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
83# from pygeodesy.errors import _ValueError # from .fmath
84from pygeodesy.fmath import fdot, hypot1, zqrt, _ValueError
85from pygeodesy.fsums import Fsum, _sum
86# from pygeodesy.internals import _dunder_nameof # from .lazily
87from pygeodesy.interns import NN, _delta_, _DOT_, _f_, _invalid_, \
88 _invokation_, _negative_, _SPACE_
89from pygeodesy.karney import _K_2_0, _norm180, _signBit, _sincos2
90from pygeodesy.lazily import _ALL_LAZY, _dunder_nameof
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__ = '24.05.13'
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 _Cs(object):
110 '''(INTERAL) Complete integrals cache.
111 '''
112 def __init__(self, **kwds):
113 self.__dict__ = kwds
116class _Dsum(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 C++/doc/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._cDEKEeps.cD
178 @Property_RO
179 def _cDEKEeps(self):
180 '''(INTERNAL) Get the complete integrals D, E, K and KE plus C{eps}.
181 '''
182 k2, kp2 = self.k2, self.kp2
183 if k2:
184 if kp2:
185 try:
186 self._iteration = 0
187 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3
188 # <https://DLMF.NIST.gov/19.25.E1>
189 D = _RD(self, _0_0, kp2, _1_0, _3_0)
190 cD = float(D)
191 # Complete elliptic integral E(k), Carlson eq. 4.2
192 # <https://DLMF.NIST.gov/19.25.E1>
193 cE = _rG2(self, kp2, _1_0, PI_=PI_2)
194 # Complete elliptic integral K(k), Carlson eq. 4.1
195 # <https://DLMF.NIST.gov/19.25.E1>
196 cK = _rF2(self, kp2, _1_0)
197 cKE = float(D.fmul(k2))
198 eps = k2 / (sqrt(kp2) + _1_0)**2
200 except Exception as e:
201 raise _ellipticError(self.reset, k2=k2, kp2=kp2, cause=e)
202 else:
203 cD = cK = cKE = INF
204 cE = _1_0
205 eps = k2
206 else:
207 cD = PI_4
208 cE = cK = PI_2
209 cKE = _0_0 # k2 * cD
210 eps = EPS
212 return _Cs(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps)
214 @Property_RO
215 def cE(self):
216 '''Get the complete integral of the second kind C{E(k)}
217 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}.
218 '''
219 return self._cDEKEeps.cE
221 @Property_RO
222 def cG(self):
223 '''Get Legendre's complete geodesic longitude integral
224 C{G(α^2, k)} (C{float}).
225 '''
226 return self._cGHPi.cG
228 @Property_RO
229 def _cGHPi(self):
230 '''(INTERNAL) Get the complete integrals G, H and Pi.
231 '''
232 alpha2, alphap2, kp2 = self.alpha2, self.alphap2, self.kp2
233 try:
234 self._iteration = 0
235 if alpha2:
236 if alphap2:
237 if kp2: # <https://DLMF.NIST.gov/19.25.E2>
238 cK = self.cK
239 Rj = _RJ(self, _0_0, kp2, _1_0, alphap2, _3_0)
240 cG = float(Rj * (alpha2 - self.k2) + cK) # G(alpha2, k)
241 cH = -float(Rj * alphap2 - cK) # H(alpha2, k)
242 cPi = float(Rj * alpha2 + cK) # Pi(alpha2, k)
243 else:
244 cG = cH = _rC(self, _1_0, alphap2)
245 cPi = INF # XXX or NAN?
246 else:
247 cG = cH = cPi = INF # XXX or NAN?
248 else:
249 cG, cPi = self.cE, self.cK
250 # H = K - D but this involves large cancellations if k2 is near 1.
251 # So write (for alpha2 = 0)
252 # H = int(cos(phi)**2 / sqrt(1-k2 * sin(phi)**2), phi, 0, pi/2)
253 # = 1 / sqrt(1-k2) * int(sin(phi)**2 / sqrt(1-k2/kp2 * sin(phi)**2,...)
254 # = 1 / kp * D(i * k/kp)
255 # and use D(k) = RD(0, kp2, 1) / 3, so
256 # H = 1/kp * RD(0, 1/kp2, 1) / 3
257 # = kp2 * RD(0, 1, kp2) / 3
258 # using <https://DLMF.NIST.gov/19.20.E18>. Equivalently
259 # RF(x, 1) - RD(0, x, 1) / 3 = x * RD(0, 1, x) / 3 for x > 0
260 # For k2 = 1 and alpha2 = 0, we have
261 # H = int(cos(phi),...) = 1
262 cH = float(_RD(self, _0_0, _1_0, kp2, _3_0 / kp2)) if kp2 else _1_0
264 except Exception as e:
265 raise _ellipticError(self.reset, kp2=kp2, alpha2 =alpha2,
266 alphap2=alphap2, cause=e)
267 return _Cs(cG=cG, cH=cH, cPi=cPi)
269 @Property_RO
270 def cH(self):
271 '''Get Cayley's complete geodesic longitude difference integral
272 C{H(α^2, k)} (C{float}).
273 '''
274 return self._cGHPi.cH
276 @Property_RO
277 def cK(self):
278 '''Get the complete integral of the first kind C{K(k)}
279 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}.
280 '''
281 return self._cDEKEeps.cK
283 @Property_RO
284 def cKE(self):
285 '''Get the difference between the complete integrals of the
286 first and second kinds, C{K(k) − E(k)} (C{float}).
287 '''
288 return self._cDEKEeps.cKE
290 @Property_RO
291 def cPi(self):
292 '''Get the complete integral of the third kind C{Pi(α^2, k)}
293 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}.
294 '''
295 return self._cGHPi.cPi
297 def deltaD(self, sn, cn, dn):
298 '''Jahnke's periodic incomplete elliptic integral.
300 @arg sn: sin(φ).
301 @arg cn: cos(φ).
302 @arg dn: sqrt(1 − k2 * sin(2φ)).
304 @return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}).
306 @raise EllipticError: Invalid invokation or no convergence.
307 '''
308 return _deltaX(sn, cn, dn, self.cD, self.fD)
310 def deltaE(self, sn, cn, dn):
311 '''The periodic incomplete integral of the second kind.
313 @arg sn: sin(φ).
314 @arg cn: cos(φ).
315 @arg dn: sqrt(1 − k2 * sin(2φ)).
317 @return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}).
319 @raise EllipticError: Invalid invokation or no convergence.
320 '''
321 return _deltaX(sn, cn, dn, self.cE, self.fE)
323 def deltaEinv(self, stau, ctau):
324 '''The periodic inverse of the incomplete integral of the second kind.
326 @arg stau: sin(τ)
327 @arg ctau: cos(τ)
329 @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}).
331 @raise EllipticError: No convergence.
332 '''
333 try:
334 if _signBit(ctau): # pi periodic
335 stau, ctau = neg_(stau, ctau)
336 t = atan2(stau, ctau)
337 return self._Einv(t * self.cE / PI_2) - t
339 except Exception as e:
340 raise _ellipticError(self.deltaEinv, stau, ctau, cause=e)
342 def deltaF(self, sn, cn, dn):
343 '''The periodic incomplete integral of the first kind.
345 @arg sn: sin(φ).
346 @arg cn: cos(φ).
347 @arg dn: sqrt(1 − k2 * sin(2φ)).
349 @return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}).
351 @raise EllipticError: Invalid invokation or no convergence.
352 '''
353 return _deltaX(sn, cn, dn, self.cK, self.fF)
355 def deltaG(self, sn, cn, dn):
356 '''Legendre's periodic geodesic longitude integral.
358 @arg sn: sin(φ).
359 @arg cn: cos(φ).
360 @arg dn: sqrt(1 − k2 * sin(2φ)).
362 @return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}).
364 @raise EllipticError: Invalid invokation or no convergence.
365 '''
366 return _deltaX(sn, cn, dn, self.cG, self.fG)
368 def deltaH(self, sn, cn, dn):
369 '''Cayley's periodic geodesic longitude difference integral.
371 @arg sn: sin(φ).
372 @arg cn: cos(φ).
373 @arg dn: sqrt(1 − k2 * sin(2φ)).
375 @return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}).
377 @raise EllipticError: Invalid invokation or no convergence.
378 '''
379 return _deltaX(sn, cn, dn, self.cH, self.fH)
381 def deltaPi(self, sn, cn, dn):
382 '''The periodic incomplete integral of the third kind.
384 @arg sn: sin(φ).
385 @arg cn: cos(φ).
386 @arg dn: sqrt(1 − k2 * sin(2φ)).
388 @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ
389 (C{float}).
391 @raise EllipticError: Invalid invokation or no convergence.
392 '''
393 return _deltaX(sn, cn, dn, self.cPi, self.fPi)
395 def _Einv(self, x):
396 '''(INTERNAL) Helper for C{.deltaEinv} and C{.fEinv}.
397 '''
398 E2 = self.cE * _2_0
399 n = floor(x / E2 + _0_5)
400 r = x - E2 * n # r in [-cE, cE)
401 # linear approximation
402 phi = PI * r / E2 # phi in [-PI_2, PI_2)
403 Phi = Fsum(phi)
404 # first order correction
405 phi = Phi.fsum_(self.eps * sin(phi * _2_0) / _N_2_0)
406 self._iteration = 0
407 # For kp2 close to zero use asin(r / cE) or J. P. Boyd,
408 # Applied Math. and Computation 218, 7005-7013 (2012)
409 # <https://DOI.org/10.1016/j.amc.2011.12.021>
410 _Phi2 = Phi.fsum2f_ # aggregate
411 _abs = fabs
412 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
413 sn, cn, dn = self._sncndn3(phi)
414 if dn:
415 sn = self.fE(sn, cn, dn)
416 phi, d = _Phi2((r - sn) / dn)
417 else: # PYCHOK no cover
418 d = _0_0 # XXX continue?
419 if _abs(d) < _TolJAC: # 3-4 trips
420 _iterations(self, i)
421 break
422 else: # PYCHOK no cover
423 raise _convergenceError(d, _TolJAC)
424 return Phi.fsum_(n * PI) if n else phi
426 @Property_RO
427 def eps(self):
428 '''Get epsilon (C{float}).
429 '''
430 return self._cDEKEeps.eps
432 def fD(self, phi_or_sn, cn=None, dn=None):
433 '''Jahnke's incomplete elliptic integral in terms of
434 Jacobi elliptic functions.
436 @arg phi_or_sn: φ or sin(φ).
437 @kwarg cn: C{None} or cos(φ).
438 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
440 @return: D(φ, k) as though φ ∈ (−π, π] (C{float}),
441 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
443 @raise EllipticError: Invalid invokation or no convergence.
444 '''
445 def _fD(sn, cn, dn):
446 r = fabs(sn)**3
447 if r:
448 r = float(_RD(self, cn**2, dn**2, _1_0, _3_0 / r))
449 return r
451 return self._fXf(phi_or_sn, cn, dn, self.cD,
452 self.deltaD, _fD)
454 def fDelta(self, sn, cn):
455 '''The C{Delta} amplitude function.
457 @arg sn: sin(φ).
458 @arg cn: cos(φ).
460 @return: sqrt(1 − k2 * sin(2φ)) (C{float}).
461 '''
462 try:
463 k2 = self.k2
464 s = (self.kp2 + cn**2 * k2) if k2 > 0 else (
465 (_1_0 - sn**2 * k2) if k2 < 0 else self.kp2)
466 return sqrt(s) if s else _0_0
468 except Exception as e:
469 raise _ellipticError(self.fDelta, sn, cn, k2=k2, cause=e)
471 def fE(self, phi_or_sn, cn=None, dn=None):
472 '''The incomplete integral of the second kind in terms of
473 Jacobi elliptic functions.
475 @arg phi_or_sn: φ or sin(φ).
476 @kwarg cn: C{None} or cos(φ).
477 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
479 @return: E(φ, k) as though φ ∈ (−π, π] (C{float}),
480 U{defined<https://DLMF.NIST.gov/19.2.E5>}.
482 @raise EllipticError: Invalid invokation or no convergence.
483 '''
484 def _fE(sn, cn, dn):
485 '''(INTERNAL) Core of C{.fE}.
486 '''
487 if sn:
488 sn2, cn2, dn2 = sn**2, cn**2, dn**2
489 kp2, k2 = self.kp2, self.k2
490 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9>
491 Ei = _RF3(self, cn2, dn2, _1_0)
492 if k2:
493 Ei -= _RD(self, cn2, dn2, _1_0, _3over(k2, sn2))
494 elif kp2 >= 0: # k2 > 0, <https://DLMF.NIST.gov/19.25.E10>
495 Ei = _over(k2 * fabs(cn), dn) # float
496 if kp2:
497 Ei += (_RD( self, cn2, _1_0, dn2, _3over(k2, sn2)) +
498 _RF3(self, cn2, dn2, _1_0)) * kp2
499 else: # kp2 < 0, <https://DLMF.NIST.gov/19.25.E11>
500 Ei = _over(dn, fabs(cn))
501 Ei -= _RD(self, dn2, _1_0, cn2, _3over(kp2, sn2))
502 Ei *= fabs(sn)
503 ei = float(Ei)
504 else: # PYCHOK no cover
505 ei = _0_0
506 return ei
508 return self._fXf(phi_or_sn, cn, dn, self.cE,
509 self.deltaE, _fE)
511 def fEd(self, deg):
512 '''The incomplete integral of the second kind with
513 the argument given in C{degrees}.
515 @arg deg: Angle (C{degrees}).
517 @return: E(π B{C{deg}} / 180, k) (C{float}).
519 @raise EllipticError: No convergence.
520 '''
521 if _K_2_0:
522 e = round((deg - _norm180(deg)) / _360_0)
523 elif fabs(deg) < _180_0:
524 e = _0_0
525 else:
526 e = ceil(deg / _360_0 - _0_5)
527 deg -= e * _360_0
528 return self.fE(radians(deg)) + e * self.cE * _4_0
530 def fEinv(self, x):
531 '''The inverse of the incomplete integral of the second kind.
533 @arg x: Argument (C{float}).
535 @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}
536 (C{float}).
538 @raise EllipticError: No convergence.
539 '''
540 try:
541 return self._Einv(x)
542 except Exception as e:
543 raise _ellipticError(self.fEinv, x, cause=e)
545 def fF(self, phi_or_sn, cn=None, dn=None):
546 '''The incomplete integral of the first kind in terms of
547 Jacobi elliptic functions.
549 @arg phi_or_sn: φ or sin(φ).
550 @kwarg cn: C{None} or cos(φ).
551 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
553 @return: F(φ, k) as though φ ∈ (−π, π] (C{float}),
554 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
556 @raise EllipticError: Invalid invokation or no convergence.
557 '''
558 def _fF(sn, cn, dn):
559 r = fabs(sn)
560 if r:
561 r = float(_RF3(self, cn**2, dn**2, _1_0).fmul(r))
562 return r
564 return self._fXf(phi_or_sn, cn, dn, self.cK,
565 self.deltaF, _fF)
567 def fG(self, phi_or_sn, cn=None, dn=None):
568 '''Legendre's geodesic longitude integral in terms of
569 Jacobi elliptic functions.
571 @arg phi_or_sn: φ or sin(φ).
572 @kwarg cn: C{None} or cos(φ).
573 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
575 @return: G(φ, k) as though φ ∈ (−π, π] (C{float}).
577 @raise EllipticError: Invalid invokation or no convergence.
579 @note: Legendre expresses the longitude of a point on the
580 geodesic in terms of this combination of elliptic
581 integrals in U{Exercices de Calcul Intégral, Vol 1
582 (1811), p 181<https://Books.Google.com/books?id=
583 riIOAAAAQAAJ&pg=PA181>}.
585 @see: U{Geodesics in terms of elliptic integrals<https://
586 GeographicLib.SourceForge.io/C++/doc/geodesic.html#geodellip>}
587 for the expression for the longitude in terms of this function.
588 '''
589 return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2,
590 self.cG, self.deltaG)
592 def fH(self, phi_or_sn, cn=None, dn=None):
593 '''Cayley's geodesic longitude difference integral in terms of
594 Jacobi elliptic functions.
596 @arg phi_or_sn: φ or sin(φ).
597 @kwarg cn: C{None} or cos(φ).
598 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
600 @return: H(φ, k) as though φ ∈ (−π, π] (C{float}).
602 @raise EllipticError: Invalid invokation or no convergence.
604 @note: Cayley expresses the longitude difference of a point
605 on the geodesic in terms of this combination of
606 elliptic integrals in U{Phil. Mag. B{40} (1870), p 333
607 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}.
609 @see: U{Geodesics in terms of elliptic integrals<https://
610 GeographicLib.SourceForge.io/C++/doc/geodesic.html#geodellip>}
611 for the expression for the longitude in terms of this function.
612 '''
613 return self._fXa(phi_or_sn, cn, dn, -self.alphap2,
614 self.cH, self.deltaH)
616 def fPi(self, phi_or_sn, cn=None, dn=None):
617 '''The incomplete integral of the third kind in terms of
618 Jacobi elliptic functions.
620 @arg phi_or_sn: φ or sin(φ).
621 @kwarg cn: C{None} or cos(φ).
622 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
624 @return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}).
626 @raise EllipticError: Invalid invokation or no convergence.
627 '''
628 if dn is None and cn is not None: # and isscalar(phi_or_sn)
629 dn = self.fDelta(phi_or_sn, cn) # in .triaxial
630 return self._fXa(phi_or_sn, cn, dn, self.alpha2,
631 self.cPi, self.deltaPi)
633 def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX):
634 '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}.
635 '''
636 def _fX(sn, cn, dn):
637 if sn:
638 cn2, dn2 = cn**2, dn**2
639 R = _RF3(self, cn2, dn2, _1_0)
640 if aX:
641 sn2 = sn**2
642 p = sn2 * self.alphap2 + cn2
643 R += _RJ(self, cn2, dn2, _1_0, p, _3over(aX, sn2))
644 R *= fabs(sn)
645 r = float(R)
646 else: # PYCHOK no cover
647 r = _0_0
648 return r
650 return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX)
652 def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX):
653 '''(INTERNAL) Helper for C{.fD}, C{.fE}, C{.fF} and C{._fXa}.
654 '''
655 self._iteration = 0 # aggregate
656 phi = sn = phi_or_sn
657 if cn is dn is None: # fX(phi) call
658 sn, cn, dn = self._sncndn3(phi)
659 if fabs(phi) >= PI:
660 return (deltaX(sn, cn, dn) + phi) * cX / PI_2
661 # fall through
662 elif cn is None or dn is None:
663 n = NN(_f_, deltaX.__name__[5:])
664 raise _ellipticError(n, sn, cn, dn)
666 if _signBit(cn): # enforce usual trig-like symmetries
667 xi = cX * _2_0 - fX(sn, cn, dn)
668 else:
669 xi = fX(sn, cn, dn) if cn > 0 else cX
670 return copysign0(xi, sn)
672 @Property_RO
673 def k2(self):
674 '''Get k^2, the square of the modulus (C{float}).
675 '''
676 return self._k2
678 @Property_RO
679 def kp2(self):
680 '''Get k'^2, the square of the complementary modulus (C{float}).
681 '''
682 return self._kp2
684 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13
685 '''Reset the modulus, parameter and the complementaries.
687 @kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1).
688 @kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1).
689 @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0).
690 @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0).
692 @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}}
693 or B{C{alphap2}}.
695 @note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and
696 C{B{alpha2} + B{alphap2} = 1}. No checking is done
697 that these conditions are met to enable accuracy to be
698 maintained, e.g., when C{k} is very close to unity.
699 '''
700 if self.__dict__:
701 _update_all(self, _Named.iteration._uname, Base=Property_RO)
703 self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0)
704 self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError)
706 self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0)
707 self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2),
708 Error=EllipticError)
710 # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
711 # K E D
712 # k = 0: pi/2 pi/2 pi/4
713 # k = 1: inf 1 inf
714 # Pi G H
715 # k = 0, alpha = 0: pi/2 pi/2 pi/4
716 # k = 1, alpha = 0: inf 1 1
717 # k = 0, alpha = 1: inf inf pi/2
718 # k = 1, alpha = 1: inf inf inf
719 #
720 # G(0, k) = Pi(0, k) = H(1, k) = E(k)
721 # H(0, k) = K(k) - D(k)
722 # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2))
723 # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1))
724 # Pi(alpha2, 1) = inf
725 # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
727 def sncndn(self, x):
728 '''The Jacobi elliptic function.
730 @arg x: The argument (C{float}).
732 @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with
733 C{*n(B{x}, k)}.
735 @raise EllipticError: No convergence.
736 '''
737 self._iteration = 0 # reset
738 try: # Bulirsch's sncndn routine, p 89.
739 if self.kp2:
740 c, d, cd, mn = self._sncndn4
741 dn = _1_0
742 sn, cn = _sincos2(x * cd)
743 if sn:
744 a = cn / sn
745 c *= a
746 for m, n in reversed(mn):
747 a *= c
748 c *= dn
749 dn = (n + a) / (m + a)
750 a = c / m
751 a = _1_0 / hypot1(c)
752 sn = neg(a) if _signBit(sn) else a
753 cn = c * sn
754 if d and _signBit(self.kp2):
755 cn, dn = dn, cn
756 sn = sn / d # /= chokes PyChecker
757 else:
758 sn = tanh(x)
759 cn = dn = _1_0 / cosh(x)
761 except Exception as e:
762 raise _ellipticError(self.sncndn, x, kp2=self.kp2, cause=e)
764 return Elliptic3Tuple(sn, cn, dn, iteration=self._iteration)
766 def _sncndn3(self, phi):
767 '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}.
768 '''
769 sn, cn = _sincos2(phi)
770 return sn, cn, self.fDelta(sn, cn)
772 @Property_RO
773 def _sncndn4(self):
774 '''(INTERNAL) Get Bulirsch' 4-tuple C{(c, d, cd, mn)}.
775 '''
776 # Bulirsch's sncndn routine, p 89.
777 d, mc = 0, self.kp2
778 if _signBit(mc):
779 d = _1_0 - mc
780 mc = neg(mc / d)
781 d = sqrt(d)
783 mn, a = [], _1_0
784 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
785 mc = sqrt(mc)
786 mn.append((a, mc))
787 c = (a + mc) * _0_5
788 r = fabs(mc - a)
789 t = _TolJAC * a
790 if r <= t: # 6 trips, quadratic
791 _iterations(self, i)
792 break
793 mc *= a
794 a = c
795 else: # PYCHOK no cover
796 raise _convergenceError(r, t)
797 cd = (c * d) if d else c
798 return c, d, cd, mn
800 @staticmethod
801 def fRC(x, y):
802 '''Degenerate symmetric integral of the first kind C{RC(x, y)}.
804 @return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}.
806 @see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and
807 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
808 '''
809 return _rC(None, x, y)
811 @staticmethod
812 def fRD(x, y, z, *over):
813 '''Degenerate symmetric integral of the third kind C{RD(x, y, z)}.
815 @return: C{RD(x, y, z) / over}, equivalent to C{RJ(x, y, z, z)
816 / over} with C{over} typically 3.
818 @see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and
819 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
820 '''
821 try:
822 return float(_RD(None, x, y, z, *over))
823 except Exception as e:
824 raise _ellipticError(Elliptic.fRD, x, y, z, *over, cause=e)
826 @staticmethod
827 def fRF(x, y, z=0):
828 '''Symmetric or complete symmetric integral of the first kind
829 C{RF(x, y, z)} respectively C{RF(x, y)}.
831 @return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}.
833 @see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and
834 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
835 '''
836 try:
837 return float(_RF3(None, x, y, z)) if z else _rF2(None, x, y)
838 except Exception as e:
839 raise _ellipticError(Elliptic.fRF, x, y, z, cause=e)
841 @staticmethod
842 def fRG(x, y, z=0):
843 '''Symmetric or complete symmetric integral of the second kind
844 C{RG(x, y, z)} respectively C{RG(x, y)}.
846 @return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}.
848 @see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>},
849 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>} and
850 U{RG<https://GeographicLib.SourceForge.io/C++/doc/
851 EllipticFunction_8cpp_source.html#l00096>} version 2.3.
852 '''
853 try:
854 return _rG2(None, x, y) if z == 0 else (
855 _rG2(None, z, x) if y == 0 else (
856 _rG2(None, y, z) if x == 0 else _rG3(None, x, y, z)))
857 except Exception as e:
858 t = _negative_ if min(x, y, z) < 0 else NN
859 raise _ellipticError(Elliptic.fRG, x, y, z, cause=e, txt=t)
861 @staticmethod
862 def fRJ(x, y, z, p):
863 '''Symmetric integral of the third kind C{RJ(x, y, z, p)}.
865 @return: C{RJ(x, y, z, p)}.
867 @see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and
868 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
869 '''
870 try:
871 return float(_RJ(None, x, y, z, p))
872 except Exception as e:
873 raise _ellipticError(Elliptic.fRJ, x, y, z, p, cause=e)
875 @staticmethod
876 def _RFRD(x, y, z, m):
877 # in .auxilats.AuxDLat.DE, .auxilats.AuxLat.Rectifying
878 try: # float(RF(x, y, z) - RD(x, y, z, 3 / m))
879 R = _RF3(None, x, y, z)
880 if m:
881 R -= _RD(None, x, y, z, _3_0 / m)
882 except Exception as e:
883 raise _ellipticError(Elliptic._RFRD, x, y, z, m, cause=e)
884 return float(R)
886_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset
889class EllipticError(_ValueError):
890 '''Elliptic function, integral, convergence or other L{Elliptic} issue.
891 '''
892 pass
895class Elliptic3Tuple(_NamedTuple):
896 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.
897 '''
898 _Names_ = ('sn', 'cn', 'dn')
899 _Units_ = ( Scalar, Scalar, Scalar)
902class _List(list):
903 '''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}.
904 '''
905 _a0 = None
906# _xyzp = ()
908 def __init__(self, *xyzp): # x, y, z [, p]
909 list.__init__(self, xyzp)
910 self._xyzp = xyzp
912 def a0(self, n):
913 '''Compute the initial C{a}.
914 '''
915 t = tuple(self)
916 m = n - len(t)
917 if m > 0:
918 t += t[-1:] * m
919 try:
920 a = Fsum(*t).fover(n)
921 except ValueError: # Fsum(NAN) exception
922 a = _sum(t) / n
923 self._a0 = a
924 return a
926 def amrs4(self, inst, y, Tol):
927 '''Yield Carlson 4-tuples C{(An, mul, lam, s)} plus sentinel, with
928 C{lam = fdot(sqrt(x), ... (z))} and C{s = (sqrt(x), ... (p))}.
929 '''
930 L = self
931 a = L.a0(5 if y else 3)
932 m = 1
933 t = max(fabs(a - _) for _ in L) / Tol
934 for i in range(_TRIPS):
935 d = fabs(a * m)
936 if d > t: # 3-6 trips
937 _iterations(inst, i)
938 break
939 s = map2(sqrt, L) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)]
940 try:
941 r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ...
942 except ValueError: # Fsum(NAN) exception
943 r = _sum(s[i] * s[(i + 1) % 3] for i in range(3))
944 L[:] = ((r + _) * _0_25 for _ in L)
945 a = (r + a) * _0_25
946 if y: # yield only if used
947 yield a, m, r, s # L[2] is next z
948 m *= 4
949 else: # PYCHOK no cover
950 raise _convergenceError(d, t, thresh=True)
951 yield a, m, None, () # sentinel: same a, next m, no r and s
953 def rescale(self, am, *xs):
954 '''Rescale C{x}, C{y}, ...
955 '''
956 # assert am
957 a0 = self._a0
958 _am = _1_0 / am
959 for x in xs:
960 yield (a0 - x) * _am
963def _ab2(inst, x, y):
964 '''(INTERNAL) Yield Carlson 2-tuples C{(xn, yn)}.
965 '''
966 a, b = sqrt(x), sqrt(y)
967 if b > a:
968 b, a = a, b
969 for i in range(_TRIPS):
970 yield a, b # xi, yi
971 d = fabs(a - b)
972 t = _TolRG0 * a
973 if d <= t: # 3-4 trips
974 _iterations(inst, i)
975 break
976 a, b = ((a + b) * _0_5), sqrt(a * b)
977 else: # PYCHOK no cover
978 raise _convergenceError(d, t)
981def _convergenceError(d, tol, **thresh):
982 '''(INTERNAL) Format a no-convergence Error.
983 '''
984 t = Fmt.no_convergence(d, tol, **thresh)
985 return ValueError(t) # txt only
988def _deltaX(sn, cn, dn, cX, fX):
989 '''(INTERNAL) Helper for C{Elliptic.deltaD} thru C{.deltaPi}.
990 '''
991 try:
992 if cn is None or dn is None:
993 raise ValueError(_invalid_)
995 if _signBit(cn):
996 sn, cn = neg_(sn, cn)
997 r = fX(sn, cn, dn) * PI_2 / cX
998 return r - atan2(sn, cn)
1000 except Exception as e:
1001 n = NN(_delta_, fX.__name__[1:])
1002 raise _ellipticError(n, sn, cn, dn, cause=e)
1005def _ellipticError(where, *args, **kwds_cause_txt):
1006 '''(INTERNAL) Format an L{EllipticError}.
1007 '''
1008 def _x_t_kwds(cause=None, txt=NN, **kwds):
1009 return cause, txt, kwds
1011 x, t, kwds = _x_t_kwds(**kwds_cause_txt)
1013 n = _dunder_nameof(where, where)
1014 n = _DOT_(Elliptic.__name__, n)
1015 n = _SPACE_(_invokation_, n)
1016 u = unstr(n, *args, **kwds)
1017 return EllipticError(u, cause=x, txt=t)
1020def _Horner(S, e1, E2, E3, E4, E5, *over):
1021 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.
1022 '''
1023 E22 = E2**2
1024 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>
1025 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52
1026 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20
1027 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)
1028 # converted to Horner-like form ...
1029 e = e1 * 4084080
1030 S *= e
1031 S += Fsum(E2 * -540540, 471240).fmul(E5)
1032 S += Fsum(E2 * 612612, E3 * -540540, -556920).fmul(E4)
1033 S += Fsum(E2 * -706860, E22 * 675675, E3 * 306306, 680680).fmul(E3)
1034 S += Fsum(E2 * 417690, E22 * -255255, -875160).fmul(E2)
1035 S += 4084080
1036 if over:
1037 e *= over[0]
1038 return S.fdiv(e) # Fsum
1041def _iterations(inst, i):
1042 '''(INTERNAL) Aggregate iterations B{C{i}}.
1043 '''
1044 if inst and i > 0:
1045 inst._iteration += i
1048def _3over(a, b):
1049 '''(INTERNAL) Return C{3 / (a * b)}.
1050 '''
1051 return _over(_3_0, a * b)
1054def _rC(unused, x, y):
1055 '''(INTERNAL) Defined only for C{y != 0} and C{x >= 0}.
1056 '''
1057 d = x - y
1058 if d < 0: # catch NaN
1059 # <https://DLMF.NIST.gov/19.2.E18>
1060 d = -d
1061 r = atan(sqrt(d / x)) if x > 0 else PI_2
1062 elif d == 0: # XXX d < EPS0? or EPS02 or _EPSmin
1063 d, r = y, _1_0
1064 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>
1065 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))
1066 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>
1067 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))
1068 else: # PYCHOK no cover
1069 raise _ellipticError(Elliptic.fRC, x, y)
1070 return r / sqrt(d) # float
1073def _RD(inst, x, y, z, *over):
1074 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.
1075 '''
1076 L = _List(x, y, z)
1077 S = _Dsum()
1078 for a, m, r, s in L.amrs4(inst, True, _TolRF):
1079 if s:
1080 S += _over(_3_0, (r + z) * s[2] * m)
1081 z = L[2] # s[2] = sqrt(z)
1082 x, y = L.rescale(-a * m, x, y)
1083 xy = x * y
1084 z = (x + y) / _3_0
1085 z2 = z**2
1086 return _Horner(S(_1_0), sqrt(a) * a * m,
1087 xy - _6_0 * z2,
1088 (xy * _3_0 - _8_0 * z2) * z,
1089 (xy - z2) * _3_0 * z2,
1090 xy * z2 * z, *over) # Fsum
1093def _rF2(inst, x, y): # 2-arg version, z=0
1094 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.
1095 '''
1096 for a, b in _ab2(inst, x, y): # PYCHOK yield
1097 pass
1098 return _over(PI, a + b) # float
1101def _RF3(inst, x, y, z): # 3-arg version
1102 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.
1103 '''
1104 L = _List(x, y, z)
1105 for a, m, _, _ in L.amrs4(inst, False, _TolRF):
1106 pass
1107 x, y = L.rescale(a * m, x, y)
1108 z = neg(x + y)
1109 xy = x * y
1110 e2 = xy - z**2
1111 e3 = xy * z
1112 e4 = e2**2
1113 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>
1114 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44
1115 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)
1116 # converted to Horner-like form ...
1117 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)
1118 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)
1119 S += 240240
1120 return S.fdiv(sqrt(a) * 240240) # Fsum
1123def _rG2(inst, x, y, PI_=PI_4): # 2-args
1124 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.
1125 '''
1126 m = -1 # neg!
1127 S = None
1128 for a, b in _ab2(inst, x, y): # PYCHOK yield
1129 if S is None: # initial
1130 S = _Dsum()
1131 S += (a + b)**2 * _0_5
1132 else:
1133 S += (a - b)**2 * m
1134 m *= 2
1135 return S(PI_).fover(a + b)
1138def _rG3(inst, x, y, z): # 3-arg version
1139 '''(INTERNAL) C{x}, C{y} and C{z} all non-zero, see C{.fRG}.
1140 '''
1141 R = _RF3(inst, x, y, z) * z
1142 rd = (x - z) * (z - y) # - (y - z)
1143 if rd: # Carlson, eq 1.7
1144 R += _RD(inst, x, y, z, _3_0 / rd)
1145 R += sqrt(x * y / z)
1146 return R.fover(_2_0)
1149def _RJ(inst, x, y, z, p, *over):
1150 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.
1151 '''
1152 def _xyzp(x, y, z, p):
1153 return (x + p) * (y + p) * (z + p)
1155 L = _List(x, y, z, p)
1156 n = neg(_xyzp(x, y, z, -p))
1157 S = _Dsum()
1158 for a, m, _, s in L.amrs4(inst, True, _TolRD):
1159 if s:
1160 d = _xyzp(*s)
1161 if d:
1162 if n:
1163 rc = _rC(inst, _1_0, n / d**2 + _1_0)
1164 n *= _1_64th # /= chokes PyChecker
1165 else:
1166 rc = _1_0 # == _rC(None, _1_0, _1_0)
1167 S += rc / (d * m)
1168 else: # PYCHOK no cover
1169 return NAN
1170 x, y, z = L.rescale(a * m, x, y, z)
1171 p = Fsum(x, y, z).fover(_N_2_0)
1172 p2 = p**2
1173 p3 = p2 * p
1174 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)
1175 E2p = E2 * p
1176 xyz = x * y * z
1177 return _Horner(S(_6_0), sqrt(a) * a * m, E2,
1178 Fsum(p3 * _4_0, xyz, E2p * _2_0),
1179 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),
1180 xyz * p2, *over) # Fsum
1182# **) MIT License
1183#
1184# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
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