# Coverage for pygeodesy/elliptic.py : 89%

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Python method names follow the C++ member functions, 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 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>} (2008-2017) 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 the 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{alpha2}. The modulus is always passed as its square 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>}, 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 from __future__ import division division = 1 / 2 # double check int division, see .datum.py, .utily.py if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.fmath import EPS, fdot, Fsum, fsum_, hypot1, \ INF, map2 from pygeodesy.lazily import _ALL_LAZY from pygeodesy.named import _Named from pygeodesy.utily import PI, PI_2, PI_4, property_RO, \ sincos2, sincos2d from math import asinh, atan, atan2, ceil, copysign, cosh, \ floor, sin, sqrt, tanh __all__ = _ALL_LAZY.elliptic __version__ = '20.01.27' _tolJAC = sqrt(EPS * 0.01) _tolRD = pow(EPS * 0.002, 0.125) _tolRF = pow(EPS * 0.030, 0.125) _tolRG0 = _tolJAC * 2.7 _TRIPS = 13 # Max depth for sncndn, etc, 5-7 might be enough class EllipticError(ValueError): '''Elliptic integral, function, convergence or other L{Elliptic} issue. ''' pass class Elliptic(_Named): '''Elliptic integrals and functions. @see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1EllipticFunction.html#details>}. ''' _alpha2 = 0 _alphap2 = 0 _eps = EPS _k2 = 0 _kp2 = 0 _trips_ = _TRIPS _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}. @keyword k2: Modulus squared (C{scalar} k^2 <= 1). @keyword alpha2: Parameter squared (C{scalar} α^2 <= 1). @keyword kp2: Complementary modulus squared (C{float} k'^2 >= 0). @keyword alphap2: Complementary parameter α'2 (C{float} α'^2 >= 0). @note: If only elliptic integrals of the first and second kinds are needed, then set B{C{alpha2}} = 0 (the default value); in this case, we have Π(φ, 0, k) = F(φ, k), G(φ, 0, k) = E(φ, k), and H(φ, 0, k) = F(φ, k) - D(φ, k). @see: Method L{reset}. ''' self.reset(k2, alpha2=alpha2, kp2=kp2, alphap2=alphap2) @property_RO def alpha2(self): '''Get the parameter B{C{alpha2}} (C{float}). ''' return self._alpha2 @property_RO def alphap2(self): '''Get the complementary parameter B{C{alphap2}} (C{float}). ''' return self._alphap2 @property_RO def cD(self): '''Get Jahnke's complete integral C{D(k)}, 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)}, 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(alpha2, k)}. ''' return self._cG @property_RO def cH(self): '''Get Cayley's complete geodesic longitude difference integral C{H(alpha2, k)}. ''' return self._cH @property_RO def cK(self): '''Get the complete integral of the first kind C{K(k)}, 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)}. ''' return self._cKE @property_RO def cPi(self): '''Get the complete integral of the third kind C{Pi(alpha2, k)}, 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. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π D(φ, k) / (2 D(k)) - φ. ''' return self._deltaX(sn, cn, dn, self.cD, self.fD) def deltaE(self, sn, cn, dn): '''The periodic incomplete integral of the second kind. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function 𝜋π E(φ, k) / (2 E(k)) - φ. ''' 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. @param stau: sin(τ) @param ctau: cos(τ) @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ. ''' # 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. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π F(φ, k) / (2 K(k)) - φ. ''' return self._deltaX(sn, cn, dn, self.cK, self.fF) def deltaG(self, sn, cn, dn): '''Legendre's periodic geodesic longitude integral. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π G(φ, k) / (2 G(k)) - φ. ''' return self._deltaX(sn, cn, dn, self.cG, self.fG) def deltaH(self, sn, cn, dn): '''Cayley's periodic geodesic longitude difference integral. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π H(φ, k) / (2 H(k)) - φ. ''' return self._deltaX(sn, cn, dn, self.cH, self.fH) def deltaPi(self, sn, cn, dn): '''The periodic incomplete integral of the third kind. @param sn: sin(φ). @param cn: cos(φ). @param dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ. ''' 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 < 0: cn, sn = -cn, -sn return fX(sn, cn, dn) * PI_2 / cX - atan2(sn, cn) def fD(self, phi_or_sn, cn=None, dn=None): '''Jahnke's incomplete elliptic integral in terms of Jacobi elliptic functions. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ). @keyword dn: sqrt(1 − k2 sin(2φ)). @return: D(φ, k) as though φ ∈ (−π, π], U{defined<https://DLMF.NIST.gov/19.2.E4>}. ''' def _fD(sn, cn, dn): return abs(sn) * sn**2 * _RD_3(cn**2, dn**2, 1) return self._fXf(phi_or_sn, cn, dn, self.cD, self.deltaD, _fD) def fDelta(self, sn, cn): '''The C{Delta} amplitude function. @param sn: sin(φ). @param cn: cos(φ). @return: C{sqrt(1 − k2 sin(2φ))}. ''' k2 = self._k2 return sqrt((1 - k2 * sn**2) if k2 < 0 else (k2 * cn**2 + self._kp2)) def fE(self, phi_or_sn, cn=None, dn=None): '''The incomplete integral of the second kind in terms of Jacobi elliptic functions. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ). @keyword dn: sqrt(1 − k2 sin(2φ)). @return: E(φ, k) as though φ ∈ (−π, π], U{defined<https://DLMF.NIST.gov/19.2.E5>}. ''' 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 and <https://DLMF.NIST.gov/19.25.E9> ei = _RF(cn2, dn2, 1) - k2 * sn2 * _RD_3(cn2, dn2, 1) elif kp2 >= 0: # <https://DLMF.NIST.gov/19.25.E10> ei = fsum_(kp2 * _RF(cn2, dn2, 1), kp2 * k2 * sn2 * _RD_3(cn2, 1, dn2), k2 * abs(cn) / dn) else: # <https://DLMF.NIST.gov/19.25.E11> ei = dn / abs(cn) - kp2 * sn2 * _RD_3(dn2, 1, 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. @param deg: Angle (C{degrees}). @return: E(π deg/180, k). ''' n = ceil(deg / 360.0 - 0.5) sn, cn = sincos2d(deg - n * 360.0) return self.fE(sn, cn, self.fDelta(sn, cn)) + 4 * self.cE * n def fEinv(self, x): '''The inverse of the incomplete integral of the second kind. @param x: Argument (C{float}). @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}. ''' 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 _ in range(self._trips_): # GEOGRAPHICLIB_PANIC sn, cn, dn = self._sncndn3(phi) phi, e = Phi.fsum2_((x - self.fE(sn, cn, dn)) / dn) if abs(e) < _tolJAC: return n * PI + phi raise EllipticError('no %s%r convergence' % ('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. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ) or C{None}. @keyword dn: sqrt(1 − k2 sin(2φ)) or C{None}. @return: F(φ, k) as though φ ∈ (−π, π], U{defined<https://DLMF.NIST.gov/19.2.E4>}. ''' def _fF(sn, cn, dn): return abs(sn) * _RF(cn**2, dn**2, 1) 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. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ) or C{None}. @keyword dn: sqrt(1 − k2 sin(2φ)) or C{None}. @return: G(φ, k) as though φ ∈ (−π, π]. @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. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ) or C{None}. @keyword dn: sqrt(1 − k2 sin(2φ)) or C{None}. @return: H(φ, k) as though φ ∈ (−π, π]. @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. @param phi_or_sn: φ or sin(φ). @keyword cn: cos(φ) or C{None}. @keyword dn: sqrt(1 − k2 sin(2φ)) or C{None}. @return: Π(φ, α2, k) as though φ ∈ (−π, π]. ''' 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) + aX * sn2 * _RJ_3(cn2, dn2, 1, 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 EllipticError('%s invalid: %s%r' % ('args', t, (sn, cn, dn))) if cn > 0: # enforce usual trig-like symmetries xi = fX(sn, cn, dn) elif cn < 0: xi = 2 * cX - fX(sn, cn, dn) else: xi = cX return copysign(xi, sn) @property_RO def k2(self): '''Get the square of the modulus (C{float}). ''' return self._k2 @property_RO def kp2(self): '''Get 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. @keyword k2: modulus squared (C{float} k2 <= 1). @keyword alpha2: parameter (C{float} α2 <= 1). @keyword kp2: complementary modulus squared (C{float} k'2 >= 0). @keyword alphap2: complementary parameter α'2 (C{float} α'2 >= 0). @note: The arguments must satisfy I{B{C{k2}} + B{C{kp2}} = 1} and I{B{C{alpha2}} + B{C{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. ''' if k2 > 1: raise EllipticError('%s invalid: %r' % ('k2', k2)) else: self._k2 = k2 = float(k2) if alpha2 > 1: raise EllipticError('%s invalid: %r' % ('alpha2', alpha2)) else: self._alpha2 = alpha2 = float(alpha2) if kp2 is None: self._kp2 = kp2 = 1 - k2 elif kp2 < 0: raise EllipticError('%s invalid: %r' % ('kp2', kp2)) else: self._kp2 = kp2 = float(kp2) if alphap2 is None: self._alphap2 = alphap2 = 1 - alpha2 elif alphap2 < 0: raise EllipticError('%s invalid: %r' % ('alphap2', alphap2)) else: self._alphap2 = alphap2 = float(alphap2) self._eps = k2 / (sqrt(kp2) + 1)**2 # 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) = E(k) # H( 0, k) = K(k) - D(k) # Pi(0, k) = K(k) # H( 1, k) = K(k) # Pi(alpha2, 0) = pi/(2*sqrt(1-alpha2)) # G( alpha2, 0) = pi/(2*sqrt(1-alpha2)) # H( alpha2, 0) = pi/(2*(1 + sqrt(1-alpha2))) # Pi(alpha2, 1) = inf # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2) if k2: if kp2: # D(k) = (K(k) - E(k))/k^2, Carlson eq.4.3 # <https://DLMF.NIST.gov/19.25.E1> self._cD = _RD_3(0, kp2, 1) self._cKE = k2 * self._cD # Complete elliptic integral E(k), Carlson eq. 4.2 # <https://DLMF.NIST.gov/19.25.E1> self._cE = _RG(kp2, 1) * 2 # Complete elliptic integral K(k), Carlson eq. 4.1 # <https://DLMF.NIST.gov/19.25.E1> self._cK = _RF(kp2, 1) else: self._cD = self._cK = self._cKE = INF self._cE = 1.0 else: self._cD = PI_4 self._cE = PI_2 self._cK = PI_2 self._cKE = 0.0 # k2 * self._cD if alpha2: if kp2: # <https://DLMF.NIST.gov/19.25.E2> if alphap2: rj_3 = _RJ_3(0, kp2, 1, alphap2) # G(alpha^2, k) self._cG = self._cK + (alpha2 - k2) * rj_3 # H(alpha^2, k) self._cH = self._cK - alphap2 * rj_3 # Pi(alpha^2, k) self._cPi = self._cK + alpha2 * rj_3 else: self._cG = self._cH = self._cPi = INF # XXX _NAN? elif alphap2: self._cG = self._cH = _RC(1, alphap2) self._cPi = INF # XXX _NAN? else: self._cG = self._cE self._cPi = self._cK # Hc = Kc - Dc but this involves large cancellations # if k2 is close to 1. So write (for alpha2 = 0) # Hc = 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 Hc = 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 # Hc = int(cos(phi),...) = 1 self._cH = kp2 * _RD_3(0, 1, kp2) if kp2 else 1.0 def sncndn(self, x): # PYCHOK x not used? '''The Jacobi elliptic function. @param x: The argument (C{float}). @return: 3-Tuple C{(sn(x, k), cn(x, k), dn(x, k))}. ''' # Bulirsch's sncndn routine, p 89. mc = self._kp2 if mc: # never negative ... if mc < 0: # PYCHOK no cover d = 1.0 - mc mc /= -d d = sqrt(d) x *= d else: d = 0 a, c, mn = 1.0, 0, [] for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC # This converges quadratically, max 6 trips mc = sqrt(mc) mn.append((a, mc)) c = (a + mc) * 0.5 if not abs(a - mc) > (_tolJAC * a): break mc *= a a = c else: raise EllipticError('no %s%r convergence' % ('sncndn', (x,))) x *= c dn = 1 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 = copysign(1.0 / hypot1(c), sn) cn = c * sn if d: cn, dn = dn, cn sn /= d else: sn = tanh(x) cn = dn = 1.0 / cosh(x) return sn, cn, dn def _sncndn3(self, phi): '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}. ''' sn, cn = sincos2(phi) return sn, cn, self.fDelta(sn, cn) 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) 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 H += 4084080 e = 4084080 * e1 H += e * e0 return H.fsum() / e 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): '''Degenerate symmetric integral of the first kind C{_RC}. @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: # d < EPS? 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 EllipticError('%s invalid: %s%r' % ('y', '_RC', (x, y))) return r / sqrt(d) def _RD_3(x, y, z): '''Degenerate symmetric integral of the third kind C{_RD} / 3. ''' return _RD(x, y, z) / 3.0 def _RD(x, y, z): '''Degenerate symmetric integral of the third kind C{_RD}. @return: C{_RD(x, y, z) = _RJ(x, y, z, z)}. @see: U{C{_RD} definition<https://DLMF.NIST.gov/19.16.E5>}. ''' # Carlson, eqs 2.28 - 2.34 m = 1.0 S = Fsum() A = fsum_(x, y, z, z, z) * 0.2 T = [A, x, y, z] Q = _Q(A, T, _tolRD) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 7 trips break t = T[3] # z0 r, s, T = _rsT(T) S += 1.0 / (m * s[2] * (t + r)) m *= 4 else: raise EllipticError('no %s%r convergence' % ('_RD', (x, y, z))) m *= T[0] # An x = (x - A) / m y = (y - A) / m z = (x + y) / 3.0 z2 = z**2 xy = x * y return _Horner(3 * S.fsum(), m * sqrt(T[0]), xy - 6 * z2, (xy * 3 - 8 * z2) * z, (xy - z2) * 3 * z2, xy * z2 * z) def _RF(x, y, z=None): '''Symmetric integral of the first kind C{_RF}. @return: C{_RF(x, y, z)}. @see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>}. ''' if z is None: # Carlson, eqs 2.36 - 2.38 a, b = sqrt(x), sqrt(y) if a < b: a, b = b, a while abs(a - b) > (_tolRG0 * a): # max 4 trips b, a = sqrt(a * b), (a + b) * 0.5 return PI / (a + b) # Carlson, eqs 2.2 - 2.7 m = 1.0 A = fsum_(x, y, z) / 3.0 T = [A, x, y, z] Q = _Q(A, T, _tolRF) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 6 trips break _, _, T = _rsT(T) m *= 4 else: raise EllipticError('no %s%r convergence' % ('_RF', (x, y, z))) m *= T[0] # An x = (A - x) / m y = (A - y) / m z = -(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 H += 240240 return H.fsum() / (240240 * sqrt(T[0])) def _RG(x, y, z=None): '''Symmetric integral of the second kind C{_RG}. @return: C{_RG(x, y, z)}. @see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and in Carlson, eq. 1.5. ''' if z is None: # 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 = -0.25 # note, negative while abs(a - b) > (_tolRG0 * a): # max 4 trips b, a = sqrt(a * b), (a + b) * 0.5 m *= 2 S += m * (a - b)**2 return S.fsum() * PI_2 / (a + b) 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), sqrt(x * y / z)) * 0.5 def _RJ_3(x, y, z, p): '''Symmetric integral of the third kind C{_RJ} / 3. ''' return _RJ(x, y, z, p) / 3.0 def _RJ(x, y, z, p): '''Symmetric integral of the third kind C{_RJ}. @return: C{_RJ(x, y, z, p)}. @see: U{C{_RJ} definition<https://DLMF.NIST.gov/19.16.E2>}. ''' def _xyzp(x, y, z, p): return (x + p) * (y + p) * (z + p) # Carlson, eqs 2.17 - 2.25 m = m3 = 1.0 S = Fsum() D = -_xyzp(x, y, z, -p) A = fsum_(x, y, z, 2 * p) * 0.2 T = [A, x, y, z, p] Q = _Q(A, T, _tolRD) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 7 trips break _, s, T = _rsT(T) d = _xyzp(*s) e = D / (m3 * d**2) S += _RC(1, 1 + e) / (m * d) m *= 4 m3 *= 64 else: raise EllipticError('no %s%r convergence' % ('_RJ', (x, y, z, p))) m *= T[0] # An x = (A - x) / m y = (A - y) / m z = (A - z) / m xyz = x * y * z p = -(x + y + z) * 0.5 p2 = p**2 e2 = fsum_(x * y, x * z, y * z, -3 * p2) return _Horner(6 * S.fsum(), m * sqrt(T[0]), e2, fsum_(xyz, 2 * p * e2, 4 * p * p2), fsum_(xyz * 2, p * e2, 3 * p * p2) * p, p2 * xyz) def _rsT(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[:] = [(t + r) * 0.25 for t in T] return r, s, T # **) MIT License # # Copyright (C) 2016-2020 -- 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. 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