphap2) 

cPi = INF # XXX or NAN? 

else: # PYCHOK no cover 

cG = cH = cPi = INF # XXX or NAN? 

else: 

cG, cPi, kp2 = self.cE, self.cK, self.kp2 

# H = K - D but this involves large cancellations if k2 is near 1. 

# So write (for alpha2 = 0) 

# H = int(cos(phi)**2/sqrt(1-k2*sin(phi)**2),phi,0,pi/2) 

# = 1/sqrt(1-k2) * int(sin(phi)**2/sqrt(1-k2/kp2*sin(phi)**2,...) 

# = 1/kp * D(i*k/kp) 

# and use D(k) = RD(0, kp2, 1) / 3 

# so H = 1/kp * RD(0, 1/kp2, 1) / 3 

# = kp2 * RD(0, 1, kp2) / 3 

# using <https://DLMF.NIST.gov/19.20.E18>. Equivalently 

# RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0 

# For k2 = 1 and alpha2 = 0, we have 

# H = int(cos(phi),...) = 1 

cH = _RD(self, _0_0, _1_0, kp2, _3_0 / kp2) if kp2 else _1_0 

 

return _Complete(cG=cG, cH=cH, cPi=cPi) 

 

@Property_RO 

def _reset4integrals(self): 

'''(INTERNAL) Get the complete integrals D, E, K and KE plus C{eps}. 

''' 

k2 = self.k2 

if k2: 

kp2 = self.kp2 

if kp2: 

self._iteration = 0 

# D(k) = (K(k) - E(k))/k2, Carlson eq.4.3 

# <https://DLMF.NIST.gov/19.25.E1> 

cD = _RD(self, _0_0, kp2, _1_0, _3_0) 

# Complete elliptic integral E(k), Carlson eq. 4.2 

# <https://DLMF.NIST.gov/19.25.E1> 

cE = _RG2(self, kp2, _1_0) 

# Complete elliptic integral K(k), Carlson eq. 4.1 

# <https://DLMF.NIST.gov/19.25.E1> 

cK = _RF2(self, kp2, _1_0) 

cKE = k2 * cD 

eps = k2 / (sqrt(kp2) + _1_0)**2 

else: # PYCHOK no cover 

cD = cK = cKE = INF 

cE = _1_0 

eps = k2 

else: # PYCHOK no cover 

cD = PI_4 

cE = cK = PI_2 

cKE = _0_0 # k2 * cD 

eps = EPS 

 

return _Complete(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps) 

 

def sncndn(self, x): 

'''The Jacobi elliptic function. 

 

@arg x: The argument (C{float}). 

 

@return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with 

C{*n(B{x}, k)}. 

 

@raise EllipticError: No convergence. 

''' 

self._iteration = 0 

# Bulirsch's sncndn routine, p 89. 

if self.kp2: 

c, d, mn_ = self._sncndnBulirsch 

dn = _1_0 

x *= (c * d) if d else c 

sn, cn = sincos2(x) 

if sn: 

a = cn / sn 

c *= a 

for m, n in mn_: 

a *= c 

c *= dn 

dn = (n + a) / (m + a) 

a = c / m 

sn = copysign0(_1_0 / hypot1(c), sn) 

cn = c * sn 

if d: # PYCHOK no cover 

cn, dn = dn, cn 

sn = sn / d # /= d chokes PyChecker 

else: 

sn = tanh(x) 

cn = dn = _1_0 / cosh(x) 

 

r = Elliptic3Tuple(sn, cn, dn) 

r._iteration = self._iteration 

return r 

 

@Property_RO 

def _sncndnBulirsch(self): 

'''(INTERNAL) Get and cache Bulirsch' 3-tuple C{(c, d, mn_)}. 

''' 

# Bulirsch's sncndn routine, p 89. 

d, mc = 0, self.kp2 

if mc < 0: # PYCHOK no cover 

d = _1_0 - mc 

mc = neg(mc / d) 

d = sqrt(d) 

 

a, mn, t = _1_0, [], _TolJAC 

for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC 

# This converges quadratically, max 6 trips 

mc = sqrt(mc) 

mn.append((a, mc)) 

c = (a + mc) * _0_5 

if abs(a - mc) <= t: 

self._iteration += i 

break 

mc *= a 

a = c 

t = _TolJAC * a 

else: # PYCHOK no cover 

raise _convergenceError(t, None, kp=self.kp, kp2=self.kp2) 

return c, d, tuple(reversed(mn)) # mn reversed! 

 

def _sncndnPhi(self, phi): 

'''(INTERNAL) Helper for C{.fEinv} and C{._fXf}. 

''' 

sn, cn = sincos2(phi) 

return Elliptic3Tuple(sn, cn, self.fDelta(sn, cn)) 

 

@staticmethod 

def fRC(x, y): 

'''Degenerate symmetric integral of the first kind C{RC(x, y)}. 

 

@return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}. 

 

@see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and 

U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. 

''' 

return _RC(None, x, y) 

 

@staticmethod 

def fRD(x, y, z): 

'''Degenerate symmetric integral of the third kind C{RD(x, y, z)}. 

 

@return: C{RD(x, y, z)}, equivalent to C{RJ(x, y, z, z)}. 

 

@see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and 

U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. 

''' 

return _RD(None, x, y, z) 

 

@staticmethod 

def fRF(x, y, *z): 

'''Symmetric or complete symmetric integral of the first kind 

C{RF(x, y, z)} respectively C{RF(x, y)}. 

 

@return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}. 

 

@see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and 

U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. 

''' 

return _RF3(None, x, y, *z) if z and z[0] else _RF2(None, x, y) 

 

@staticmethod 

def fRG(x, y, *z): 

'''Symmetric or complete symmetric integral of the second kind 

C{RG(x, y, z)} respectively C{RG(x, y)}. 

 

@return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}. 

 

@see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>} and 

U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. 

''' 

return _RG3(None, x, y, *z) if z and z[0] else (_RG2(None,x, y) * _0_5) 

 

@staticmethod 

def fRJ(x, y, z, p): 

'''Symmetric integral of the third kind C{RJ(x, y, z, p)}. 

 

@return: C{RJ(x, y, z, p)}. 

 

@see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and 

U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. 

''' 

return _RJ(None, x, y, z, p) 

 

 

class EllipticError(_ValueError): 

'''Elliptic integral, function, convergence or other L{Elliptic} issue. 

''' 

pass 

 

 

class Elliptic3Tuple(_NamedTuple): 

'''3-Tuple C{(sn, cn, dn)} all C{scalar}. 

''' 

_Names_ = ('sn', 'cn', 'dn') 

_Units_ = ( Scalar, Scalar, Scalar) 

 

 

def _callError(name, *args): # PYCHOK no cover 

'''(INTERNAL) Return an L{EllipticError}. 

''' 

n = _DOT_(Elliptic.__name__, name) 

n = _SPACE_(_invokation_, n) 

return EllipticError(NN(n, repr(args))) # unstr 

 

 

def _convergenceError(tol, where, *args, **kwds): # PYCHOK no cover 

'''(INTERNAL) Return an L{EllipticError}. 

''' 

n = Elliptic.__name__ 

if where: 

n = _DOT_(n, where.__name__) 

t = unstr(n, *args, **kwds) 

return EllipticError(_no_(_convergence_), tol, txt=t) 

 

 

def _horner(S, e1, E2, E3, E4, E5, *over): 

'''(INTERNAL) Horner form for C{_RD} and C{_RJ} below. 

''' 

E22 = E2**2 

# Polynomial is <https://DLMF.NIST.gov/19.36.E2> 

# (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52 

# + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20 

# + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68) 

# converted to Horner form ... 

e = e1 * 4084080 

S *= e 

S += Fsum( E2 * -540540, 471240).fmul(E5) 

S += Fsum( E3 * -540540, E2 * 612612, -556920).fmul(E4) 

S += Fsum(E22 * 675675, E3 * 306306, E2 * -706860, 680680).fmul(E3) 

S += Fsum(E22 * -255255, E2 * 417690, -875160).fmul(E2) 

return S.fadd_(4084080).fover((e * over[0]) if over else e) 

 

 

def _iterations(inst, i): 

'''(INTERNAL) Aggregate iterations B{C{i}}. 

''' 

if inst: 

inst._iteration += i 

 

 

def _Q(_Tol, A0, *ts): 

'''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}. 

''' 

return max(abs(A0 - t) for t in ts) / _Tol 

 

 

def _RC(unused, x, y): 

'''(INTERNAL) Defined only for y != 0 and x >= 0. 

''' 

d = x - y 

if d < 0: # catch _NaN 

# <https://DLMF.NIST.gov/19.2.E18> 

d = -d 

r = atan(sqrt(d / x)) if x > 0 else PI_2 

elif d == _0_0: # XXX d < EPS0? or EPS02 or _EPSmin 

r, d = _1_0, y 

elif y > 0: # <https://DLMF.NIST.gov/19.2.E19> 

r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x)) 

elif y < 0: # <https://DLMF.NIST.gov/19.2.E20> 

r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y))) 

else: 

raise _callError(Elliptic.fRC.__name__, x, y) 

return r / sqrt(d) 

 

 

def _RD(inst, x, y, z, *over): 

'''(INTERNAL) Carlson, eqs 2.28 - 2.34. 

''' 

A0 = Fsum(x, y, _3_0 * z).fover(_5_0) 

T = (A0, x, y, z) 

Q = _Q(_TolRF, *T) 

S = Fsum() 

m = 1 

for i in range(_TRIPS): 

An = T[0] 

Am = An * m 

if Q < abs(Am): # max 7 trips 

_iterations(inst, i) 

break 

t = T[3] # z0...n 

T, s, r = _Tsr3(T) 

S += _1_0 / ((t + r) * s[2] * m) 

m *= 4 

else: # PYCHOK no cover 

raise _convergenceError(Q, Elliptic.fRD, x, y, z) 

 

x, y = _xyz(A0, -Am, x, y) 

z = (x + y) / _3_0 

z2 = z**2 

xy = x * y 

S *= _3_0 

return _horner(S, Am * sqrt(An), 

xy - _6_0 * z2, 

(xy * _3_0 - _8_0 * z2) * z, 

(xy - z2) * _3_0 * z2, 

xy * z2 * z, *over) 

 

 

def _3rd(a, b): 

'''(INTERNAL) Return _horner C{over} value. 

''' 

return _3_0 / (a * b) 

 

 

def _RF2(inst, x, y): # 2-arg version, z=0 

'''(INTERNAL) Carlson, eqs 2.36 - 2.38. 

''' 

a, b = sqrt(x), sqrt(y) 

if a < b: 

a, b = b, a 

for i in range(_TRIPS): 

t = _TolRG0 * a 

if abs(a - b) <= t: # max 4 trips 

_iterations(inst, i) 

return (PI / (a + b)) 

b, a = sqrt(a * b), (a + b) * _0_5 

else: # PYCHOK no cover 

raise _convergenceError(t, Elliptic.fRF, x, y) 

 

 

def _RF3(inst, x, y, z): # 3-arg version 

'''(INTERNAL) Carlson, eqs 2.2 - 2.7. 

''' 

A0 = Fsum(x, y, z).fover(_3_0) 

T = (A0, x, y, z) 

Q = _Q(_TolRF, *T) 

m = 1 

for i in range(_TRIPS): 

An = T[0] 

Am = An * m 

if Q < abs(Am): # max 6 trips 

_iterations(inst, i) 

break 

T, _, _ = _Tsr3(T) 

m *= 4 

else: # PYCHOK no cover 

raise _convergenceError(Q, Elliptic.fRF, x, y, z) 

 

x, y = _xyz(A0, Am, x, y) 

z = neg(x + y) 

e2 = x * y - z**2 

e3 = x * y * z 

e4 = e2**2 

# Polynomial is <https://DLMF.NIST.gov/19.36.E1> 

# (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44 

# - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16) 

# converted to Horner form ... 

S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3) 

S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2) 

return S.fadd_(240240).fover(sqrt(An) * 240240) 

 

 

def _RG2(inst, x, y): # 2-args and I{doubled} 

'''(INTERNAL) Carlson, eqs 2.36 - 2.39. 

''' 

a, b = sqrt(x), sqrt(y) 

if a < b: 

a, b = b, a 

ab = a - b # abs(a - b) 

S = Fsum(_0_5 * (a + b)**2) 

m = -1 

for i in range(_TRIPS): # max 4 trips 

t = _TolRG0 * a 

if ab <= t: 

_iterations(inst, i) 

return S.fover((a + b) / PI_2) 

a, b = ((a + b) * _0_5), sqrt(a * b) 

ab = abs(a - b) 

S += ab**2 * m 

m *= 2 

else: # PYCHOK no cover 

raise _convergenceError(t, Elliptic.fRG, x, y) 

 

 

def _RG3(inst, x, y, z): # 3-arg version 

'''(INTERNAL) Never called with zero B{C{z}}, see C{.fRG}. 

''' 

# if not z: 

# y, z = z, y 

rd = (x - z) * (z - y) # - (y - z) 

if rd: # Carlson, eq 1.7 

rd = _RD(inst, x, y, z, _3_0 * z / rd) 

xyz = x * y 

if xyz: 

xyz = sqrt(xyz / z**3) 

return Fsum(_RF3(inst, x, y, z), rd, xyz).fover(_2_0 / z) 

 

 

def _RJ(inst, x, y, z, p, *over): 

'''(INTERNAL) Carlson, eqs 2.17 - 2.25. 

''' 

def _xyzp(x, y, z, p): 

return (x + p) * (y + p) * (z + p) 

 

A0 = Fsum(x, y, z, _2_0 * p).fover(_5_0) 

T = (A0, x, y, z, p) 

Q = _Q(_TolRD, *T) 

S = Fsum() 

m = 1 

Dn = neg(_xyzp(x, y, z, -p)) 

for i in range(_TRIPS): 

An = T[0] 

Am = An * m 

if Q < abs(Am): # max 7 trips 

_iterations(inst, i) 

break 

T, s, _ = _Tsr3(T) 

d = _xyzp(*s) 

if Dn: 

rc = _RC(inst, _1_0, Dn / d**2 + _1_0) 

Dn *= _1_64th 

else: 

rc = _1_0 # == _RC(None, _1_0, _1_0) 

S += rc / (d * m) 

m *= 4 

else: # PYCHOK no cover 

raise _convergenceError(Q, Elliptic.fRJ, x, y, z, p) 

 

x, y, z = _xyz(A0, Am, x, y, z) 

xyz = x * y * z 

p = -Fsum(x, y, z).fover(_2_0) 

p2 = p**2 

p3 = p**3 

E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0) 

E2p = E2 * p 

S *= _6_0 

return _horner(S, Am * sqrt(An), E2, 

Fsum(_4_0 * p3, xyz, E2p * _2_0), 

Fsum(_3_0 * p3, E2p, xyz * _2_0).fmul(p), 

xyz * p2, *over) 

 

 

def _Tsr3(T): 

'''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}. 

''' 

s = map2(sqrt, T[1:]) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)] 

r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ... 

T = tuple((t + r) * _0_25 for t in T) # T[:] = ... 

return T, s, r 

 

 

def _xyz(A0, Am, *xyz): 

'''(INTERNAL) Rescale any C{xys}. 

''' 

return tuple((A0 - x) / Am for x in xyz) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2022 -- mrJean1 at Gmail -- All Rights Reserved. 

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