Coverage for pygeodesy/elliptic.py: 96%
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2# -*- coding: utf-8 -*-
4u'''I{Karney}'s elliptic functions and integrals.
6Class L{Elliptic} transcoded from I{Charles Karney}'s C++ class U{EllipticFunction
7<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1EllipticFunction.html>}
8to pure Python, including symmetric integrals L{Elliptic.fRC}, L{Elliptic.fRD},
9L{Elliptic.fRF}, L{Elliptic.fRG} and L{Elliptic.fRJ} as C{static methods}.
11Python method names follow the C++ member functions, I{except}:
13 - member functions I{without arguments} are mapped to Python properties
14 prefixed with C{"c"}, for example C{E()} is property C{cE},
16 - member functions with 1 or 3 arguments are renamed to Python methods
17 starting with an C{"f"}, example C{E(psi)} to C{fE(psi)} and C{E(sn,
18 cn, dn)} to C{fE(sn, cn, dn)},
20 - other Python method names conventionally start with a lower-case
21 letter or an underscore if private.
23Following is a copy of I{Karney}'s U{EllipticFunction.hpp
24<https://GeographicLib.SourceForge.io/C++/doc/EllipticFunction_8hpp_source.html>}
25file C{Header}.
27Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023)
28and licensed under the MIT/X11 License. For more information, see the
29U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.
31B{Elliptic integrals and functions.}
33This provides the elliptic functions and integrals needed for
34C{Ellipsoid}, C{GeodesicExact}, and C{TransverseMercatorExact}. Two
35categories of function are provided:
37 - functions to compute U{symmetric elliptic integrals
38 <https://DLMF.NIST.gov/19.16.i>}
40 - methods to compute U{Legrendre's elliptic integrals
41 <https://DLMF.NIST.gov/19.2.ii>} and U{Jacobi elliptic
42 functions<https://DLMF.NIST.gov/22.2>}.
44In the latter case, an object is constructed giving the modulus
45C{k} (and optionally the parameter C{alpha}). The modulus (and
46parameter) are always passed as squares which allows C{k} to be
47pure imaginary. (Confusingly, Abramowitz and Stegun call C{m = k**2}
48the "parameter" and C{n = alpha**2} the "characteristic".)
50In geodesic applications, it is convenient to separate the incomplete
51integrals into secular and periodic components, e.g.
53I{C{E(phi, k) = (2 E(k) / pi) [ phi + delta E(phi, k) ]}}
55where I{C{delta E(phi, k)}} is an odd periodic function with
56period I{C{pi}}.
58The computation of the elliptic integrals uses the algorithms given
59in U{B. C. Carlson, Computation of real or complex elliptic integrals
60<https://DOI.org/10.1007/BF02198293>} (also available U{here
61<https://ArXiv.org/pdf/math/9409227.pdf>}), Numerical Algorithms 10,
6213--26 (1995) with the additional optimizations given U{here
63<https://DLMF.NIST.gov/19.36.i>}.
65The computation of the Jacobi elliptic functions uses the algorithm
66given in U{R. Bulirsch, Numerical Calculation of Elliptic Integrals
67and Elliptic Functions<https://DOI.org/10.1007/BF01397975>},
68Numerische Mathematik 7, 78--90 (1965).
70The notation follows U{NIST Digital Library of Mathematical Functions
71<https://DLMF.NIST.gov>} chapters U{19<https://DLMF.NIST.gov/19>} and
72U{22<https://DLMF.NIST.gov/22>}.
73'''
74# make sure int/int division yields float quotient, see .basics
75from __future__ import division as _; del _ # PYCHOK semicolon
77from pygeodesy.basics import copysign0, map2, neg, neg_
78from pygeodesy.constants import EPS, INF, NAN, PI, PI_2, PI_4, \
79 _EPStol as _TolJAC, _0_0, _1_64th, \
80 _0_25, _0_5, _1_0, _2_0, _N_2_0, \
81 _3_0, _4_0, _6_0, _8_0, _180_0, \
82 _360_0, _over
83from pygeodesy.errors import _ValueError, _xattr, _xkwds_pop
84from pygeodesy.fmath import fdot, hypot1, zqrt
85from pygeodesy.fsums import Fsum, _sum
86from pygeodesy.interns import NN, _delta_, _DOT_, _f_, _invalid_, \
87 _invokation_, _negative_, _SPACE_
88from pygeodesy.karney import _K_2_0, _norm180, _signBit, _sincos2, \
89 _ALL_LAZY
90# from pygeodesy.lazily import _ALL_LAZY # from .karney
91from pygeodesy.named import _Named, _NamedTuple, Fmt, unstr
92from pygeodesy.props import _allPropertiesOf_n, Property_RO, _update_all
93# from pygeodesy.streprs import Fmt, unstr # from .named
94from pygeodesy.units import Scalar, Scalar_
95# from pygeodesy.utily import sincos2 as _sincos2 # from .karney
97from math import asinh, atan, atan2, ceil, cosh, fabs, floor, \
98 radians, sin, sqrt, tanh
100__all__ = _ALL_LAZY.elliptic
101__version__ = '23.08.31'
103_TolRD = zqrt(EPS * 0.002)
104_TolRF = zqrt(EPS * 0.030)
105_TolRG0 = _TolJAC * 2.7
106_TRIPS = 21 # Max depth, 7 might be sufficient
109class _CIs(object):
110 '''(INTERAL) Complete integrals cache.
111 '''
112 def __init__(self, **kwds):
113 self.__dict__ = kwds
116class _D(list):
117 '''(INTERNAL) Deferred C{Fsum}.
118 '''
119 def __call__(self, s):
120 try: # Fsum *= s
121 return Fsum(*self).fmul(s)
122 except ValueError: # Fsum(NAN) exception
123 return _sum(self) * s
125 def __iadd__(self, x):
126 list.append(self, x)
127 return self
130class Elliptic(_Named):
131 '''Elliptic integrals and functions.
133 @see: I{Karney}'s U{Detailed Description<https://GeographicLib.SourceForge.io/
134 html/classGeographicLib_1_1EllipticFunction.html#details>}.
135 '''
136# _alpha2 = 0
137# _alphap2 = 0
138# _eps = EPS
139# _k2 = 0
140# _kp2 = 0
142 def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None, name=NN):
143 '''Constructor, specifying the C{modulus} and C{parameter}.
145 @kwarg name: Optional name (C{str}).
147 @see: Method L{Elliptic.reset} for further details.
149 @note: If only elliptic integrals of the first and second kinds
150 are needed, use C{B{alpha2}=0}, the default value. In
151 that case, we have C{Π(φ, 0, k) = F(φ, k), G(φ, 0, k) =
152 E(φ, k)} and C{H(φ, 0, k) = F(φ, k) - D(φ, k)}.
153 '''
154 self.reset(k2=k2, alpha2=alpha2, kp2=kp2, alphap2=alphap2)
156 if name:
157 self.name = name
159 @Property_RO
160 def alpha2(self):
161 '''Get α^2, the square of the parameter (C{float}).
162 '''
163 return self._alpha2
165 @Property_RO
166 def alphap2(self):
167 '''Get α'^2, the square of the complementary parameter (C{float}).
168 '''
169 return self._alphap2
171 @Property_RO
172 def cD(self):
173 '''Get Jahnke's complete integral C{D(k)} (C{float}),
174 U{defined<https://DLMF.NIST.gov/19.2.E6>}.
175 '''
176 return self._reset_cDEKEeps.cD
178 @Property_RO
179 def cE(self):
180 '''Get the complete integral of the second kind C{E(k)}
181 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}.
182 '''
183 return self._reset_cDEKEeps.cE
185 @Property_RO
186 def cG(self):
187 '''Get Legendre's complete geodesic longitude integral
188 C{G(α^2, k)} (C{float}).
189 '''
190 return self._reset_cGHPi.cG
192 @Property_RO
193 def cH(self):
194 '''Get Cayley's complete geodesic longitude difference integral
195 C{H(α^2, k)} (C{float}).
196 '''
197 return self._reset_cGHPi.cH
199 @Property_RO
200 def cK(self):
201 '''Get the complete integral of the first kind C{K(k)}
202 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}.
203 '''
204 return self._reset_cDEKEeps.cK
206 @Property_RO
207 def cKE(self):
208 '''Get the difference between the complete integrals of the
209 first and second kinds, C{K(k) − E(k)} (C{float}).
210 '''
211 return self._reset_cDEKEeps.cKE
213 @Property_RO
214 def cPi(self):
215 '''Get the complete integral of the third kind C{Pi(α^2, k)}
216 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}.
217 '''
218 return self._reset_cGHPi.cPi
220 def deltaD(self, sn, cn, dn):
221 '''The periodic Jahnke's incomplete elliptic integral.
223 @arg sn: sin(φ).
224 @arg cn: cos(φ).
225 @arg dn: sqrt(1 − k2 * sin(2φ)).
227 @return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}).
229 @raise EllipticError: Invalid invokation or no convergence.
230 '''
231 return _deltaX(sn, cn, dn, self.cD, self.fD)
233 def deltaE(self, sn, cn, dn):
234 '''The periodic incomplete integral of the second kind.
236 @arg sn: sin(φ).
237 @arg cn: cos(φ).
238 @arg dn: sqrt(1 − k2 * sin(2φ)).
240 @return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}).
242 @raise EllipticError: Invalid invokation or no convergence.
243 '''
244 return _deltaX(sn, cn, dn, self.cE, self.fE)
246 def deltaEinv(self, stau, ctau):
247 '''The periodic inverse of the incomplete integral of the second kind.
249 @arg stau: sin(τ)
250 @arg ctau: cos(τ)
252 @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}).
254 @raise EllipticError: No convergence.
255 '''
256 try:
257 if _signBit(ctau): # pi periodic
258 stau, ctau = neg_(stau, ctau)
259 t = atan2(stau, ctau)
260 return self._Einv(t * self.cE / PI_2) - t
262 except Exception as e:
263 raise _ellipticError(self.deltaEinv, stau, ctau, cause=e)
265 def deltaF(self, sn, cn, dn):
266 '''The periodic incomplete integral of the first kind.
268 @arg sn: sin(φ).
269 @arg cn: cos(φ).
270 @arg dn: sqrt(1 − k2 * sin(2φ)).
272 @return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}).
274 @raise EllipticError: Invalid invokation or no convergence.
275 '''
276 return _deltaX(sn, cn, dn, self.cK, self.fF)
278 def deltaG(self, sn, cn, dn):
279 '''Legendre's periodic geodesic longitude integral.
281 @arg sn: sin(φ).
282 @arg cn: cos(φ).
283 @arg dn: sqrt(1 − k2 * sin(2φ)).
285 @return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}).
287 @raise EllipticError: Invalid invokation or no convergence.
288 '''
289 return _deltaX(sn, cn, dn, self.cG, self.fG)
291 def deltaH(self, sn, cn, dn):
292 '''Cayley's periodic geodesic longitude difference integral.
294 @arg sn: sin(φ).
295 @arg cn: cos(φ).
296 @arg dn: sqrt(1 − k2 * sin(2φ)).
298 @return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}).
300 @raise EllipticError: Invalid invokation or no convergence.
301 '''
302 return _deltaX(sn, cn, dn, self.cH, self.fH)
304 def deltaPi(self, sn, cn, dn):
305 '''The periodic incomplete integral of the third kind.
307 @arg sn: sin(φ).
308 @arg cn: cos(φ).
309 @arg dn: sqrt(1 − k2 * sin(2φ)).
311 @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ
312 (C{float}).
314 @raise EllipticError: Invalid invokation or no convergence.
315 '''
316 return _deltaX(sn, cn, dn, self.cPi, self.fPi)
318 def _Einv(self, x):
319 '''(INTERNAL) Helper for C{.deltaEinv} and C{.fEinv}.
320 '''
321 E2 = self.cE * _2_0
322 n = floor(x / E2 + _0_5)
323 r = x - E2 * n # r in [-cE, cE)
324 # linear approximation
325 phi = PI * r / E2 # phi in [-PI_2, PI_2)
326 Phi = Fsum(phi)
327 # first order correction
328 phi = Phi.fsum_(self.eps * sin(phi * _2_0) / _N_2_0)
329 # For kp2 close to zero use asin(r / cE) or J. P. Boyd,
330 # Applied Math. and Computation 218, 7005-7013 (2012)
331 # <https://DOI.org/10.1016/j.amc.2011.12.021>
332 _Phi2, self._iteration = Phi.fsum2_, 0 # aggregate
333 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
334 sn, cn, dn = self._sncndn3(phi)
335 if dn:
336 sn = self.fE(sn, cn, dn)
337 phi, d = _Phi2((r - sn) / dn)
338 else: # PYCHOK no cover
339 d = _0_0 # XXX continue?
340 if fabs(d) < _TolJAC: # 3-4 trips
341 _iterations(self, i)
342 break
343 else: # PYCHOK no cover
344 raise _convergenceError(d, _TolJAC)
345 return Phi.fsum_(n * PI) if n else phi
347 @Property_RO
348 def eps(self):
349 '''Get epsilon (C{float}).
350 '''
351 return self._reset_cDEKEeps.eps
353 def fD(self, phi_or_sn, cn=None, dn=None):
354 '''Jahnke's incomplete elliptic integral in terms of
355 Jacobi elliptic functions.
357 @arg phi_or_sn: φ or sin(φ).
358 @kwarg cn: C{None} or cos(φ).
359 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
361 @return: D(φ, k) as though φ ∈ (−π, π] (C{float}),
362 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
364 @raise EllipticError: Invalid invokation or no convergence.
365 '''
366 def _fD(sn, cn, dn):
367 r = fabs(sn)**3
368 if r:
369 r = float(_RD(self, cn**2, dn**2, _1_0, _3_0 / r))
370 return r
372 return self._fXf(phi_or_sn, cn, dn, self.cD,
373 self.deltaD, _fD)
375 def fDelta(self, sn, cn):
376 '''The C{Delta} amplitude function.
378 @arg sn: sin(φ).
379 @arg cn: cos(φ).
381 @return: sqrt(1 − k2 * sin(2φ)) (C{float}).
382 '''
383 try:
384 k2 = self.k2
385 s = (self.kp2 + cn**2 * k2) if k2 > 0 else (
386 (_1_0 - sn**2 * k2) if k2 < 0 else self.kp2)
387 return sqrt(s) if s else _0_0
389 except Exception as e:
390 raise _ellipticError(self.fDelta, sn, cn, k2=k2, cause=e)
392 def fE(self, phi_or_sn, cn=None, dn=None):
393 '''The incomplete integral of the second kind in terms of
394 Jacobi elliptic functions.
396 @arg phi_or_sn: φ or sin(φ).
397 @kwarg cn: C{None} or cos(φ).
398 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
400 @return: E(φ, k) as though φ ∈ (−π, π] (C{float}),
401 U{defined<https://DLMF.NIST.gov/19.2.E5>}.
403 @raise EllipticError: Invalid invokation or no convergence.
404 '''
405 def _fE(sn, cn, dn):
406 '''(INTERNAL) Core of C{.fE}.
407 '''
408 if sn:
409 sn2, cn2, dn2 = sn**2, cn**2, dn**2
410 kp2, k2 = self.kp2, self.k2
411 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9>
412 Ei = _RF3(self, cn2, dn2, _1_0)
413 if k2:
414 Ei -= _RD(self, cn2, dn2, _1_0, _3over(k2, sn2))
415 elif kp2 >= 0: # k2 > 0, <https://DLMF.NIST.gov/19.25.E10>
416 Ei = _over(k2 * fabs(cn), dn) # float
417 if kp2:
418 Ei += (_RD( self, cn2, _1_0, dn2, _3over(k2, sn2)) +
419 _RF3(self, cn2, dn2, _1_0)) * kp2
420 else: # kp2 < 0, <https://DLMF.NIST.gov/19.25.E11>
421 Ei = _over(dn, fabs(cn))
422 Ei -= _RD(self, dn2, _1_0, cn2, _3over(kp2, sn2))
423 Ei *= fabs(sn)
424 ei = float(Ei)
425 else: # PYCHOK no cover
426 ei = _0_0
427 return ei
429 return self._fXf(phi_or_sn, cn, dn, self.cE,
430 self.deltaE, _fE)
432 def fEd(self, deg):
433 '''The incomplete integral of the second kind with
434 the argument given in C{degrees}.
436 @arg deg: Angle (C{degrees}).
438 @return: E(π B{C{deg}} / 180, k) (C{float}).
440 @raise EllipticError: No convergence.
441 '''
442 if _K_2_0:
443 e = round((deg - _norm180(deg)) / _360_0)
444 elif fabs(deg) < _180_0:
445 e = _0_0
446 else:
447 e = ceil(deg / _360_0 - _0_5)
448 deg -= e * _360_0
449 return self.fE(radians(deg)) + e * self.cE * _4_0
451 def fEinv(self, x):
452 '''The inverse of the incomplete integral of the second kind.
454 @arg x: Argument (C{float}).
456 @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}
457 (C{float}).
459 @raise EllipticError: No convergence.
460 '''
461 try:
462 return self._Einv(x)
463 except Exception as e:
464 raise _ellipticError(self.fEinv, x, cause=e)
466 def fF(self, phi_or_sn, cn=None, dn=None):
467 '''The incomplete integral of the first kind in terms of
468 Jacobi elliptic functions.
470 @arg phi_or_sn: φ or sin(φ).
471 @kwarg cn: C{None} or cos(φ).
472 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
474 @return: F(φ, k) as though φ ∈ (−π, π] (C{float}),
475 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
477 @raise EllipticError: Invalid invokation or no convergence.
478 '''
479 def _fF(sn, cn, dn):
480 r = fabs(sn)
481 if r:
482 r = float(_RF3(self, cn**2, dn**2, _1_0).fmul(r))
483 return r
485 return self._fXf(phi_or_sn, cn, dn, self.cK,
486 self.deltaF, _fF)
488 def fG(self, phi_or_sn, cn=None, dn=None):
489 '''Legendre's geodesic longitude integral in terms of
490 Jacobi elliptic functions.
492 @arg phi_or_sn: φ or sin(φ).
493 @kwarg cn: C{None} or cos(φ).
494 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
496 @return: G(φ, k) as though φ ∈ (−π, π] (C{float}).
498 @raise EllipticError: Invalid invokation or no convergence.
500 @note: Legendre expresses the longitude of a point on the
501 geodesic in terms of this combination of elliptic
502 integrals in U{Exercices de Calcul Intégral, Vol 1
503 (1811), p 181<https://Books.Google.com/books?id=
504 riIOAAAAQAAJ&pg=PA181>}.
506 @see: U{Geodesics in terms of elliptic integrals<https://
507 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
508 for the expression for the longitude in terms of this function.
509 '''
510 return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2,
511 self.cG, self.deltaG)
513 def fH(self, phi_or_sn, cn=None, dn=None):
514 '''Cayley's geodesic longitude difference integral in terms of
515 Jacobi elliptic functions.
517 @arg phi_or_sn: φ or sin(φ).
518 @kwarg cn: C{None} or cos(φ).
519 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
521 @return: H(φ, k) as though φ ∈ (−π, π] (C{float}).
523 @raise EllipticError: Invalid invokation or no convergence.
525 @note: Cayley expresses the longitude difference of a point
526 on the geodesic in terms of this combination of
527 elliptic integrals in U{Phil. Mag. B{40} (1870), p 333
528 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}.
530 @see: U{Geodesics in terms of elliptic integrals<https://
531 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
532 for the expression for the longitude in terms of this function.
533 '''
534 return self._fXa(phi_or_sn, cn, dn, -self.alphap2,
535 self.cH, self.deltaH)
537 def fPi(self, phi_or_sn, cn=None, dn=None):
538 '''The incomplete integral of the third kind in terms of
539 Jacobi elliptic functions.
541 @arg phi_or_sn: φ or sin(φ).
542 @kwarg cn: C{None} or cos(φ).
543 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
545 @return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}).
547 @raise EllipticError: Invalid invokation or no convergence.
548 '''
549 if dn is None and cn is not None: # and isscalar(phi_or_sn)
550 dn = self.fDelta(phi_or_sn, cn) # in .triaxial
551 return self._fXa(phi_or_sn, cn, dn, self.alpha2,
552 self.cPi, self.deltaPi)
554 def _fXa(self, phi_or_sn, cn, dn, aX, cX, _deltaX):
555 '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}.
556 '''
557 def _fX(sn, cn, dn):
558 if sn:
559 cn2, dn2 = cn**2, dn**2
560 R = _RF3(self, cn2, dn2, _1_0)
561 if aX:
562 sn2 = sn**2
563 p = sn2 * self.alphap2 + cn2
564 R += _RJ(self, cn2, dn2, _1_0, p, _3over(aX, sn2))
565 r = float(R.fmul(fabs(sn)))
566 else: # PYCHOK no cover
567 r = _0_0
568 return r
570 return self._fXf(phi_or_sn, cn, dn, cX, _deltaX, _fX)
572 def _fXf(self, phi_or_sn, cn, dn, cX, _deltaX, _fX):
573 '''(INTERNAL) Helper for C{.fD}, C{.fE}, C{.fF} and C{._fXa}.
574 '''
575 self._iteration = 0 # aggregate
576 phi = sn = phi_or_sn
577 if cn is dn is None: # fX(phi) call
578 sn, cn, dn = self._sncndn3(phi)
579 if fabs(phi) >= PI:
580 if cX:
581 cX *= (_deltaX(sn, cn, dn) + phi) / PI_2
582 return cX
583 # fall through
584 elif cn is None or dn is None:
585 n = NN(_f_, _deltaX.__name__[5:])
586 raise _ellipticError(n, sn, cn, dn)
588 if _signBit(cn): # enforce usual trig-like symmetries
589 xi = cX * _2_0 - _fX(sn, cn, dn)
590 elif cn > 0:
591 xi = _fX(sn, cn, dn)
592 else:
593 xi = cX
594 return copysign0(xi, sn)
596 @Property_RO
597 def k2(self):
598 '''Get k^2, the square of the modulus (C{float}).
599 '''
600 return self._k2
602 @Property_RO
603 def kp2(self):
604 '''Get k'^2, the square of the complementary modulus (C{float}).
605 '''
606 return self._kp2
608 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13
609 '''Reset the modulus, parameter and the complementaries.
611 @kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1).
612 @kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1).
613 @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0).
614 @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0).
616 @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}}
617 or B{C{alphap2}}.
619 @note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and
620 C{B{alpha2} + B{alphap2} = 1}. No checking is done
621 that these conditions are met to enable accuracy to be
622 maintained, e.g., when C{k} is very close to unity.
623 '''
624 if self.__dict__:
625 _update_all(self, _Named.iteration._uname, Base=Property_RO)
627 self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0)
628 self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError)
630 self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0)
631 self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2),
632 Error=EllipticError)
634 # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
635 # K E D
636 # k = 0: pi/2 pi/2 pi/4
637 # k = 1: inf 1 inf
638 # Pi G H
639 # k = 0, alpha = 0: pi/2 pi/2 pi/4
640 # k = 1, alpha = 0: inf 1 1
641 # k = 0, alpha = 1: inf inf pi/2
642 # k = 1, alpha = 1: inf inf inf
643 #
644 # G(0, k) = Pi(0, k) = H(1, k) = E(k)
645 # H(0, k) = K(k) - D(k)
646 # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2))
647 # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1))
648 # Pi(alpha2, 1) = inf
649 # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
651 @Property_RO
652 def _reset_cDEKEeps(self):
653 '''(INTERNAL) Get the complete integrals D, E, K and KE plus C{eps}.
654 '''
655 k2, kp2 = self.k2, self.kp2
656 if k2:
657 if kp2:
658 try:
659 self._iteration = 0
660 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3
661 # <https://DLMF.NIST.gov/19.25.E1>
662 D = _RD(self, _0_0, kp2, _1_0, _3_0)
663 cD = float(D)
664 # Complete elliptic integral E(k), Carlson eq. 4.2
665 # <https://DLMF.NIST.gov/19.25.E1>
666 cE = _rG2(self, kp2, _1_0, PI_=PI_2)
667 # Complete elliptic integral K(k), Carlson eq. 4.1
668 # <https://DLMF.NIST.gov/19.25.E1>
669 cK = _rF2(self, kp2, _1_0)
670 cKE = float(D.fmul(k2))
671 eps = k2 / (sqrt(kp2) + _1_0)**2
673 except Exception as e:
674 raise _ellipticError(self.reset, k2=k2, kp2=kp2, cause=e)
675 else:
676 cD = cK = cKE = INF
677 cE = _1_0
678 eps = k2
679 else:
680 cD = PI_4
681 cE = cK = PI_2
682 cKE = _0_0 # k2 * cD
683 eps = EPS
685 return _CIs(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps)
687 @Property_RO
688 def _reset_cGHPi(self):
689 '''(INTERNAL) Get the complete integrals G, H and Pi.
690 '''
691 alpha2, alphap2, kp2 = self.alpha2, self.alphap2, self.kp2
692 try:
693 self._iteration = 0
694 if alpha2:
695 if alphap2:
696 if kp2: # <https://DLMF.NIST.gov/19.25.E2>
697 cK = self.cK
698 Rj = _RJ(self, _0_0, kp2, _1_0, alphap2, _3_0)
699 cG = float(Rj * (alpha2 - self.k2) + cK) # G(alpha2, k)
700 cH = -float(Rj * alphap2 - cK) # H(alpha2, k)
701 cPi = float(Rj * alpha2 + cK) # Pi(alpha2, k)
702 else:
703 cG = cH = _rC(self, _1_0, alphap2)
704 cPi = INF # XXX or NAN?
705 else:
706 cG = cH = cPi = INF # XXX or NAN?
707 else:
708 cG, cPi = self.cE, self.cK
709 # H = K - D but this involves large cancellations if k2 is near 1.
710 # So write (for alpha2 = 0)
711 # H = int(cos(phi)**2 / sqrt(1-k2 * sin(phi)**2), phi, 0, pi/2)
712 # = 1 / sqrt(1-k2) * int(sin(phi)**2 / sqrt(1-k2/kp2 * sin(phi)**2,...)
713 # = 1 / kp * D(i * k/kp)
714 # and use D(k) = RD(0, kp2, 1) / 3, so
715 # H = 1/kp * RD(0, 1/kp2, 1) / 3
716 # = kp2 * RD(0, 1, kp2) / 3
717 # using <https://DLMF.NIST.gov/19.20.E18>. Equivalently
718 # RF(x, 1) - RD(0, x, 1) / 3 = x * RD(0, 1, x) / 3 for x > 0
719 # For k2 = 1 and alpha2 = 0, we have
720 # H = int(cos(phi),...) = 1
721 cH = float(_RD(self, _0_0, _1_0, kp2, _3_0 / kp2)) if kp2 else _1_0
723 except Exception as e:
724 raise _ellipticError(self.reset, kp2=kp2, alpha2 =alpha2,
725 alphap2=alphap2, cause=e)
726 return _CIs(cG=cG, cH=cH, cPi=cPi)
728 def sncndn(self, x):
729 '''The Jacobi elliptic function.
731 @arg x: The argument (C{float}).
733 @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with
734 C{*n(B{x}, k)}.
736 @raise EllipticError: No convergence.
737 '''
738 self._iteration = 0 # reset
739 try: # Bulirsch's sncndn routine, p 89.
740 if self.kp2:
741 c, d, cd, mn = self._sncndn4
742 dn = _1_0
743 sn, cn = _sincos2(x * cd)
744 if sn:
745 a = cn / sn
746 c *= a
747 for m, n in reversed(mn):
748 a *= c
749 c *= dn
750 dn = (n + a) / (m + a)
751 a = c / m
752 a = _1_0 / hypot1(c)
753 sn = neg(a) if _signBit(sn) else a
754 cn = c * sn
755 if d and _signBit(self.kp2):
756 cn, dn = dn, cn
757 sn = sn / d # /= chokes PyChecker
758 else:
759 sn = tanh(x)
760 cn = dn = _1_0 / cosh(x)
762 except Exception as e:
763 raise _ellipticError(self.sncndn, x, kp2=self.kp2, cause=e)
765 return Elliptic3Tuple(sn, cn, dn, iteration=self._iteration)
767 def _sncndn3(self, phi):
768 '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}.
769 '''
770 sn, cn = _sincos2(phi)
771 return sn, cn, self.fDelta(sn, cn)
773 @Property_RO
774 def _sncndn4(self):
775 '''(INTERNAL) Get Bulirsch' 4-tuple C{(c, d, cd, mn_)}.
776 '''
777 # Bulirsch's sncndn routine, p 89.
778 d, mc = 0, self.kp2
779 if _signBit(mc):
780 d = _1_0 - mc
781 mc = neg(mc / d)
782 d = sqrt(d)
784 mn, a = [], _1_0
785 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
786 mc = sqrt(mc)
787 mn.append((a, mc))
788 c = (a + mc) * _0_5
789 r = fabs(mc - a)
790 t = _TolJAC * a
791 if r <= t: # 6 trips, quadratic
792 _iterations(self, i)
793 break
794 mc *= a
795 a = c
796 else: # PYCHOK no cover
797 raise _convergenceError(r, t)
798 cd = (c * d) if d else c
799 return c, d, cd, mn
801 @staticmethod
802 def fRC(x, y):
803 '''Degenerate symmetric integral of the first kind C{RC(x, y)}.
805 @return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}.
807 @see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and
808 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
809 '''
810 return _rC(None, x, y)
812 @staticmethod
813 def fRD(x, y, z, *over):
814 '''Degenerate symmetric integral of the third kind C{RD(x, y, z)}.
816 @return: C{RD(x, y, z) / over}, equivalent to C{RJ(x, y, z, z)
817 / over} with C{over} typically 3.
819 @see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and
820 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
821 '''
822 try:
823 return float(_RD(None, x, y, z, *over))
824 except Exception as e:
825 raise _ellipticError(Elliptic.fRD, x, y, z, *over, cause=e)
827 @staticmethod
828 def fRF(x, y, z=0):
829 '''Symmetric or complete symmetric integral of the first kind
830 C{RF(x, y, z)} respectively C{RF(x, y)}.
832 @return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}.
834 @see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and
835 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
836 '''
837 try:
838 return float(_RF3(None, x, y, z)) if z else _rF2(None, x, y)
839 except Exception as e:
840 raise _ellipticError(Elliptic.fRF, x, y, z, cause=e)
842 @staticmethod
843 def fRG(x, y, z=0):
844 '''Symmetric or complete symmetric integral of the second kind
845 C{RG(x, y, z)} respectively C{RG(x, y)}.
847 @return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}.
849 @see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>},
850 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>} and
851 U{RG<https://GeographicLib.SourceForge.io/C++/doc/
852 EllipticFunction_8cpp_source.html#l00096>} version 2.3.
853 '''
854 try:
855 return _rG2(None, x, y) if z == 0 else (
856 _rG2(None, z, x) if y == 0 else (
857 _rG2(None, y, z) if x == 0 else _rG3(None, x, y, z)))
859 except Exception as e:
860 t = _negative_ if min(x, y, z) < 0 else NN
861 raise _ellipticError(Elliptic.fRG, x, y, z, cause=e, txt=t)
863 @staticmethod
864 def fRJ(x, y, z, p):
865 '''Symmetric integral of the third kind C{RJ(x, y, z, p)}.
867 @return: C{RJ(x, y, z, p)}.
869 @see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and
870 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
871 '''
872 try:
873 return float(_RJ(None, x, y, z, p))
874 except Exception as e:
875 raise _ellipticError(Elliptic.fRF, x, y, z, p, cause=e)
877_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset
880class EllipticError(_ValueError):
881 '''Elliptic function, integral, convergence or other L{Elliptic} issue.
882 '''
883 pass
886class Elliptic3Tuple(_NamedTuple):
887 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.
888 '''
889 _Names_ = ('sn', 'cn', 'dn')
890 _Units_ = ( Scalar, Scalar, Scalar)
893class _L(list):
894 '''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}.
895 '''
896 _a0 = None
897# _xyzp = ()
899 def __init__(self, *xyzp): # x, y, z [, p]
900 list.__init__(self, xyzp)
901 self._xyzp = xyzp
903 def a0(self, n):
904 '''Compute the initial C{a}.
905 '''
906 t = tuple(self)
907 m = n - len(t)
908 if m > 0:
909 t += t[-1:] * m
910 try:
911 a = Fsum(*t).fover(n)
912 except ValueError: # Fsum(NAN) exception
913 a = _sum(t) / n
914 self._a0 = a
915 return a
917 def amrs4(self, inst, n, Tol):
918 '''Yield Carlson 4-tuples C{(An, mul, lam, s)} plus sentinel, with
919 C{lam = fdot(sqrt(x), ... (z))} and C{s = (sqrt(x), ... (p))}.
920 '''
921 L = self
922 a = L.a0(n)
923 m = 1
924 t = max(fabs(a - _) for _ in L) / Tol
925 for i in range(_TRIPS):
926 d = fabs(a * m)
927 if d > t: # 5-6 trips
928 _iterations(inst, i)
929 break
930 s = map2(sqrt, L) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)]
931 try:
932 r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ...
933 except ValueError: # Fsum(NAN) exception
934 r = _sum(s[i] * s[(i + 1) % 3] for i in range(3))
935 L[:] = [(r + _) * _0_25 for _ in L]
936 a = (r + a) * _0_25
937 yield a, m, r, s # L[2] is next z
938 m *= 4
939 else: # PYCHOK no cover
940 raise _convergenceError(d, t, thresh=True)
941 yield a, m, None, () # sentinel: same a, next m, no r and s
943 def rescale(self, am, *xs):
944 '''Rescale C{x}, C{y}, ...
945 '''
946 # assert am
947 a0 = self._a0
948 for x in xs:
949 yield (a0 - x) / am
952def _ab2(inst, x, y):
953 '''(INTERNAL) Yield Carlson 2-tuples C{(xn, yn)}.
954 '''
955 a, b = sqrt(x), sqrt(y)
956 if b > a:
957 a, b = b, a
958 yield a, b # initial x0, y0
959 for i in range(1, _TRIPS):
960 d = fabs(a - b)
961 t = _TolRG0 * a
962 if d <= t: # 3-4 trips
963 _iterations(inst, i - 1)
964 break
965 a, b = ((a + b) * _0_5), sqrt(a * b)
966 yield a, b # xn, yn
967 else: # PYCHOK no cover
968 raise _convergenceError(d, t)
971def _convergenceError(d, tol, **thresh):
972 '''(INTERNAL) Format a no-convergence Error.
973 '''
974 t = Fmt.no_convergence(d, tol, **thresh)
975 return ValueError(t) # txt only
978def _deltaX(sn, cn, dn, cX, _fX):
979 '''(INTERNAL) Helper for C{Elliptic.deltaD} thru C{.deltaPi}.
980 '''
981 try:
982 if cn is None or dn is None:
983 raise ValueError(_invalid_)
985 if _signBit(cn):
986 sn, cn = neg_(sn, cn)
987 r = _fX(sn, cn, dn) * PI_2 / cX
988 return r - atan2(sn, cn)
990 except Exception as e:
991 n = NN(_delta_, _fX.__name__[1:])
992 raise _ellipticError(n, sn, cn, dn, cause=e)
995def _ellipticError(where, *args, **kwds_cause_txt):
996 '''(INTERNAL) Format an L{EllipticError}.
997 '''
998 c = _xkwds_pop(kwds_cause_txt, cause=None)
999 t = _xkwds_pop(kwds_cause_txt, txt=NN)
1000 n = _xattr(where, __name__=where)
1001 n = _DOT_(Elliptic.__name__, n)
1002 n = _SPACE_(_invokation_, n)
1003 u = unstr(n, *args, **kwds_cause_txt)
1004 return EllipticError(u, cause=c, txt=t)
1007def _Horner(S, e1, E2, E3, E4, E5, *over):
1008 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.
1009 '''
1010 E22 = E2**2
1011 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>
1012 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52
1013 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20
1014 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)
1015 # converted to Horner-like form ...
1016 F = Fsum
1017 e = e1 * 4084080
1018 S *= e
1019 S += F(E2 * -540540, 471240).fmul(E5)
1020 S += F(E2 * 612612, E3 * -540540, -556920).fmul(E4)
1021 S += F(E2 * -706860, E22 * 675675, E3 * 306306, 680680).fmul(E3)
1022 S += F(E2 * 417690, E22 * -255255, -875160).fmul(E2)
1023 S += 4084080
1024 return S.fdiv((over[0] * e) if over else e) # Fsum
1027def _iterations(inst, i):
1028 '''(INTERNAL) Aggregate iterations B{C{i}}.
1029 '''
1030 if inst and i > 0:
1031 inst._iteration += i
1034def _3over(a, b):
1035 '''(INTERNAL) Return C{3 / (a * b)}.
1036 '''
1037 return _over(_3_0, a * b)
1040def _rC(unused, x, y):
1041 '''(INTERNAL) Defined only for C{y != 0} and C{x >= 0}.
1042 '''
1043 d = x - y
1044 if d < 0: # catch NaN
1045 # <https://DLMF.NIST.gov/19.2.E18>
1046 d = -d
1047 r = atan(sqrt(d / x)) if x > 0 else PI_2
1048 elif d == 0: # XXX d < EPS0? or EPS02 or _EPSmin
1049 d, r = y, _1_0
1050 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>
1051 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))
1052 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>
1053 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))
1054 else: # PYCHOK no cover
1055 raise _ellipticError(Elliptic.fRC, x, y)
1056 return r / sqrt(d) # float
1059def _RD(inst, x, y, z, *over):
1060 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.
1061 '''
1062 L = _L(x, y, z)
1063 S = _D()
1064 for a, m, r, s in L.amrs4(inst, 5, _TolRF):
1065 if s:
1066 S += _over(_3_0, (r + z) * s[2] * m)
1067 z = L[2] # s[2] = sqrt(z)
1068 x, y = L.rescale(-a * m, x, y)
1069 xy = x * y
1070 z = (x + y) / _3_0
1071 z2 = z**2
1072 return _Horner(S(_1_0), sqrt(a) * a * m,
1073 xy - _6_0 * z2,
1074 (xy * _3_0 - _8_0 * z2) * z,
1075 (xy - z2) * _3_0 * z2,
1076 xy * z2 * z, *over) # Fsum
1079def _rF2(inst, x, y): # 2-arg version, z=0
1080 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.
1081 '''
1082 for a, b in _ab2(inst, x, y): # PYCHOK yield
1083 pass
1084 return _over(PI, a + b) # float
1087def _RF3(inst, x, y, z): # 3-arg version
1088 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.
1089 '''
1090 L = _L(x, y, z)
1091 for a, m, _, _ in L.amrs4(inst, 3, _TolRF):
1092 pass
1093 x, y = L.rescale(a * m, x, y)
1094 z = neg(x + y)
1095 xy = x * y
1096 e2 = xy - z**2
1097 e3 = xy * z
1098 e4 = e2**2
1099 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>
1100 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44
1101 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)
1102 # converted to Horner-like form ...
1103 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)
1104 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)
1105 S += 240240
1106 return S.fdiv(sqrt(a) * 240240) # Fsum
1109def _rG2(inst, x, y, PI_=PI_4): # 2-args
1110 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.
1111 '''
1112 m = -1 # neg!
1113 S = _D()
1114 # assert not S
1115 for a, b in _ab2(inst, x, y): # PYCHOK yield
1116 if S:
1117 S += (a - b)**2 * m
1118 m *= 2
1119 else: # initial
1120 S += (a + b)**2 * _0_5
1121 return S(PI_).fover(a + b)
1124def _rG3(inst, x, y, z): # 3-arg version
1125 '''(INTERNAL) C{x}, C{y} and C{z} all non-zero, see C{.fRG}.
1126 '''
1127 R = _RF3(inst, x, y, z) * z
1128 rd = (x - z) * (z - y) # - (y - z)
1129 if rd: # Carlson, eq 1.7
1130 R += _RD(inst, x, y, z, _3_0 / rd)
1131 R += sqrt(x * y / z)
1132 return R.fover(_2_0)
1135def _RJ(inst, x, y, z, p, *over):
1136 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.
1137 '''
1138 def _xyzp(x, y, z, p):
1139 return (x + p) * (y + p) * (z + p)
1141 L = _L(x, y, z, p)
1142 n = neg(_xyzp(x, y, z, -p))
1143 S = _D()
1144 for a, m, _, s in L.amrs4(inst, 5, _TolRD):
1145 if s:
1146 d = _xyzp(*s)
1147 if d:
1148 if n:
1149 rc = _rC(inst, _1_0, n / d**2 + _1_0)
1150 n *= _1_64th # /= chokes PyChecker
1151 else:
1152 rc = _1_0 # == _rC(None, _1_0, _1_0)
1153 S += rc / (d * m)
1154 else: # PYCHOK no cover
1155 return NAN
1156 x, y, z = L.rescale(a * m, x, y, z)
1157 xyz = x * y * z
1158 p = Fsum(x, y, z).fover(_N_2_0)
1159 p2 = p**2
1160 p3 = p2 * p
1161 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)
1162 E2p = E2 * p
1163 return _Horner(S(_6_0), sqrt(a) * a * m, E2,
1164 Fsum(p3 * _4_0, xyz, E2p * _2_0),
1165 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),
1166 xyz * p2, *over) # Fsum
1168# **) MIT License
1169#
1170# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1171#
1172# Permission is hereby granted, free of charge, to any person obtaining a
1173# copy of this software and associated documentation files (the "Software"),
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1177# Software is furnished to do so, subject to the following conditions:
1178#
1179# The above copyright notice and this permission notice shall be included
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1181#
1182# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
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