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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. 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 pygeodesy.fmath import fdot, fmean_, hypot1 from pygeodesy.fsums import Fsum, fsum_ from pygeodesy.interns import EPS, INF, NN, PI, PI_2, PI_4, \ _EPStol as _TolJAC, _convergence_, \ _no_, _SPACE_, _0_0, _0_125, _0_25, \ _0_5, _1_0, _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 from pygeodesy.props import Property_RO, property_RO, _update_all # from pygeodesy.streprs import unstr 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__ = '22.01.17' _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 enough, by .etm 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) 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 # only .fEinv and .sncndn _k2 = 0 _kp2 = 0 _cD = INF _cE = _1_0 _cG = INF _cH = INF _cK = INF _cKE = INF _cPi = INF def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None): '''Constructor, specifying the C{modulus} and C{parameter}. @kwarg k2: Modulus squared (C{float}, 0 <= k^2 <= 1). @kwarg alpha2: Parameter squared (C{float}, 0 <= α^2 <= 1). @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0). @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0). @see: Method L{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._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._cE @Property_RO def cG(self): '''Get Legendre's complete geodesic longitude integral C{G(α^2, k)} (C{float}). ''' return self._cG @Property_RO def cH(self): '''Get Cayley's complete geodesic longitude difference integral C{H(α^2, k)} (C{float}). ''' return self._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._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._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._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 None in (cn, dn): t = self.classname + '.delta' + fX.__name__[1:] raise _invokationError(t, 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._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): return abs(sn) * sn**2 * _RD_3(cn**2, dn**2, _1_0) 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 = self.k2 return sqrt((_1_0 - k2 * sn**2) if k2 < 0 else (self.kp2 + k2 * cn**2)) 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): 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 = _RF(cn2, dn2, _1_0) - k2 * sn2 * _RD_3(cn2, dn2, _1_0) elif kp2 >= 0: # <https://DLMF.NIST.gov/19.25.E10> ei = fsum_(kp2 * _RF(cn2, dn2, _1_0), kp2 * k2 * sn2 * _RD_3(cn2, _1_0, dn2), k2 * abs(cn) / dn) else: # <https://DLMF.NIST.gov/19.25.E11> ei = dn / abs(cn) - kp2 * sn2 * _RD_3(dn2, _1_0, cn2) return ei * abs(sn) 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: n = ceil(deg / _360_0 - _0_5) deg -= n * _360_0 e = n * _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) x -= E2 * n # x now in [-ec, ec) # linear approximation phi = PI * x / E2 # phi in [-pi/2, pi/2) Phi = Fsum(phi) # first order correction phi = Phi.fsum_(-self.eps * sin(2 * phi) * _0_5) # 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> for self._iteration in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC sn, cn, dn = self._sncndn3(phi) phi, e = Phi.fsum2_((x - self.fE(sn, cn, dn)) / dn) if abs(e) < _TolJAC: if n: phi = Phi.fsum_(n * PI) return phi raise _convergenceError(self.fEinv, x) 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): return abs(sn) * _RF(cn**2, dn**2, _1_0) 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): cn2, sn2, dn2 = cn**2, sn**2, dn**2 return abs(sn) * (_RF(cn2, dn2, _1_0) + aX * sn2 * _RJ_3(cn2, dn2, _1_0, cn2 + self.alphap2 * sn2)) 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}. ''' phi = sn = phi_or_sn if cn is dn is None: # fX(phi) call sn, cn, dn = self._sncndn3(phi) if abs(phi) >= PI: return (deltaX(sn, cn, dn) + phi) * cX / PI_2 # fall through elif None in (cn, dn): t = self.classname + '.f' + deltaX.__name__[5:] raise _invokationError(t, 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. ''' 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 and parameter. @kwarg k2: modulus squared (C{float}, -INF <= k^2<= 1). @kwarg alpha2: parameter (C{float}, -INF <= 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}} or no convergence. @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) self._k2 = k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0) self._alpha2 = alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0) self._kp2 = kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError) self._alphap2 = 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) if k2: if kp2: self._eps = k2 / (sqrt(kp2) + _1_0)**2 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3 # <https://DLMF.NIST.gov/19.25.E1> self._cD = _RD_3(_0_0, kp2, _1_0) # Complete elliptic integral E(k), Carlson eq. 4.2 # <https://DLMF.NIST.gov/19.25.E1> self._cE = _RG_(kp2, _1_0) * _2_0 # Complete elliptic integral K(k), Carlson eq. 4.1 # <https://DLMF.NIST.gov/19.25.E1> self._cK = _RF_(kp2, _1_0) self._cKE = k2 * self.cD else: self._eps = k2 self._cD = self._cK = self._cKE = INF self._cE = _1_0 else: self._eps = EPS # delattr(self, '_eps') self._cD = PI_4 self._cE = self._cK = PI_2 self._cKE = _0_0 # k2 * self._cD if alpha2: if alphap2: # <https://DLMF.NIST.gov/19.25.E2> if kp2: rj_3 = _RJ_3(_0_0, kp2, _1_0, alphap2) # G(alpha2, k) self._cG = self.cK + (alpha2 - k2) * rj_3 # H(alpha2, k) self._cH = self.cK - alphap2 * rj_3 # Pi(alpha2, k) self._cPi = self.cK + alpha2 * rj_3 else: self._cG = self._cH = _RC(_1_0, alphap2) self._cPi = INF # XXX or NAN? else: self._cG = self._cH = self._cPi = INF # XXX or NAN? else: self._cG = self.cE self._cPi = self.cK # cH = cK - cD but this involves large cancellations # if k2 is close to 1. So write (for alpha2 = 0) # cH = 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 cH = 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 # cH = int(cos(phi),...) = 1 self._cH = kp2 * _RD_3(_0_0, _1_0, kp2) if kp2 else _1_0 # self._iteration = 0 def sncndn(self, x): # PYCHOK x used! '''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. ''' # Bulirsch's sncndn routine, p 89. mc = self.kp2 if mc: # never negative ... if mc < 0: # PYCHOK no cover d = _1_0 - mc mc = neg(mc / d) # /= d chokes PyChecker d = sqrt(d) x *= d else: d = 0 a, mn = _1_0, [] for self._iteration in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC # This converges quadratically, max 6 trips mc = sqrt(mc) # XXX sqrt0? mn.append((a, mc)) c = (a + mc) * _0_5 if abs(a - mc) <= (_TolJAC * a): break mc *= a a = c else: raise _convergenceError(self.sncndn, x) x *= c dn = _1_0 sn, cn = sincos2(x) if sn: a = cn / sn c *= a while mn: m, n = mn.pop() 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: self._iteration = 0 sn = tanh(x) cn = dn = _1_0 / cosh(x) r = Elliptic3Tuple(sn, cn, dn) r._iteration = self._iteration return r def _sncndn3(self, phi): '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}. ''' sn, cn = sincos2(phi) return Elliptic3Tuple(sn, cn, self.fDelta(sn, cn)) def _convergenceError(where, *args): # PYCHOK no cover '''(INTERNAL) Return an L{EllipticError}. ''' t = _SPACE_(where.__name__, repr(args)) return EllipticError(_no_(_convergence_), txt=t) # unstr def _horner(e0, e1, e2, e3, e4, e5): '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below. ''' # 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 ... H = Fsum(471240, -540540 * e2) * e5 H += Fsum(612612 * e2, -540540 * e3, -556920) * e4 H += Fsum(306306 * e3, 675675 * e2**2, -706860 * e2, 680680) * e3 H += Fsum(417690 * e2, -255255 * e2**2, -875160) * e2 e = 4084080 * e1 return H.fsum_(4084080, e * e0) / e def _invokationError(name, *args): # PYCHOK no cover '''(INTERNAL) Return an L{EllipticError}. ''' return EllipticError(_SPACE_('invokation', NN(name, repr(args)))) # unstr def _Q(A, T, tol): '''(INTERNAL) Helper for C{_RD}, C{_RF} and C{_RJ}. ''' return max(abs(A - t) for t in T[1:]) / tol def _RC(x, y): # used by testElliptic.py '''Degenerate symmetric integral of the first kind C{_RC(x, y)}. @return: C{_RC(x, y) = _RF(x, y, y)}. @see: U{C{_RC} definition<https://DLMF.NIST.gov/19.2.E17>}. ''' # 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? 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 _invokationError(_RC.__name__, x, y) return r / sqrt(d) def _RD_3(x, y, z): '''Degenerate symmetric integral of the third kind C{_RD(x, y, z) / 3}. ''' return _RD(x, y, z) / _3_0 def _RD(x, y, z): # used by testElliptic.py '''Degenerate symmetric integral of the third kind C{_RD(x, y, z)}. @return: C{_RD(x, y, z) = _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>}. ''' # Carlson, eqs 2.28 - 2.34 A = fsum_(x, y, _3_0 * z) / _5_0 T = (A, x, y, z) Q = _Q(A, T, _TolRD) S = Fsum() m = _1_0 for _ in range(_TRIPS): An = T[0] if Q < abs(m * An): # max 7 trips break t = T[3] # z0 r, s, T = _rsT3(T) S += _1_0 / (m * s[2] * (t + r)) m *= _4_0 else: raise _convergenceError(_RD, x, y, z) m *= An x = (x - A) / m y = (y - A) / m z = (x + y) / _3_0 z2 = z**2 xy = x * y S *= _3_0 return _horner(S.fsum(), m * sqrt(An), xy - _6_0 * z2, (xy * _3_0 - _8_0 * z2) * z, (xy - z2) * _3_0 * z2, xy * z2 * z) def _RF_(x, y): '''Symmetric integral of the first kind C{_RF_(x, y)}. @return: C{_RF(x, y)}. @see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>}. ''' # Carlson, eqs 2.36 - 2.38 a, b = sqrt(x), sqrt(y) if a < b: a, b = b, a for _ in range(_TRIPS): if abs(a - b) <= (_TolRG0 * a): # max 4 trips return PI / (a + b) b, a = sqrt(a * b), (a + b) * _0_5 raise _convergenceError(_RF_, x, y) def _RF(x, y, z): # used by testElliptic.py '''Symmetric integral of the first kind C{_RF(x, y, z)}. @return: C{_RF(x, y, z)}. @see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>} and U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. ''' # Carlson, eqs 2.2 - 2.7 A = fmean_(x, y, z) T = (A, x, y, z) Q = _Q(A, T, _TolRF) m = _1_0 for _ in range(_TRIPS): An = T[0] if Q < abs(m * An): # max 6 trips break _, _, T = _rsT3(T) m *= _4_0 else: raise _convergenceError(_RF, x, y, z) m *= An x = (A - x) / m y = (A - y) / m z = neg(x + y) e2 = x * y - z**2 e3 = x * y * z # 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 ... H = Fsum( 6930 * e3, 15015 * e2**2, -16380 * e2, 17160) * e3 H += Fsum(10010 * e2, -5775 * e2**2, -24024) * e2 return H.fsum_(240240) / (240240 * sqrt(An)) def _RG_(x, y): '''Symmetric integral of the second kind C{_RG_(x, y)}. @return: C{_RG(x, y)}. @see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. ''' # Carlson, eqs 2.36 - 2.39 a, b = sqrt(x), sqrt(y) if a < b: a, b = b, a S = Fsum(_0_25 * (a + b)**2) m = _0_5 for _ in range(_TRIPS): # max 4 trips if abs(a - b) <= (_TolRG0 * a): S *= PI_2 / (a + b) return S.fsum() b, a = sqrt(a * b), (a + b) * _0_5 S -= m * (a - b)**2 m *= _2_0 raise _convergenceError(_RG_, x, y) def _RG(x, y, z): # used by testElliptic.py '''Symmetric integral of the second kind C{_RG(x, y, z)}. @return: C{_RG(x, y, z)}. @see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}. ''' if not z: y, z = z, y # Carlson, eq 1.7 return fsum_(_RF(x, y, z) * z, _RD_3(x, y, z) * (x - z) * (z - y), # - (y - z) sqrt(x * y / z)) * _0_5 def _RJ_3(x, y, z, p): '''Symmetric integral of the third kind C{_RJ(x, y, z, p) / 3}. ''' return _RJ(x, y, z, p) / _3_0 def _RJ(x, y, z, p): # used by testElliptic.py '''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>}. ''' def _xyzp(x, y, z, p): return (x + p) * (y + p) * (z + p) # Carlson, eqs 2.17 - 2.25 D = neg(_xyzp(x, y, z, -p)) A = fsum_(x, y, z, _2_0 * p) / _5_0 T = (A, x, y, z, p) Q = _Q(A, T, _TolRD) S = Fsum() m = m3 = _1_0 for _ in range(_TRIPS): An = T[0] if Q < abs(m * An): # max 7 trips break _, s, T = _rsT3(T) d = _xyzp(*s) e = D / (m3 * d**2) S += _RC(_1_0, _1_0 + e) / (m * d) m *= _4_0 m3 *= 64 else: raise _convergenceError(_RJ, x, y, z, p) m *= An x = (A - x) / m y = (A - y) / m z = (A - z) / m xyz = x * y * z p = neg(x + y + z) * _0_5 p2 = p**2 p3 = p * p2 S *= _6_0 e2 = fsum_(x * y, x * z, y * z, -p2 * _3_0) return _horner(S.fsum(), m * sqrt(An), e2, fsum_(xyz, _2_0 * p * e2, _4_0 * p3), fsum_(xyz * _2_0, p * e2, _3_0 * p3) * p, p2 * xyz) def _rsT3(T): '''(INTERNAL) Helper for C{_RD}, C{_RF} and C{_RJ}. ''' s = map2(sqrt, T[1:]) r = fdot(s[:3], s[1], s[2], s[0]) T = tuple((t + r) * _0_25 for t in T) return r, s, T # **) MIT License # # Copyright (C) 2016-2022 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.