Coverage for pygeodesy/elliptic.py: 96%
480 statements
« prev ^ index » next coverage.py v7.2.2, created at 2024-02-05 16:22 -0500
« prev ^ index » next coverage.py v7.2.2, created at 2024-02-05 16:22 -0500
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__ = '23.09.18'
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 _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._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 # For kp2 close to zero use asin(r / cE) or J. P. Boyd,
407 # Applied Math. and Computation 218, 7005-7013 (2012)
408 # <https://DOI.org/10.1016/j.amc.2011.12.021>
409 _Phi2, self._iteration = Phi.fsum2_, 0 # aggregate
410 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
411 sn, cn, dn = self._sncndn3(phi)
412 if dn:
413 sn = self.fE(sn, cn, dn)
414 phi, d = _Phi2((r - sn) / dn)
415 else: # PYCHOK no cover
416 d = _0_0 # XXX continue?
417 if fabs(d) < _TolJAC: # 3-4 trips
418 _iterations(self, i)
419 break
420 else: # PYCHOK no cover
421 raise _convergenceError(d, _TolJAC)
422 return Phi.fsum_(n * PI) if n else phi
424 @Property_RO
425 def eps(self):
426 '''Get epsilon (C{float}).
427 '''
428 return self._cDEKEeps.eps
430 def fD(self, phi_or_sn, cn=None, dn=None):
431 '''Jahnke's incomplete elliptic integral in terms of
432 Jacobi elliptic functions.
434 @arg phi_or_sn: φ or sin(φ).
435 @kwarg cn: C{None} or cos(φ).
436 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
438 @return: D(φ, k) as though φ ∈ (−π, π] (C{float}),
439 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
441 @raise EllipticError: Invalid invokation or no convergence.
442 '''
443 def _fD(sn, cn, dn):
444 r = fabs(sn)**3
445 if r:
446 r = float(_RD(self, cn**2, dn**2, _1_0, _3_0 / r))
447 return r
449 return self._fXf(phi_or_sn, cn, dn, self.cD,
450 self.deltaD, _fD)
452 def fDelta(self, sn, cn):
453 '''The C{Delta} amplitude function.
455 @arg sn: sin(φ).
456 @arg cn: cos(φ).
458 @return: sqrt(1 − k2 * sin(2φ)) (C{float}).
459 '''
460 try:
461 k2 = self.k2
462 s = (self.kp2 + cn**2 * k2) if k2 > 0 else (
463 (_1_0 - sn**2 * k2) if k2 < 0 else self.kp2)
464 return sqrt(s) if s else _0_0
466 except Exception as e:
467 raise _ellipticError(self.fDelta, sn, cn, k2=k2, cause=e)
469 def fE(self, phi_or_sn, cn=None, dn=None):
470 '''The incomplete integral of the second kind in terms of
471 Jacobi elliptic functions.
473 @arg phi_or_sn: φ or sin(φ).
474 @kwarg cn: C{None} or cos(φ).
475 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
477 @return: E(φ, k) as though φ ∈ (−π, π] (C{float}),
478 U{defined<https://DLMF.NIST.gov/19.2.E5>}.
480 @raise EllipticError: Invalid invokation or no convergence.
481 '''
482 def _fE(sn, cn, dn):
483 '''(INTERNAL) Core of C{.fE}.
484 '''
485 if sn:
486 sn2, cn2, dn2 = sn**2, cn**2, dn**2
487 kp2, k2 = self.kp2, self.k2
488 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9>
489 Ei = _RF3(self, cn2, dn2, _1_0)
490 if k2:
491 Ei -= _RD(self, cn2, dn2, _1_0, _3over(k2, sn2))
492 elif kp2 >= 0: # k2 > 0, <https://DLMF.NIST.gov/19.25.E10>
493 Ei = _over(k2 * fabs(cn), dn) # float
494 if kp2:
495 Ei += (_RD( self, cn2, _1_0, dn2, _3over(k2, sn2)) +
496 _RF3(self, cn2, dn2, _1_0)) * kp2
497 else: # kp2 < 0, <https://DLMF.NIST.gov/19.25.E11>
498 Ei = _over(dn, fabs(cn))
499 Ei -= _RD(self, dn2, _1_0, cn2, _3over(kp2, sn2))
500 Ei *= fabs(sn)
501 ei = float(Ei)
502 else: # PYCHOK no cover
503 ei = _0_0
504 return ei
506 return self._fXf(phi_or_sn, cn, dn, self.cE,
507 self.deltaE, _fE)
509 def fEd(self, deg):
510 '''The incomplete integral of the second kind with
511 the argument given in C{degrees}.
513 @arg deg: Angle (C{degrees}).
515 @return: E(π B{C{deg}} / 180, k) (C{float}).
517 @raise EllipticError: No convergence.
518 '''
519 if _K_2_0:
520 e = round((deg - _norm180(deg)) / _360_0)
521 elif fabs(deg) < _180_0:
522 e = _0_0
523 else:
524 e = ceil(deg / _360_0 - _0_5)
525 deg -= e * _360_0
526 return self.fE(radians(deg)) + e * self.cE * _4_0
528 def fEinv(self, x):
529 '''The inverse of the incomplete integral of the second kind.
531 @arg x: Argument (C{float}).
533 @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}
534 (C{float}).
536 @raise EllipticError: No convergence.
537 '''
538 try:
539 return self._Einv(x)
540 except Exception as e:
541 raise _ellipticError(self.fEinv, x, cause=e)
543 def fF(self, phi_or_sn, cn=None, dn=None):
544 '''The incomplete integral of the first kind in terms of
545 Jacobi elliptic functions.
547 @arg phi_or_sn: φ or sin(φ).
548 @kwarg cn: C{None} or cos(φ).
549 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
551 @return: F(φ, k) as though φ ∈ (−π, π] (C{float}),
552 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
554 @raise EllipticError: Invalid invokation or no convergence.
555 '''
556 def _fF(sn, cn, dn):
557 r = fabs(sn)
558 if r:
559 r = float(_RF3(self, cn**2, dn**2, _1_0).fmul(r))
560 return r
562 return self._fXf(phi_or_sn, cn, dn, self.cK,
563 self.deltaF, _fF)
565 def fG(self, phi_or_sn, cn=None, dn=None):
566 '''Legendre's geodesic longitude integral in terms of
567 Jacobi elliptic functions.
569 @arg phi_or_sn: φ or sin(φ).
570 @kwarg cn: C{None} or cos(φ).
571 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
573 @return: G(φ, k) as though φ ∈ (−π, π] (C{float}).
575 @raise EllipticError: Invalid invokation or no convergence.
577 @note: Legendre expresses the longitude of a point on the
578 geodesic in terms of this combination of elliptic
579 integrals in U{Exercices de Calcul Intégral, Vol 1
580 (1811), p 181<https://Books.Google.com/books?id=
581 riIOAAAAQAAJ&pg=PA181>}.
583 @see: U{Geodesics in terms of elliptic integrals<https://
584 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
585 for the expression for the longitude in terms of this function.
586 '''
587 return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2,
588 self.cG, self.deltaG)
590 def fH(self, phi_or_sn, cn=None, dn=None):
591 '''Cayley's geodesic longitude difference integral in terms of
592 Jacobi elliptic functions.
594 @arg phi_or_sn: φ or sin(φ).
595 @kwarg cn: C{None} or cos(φ).
596 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
598 @return: H(φ, k) as though φ ∈ (−π, π] (C{float}).
600 @raise EllipticError: Invalid invokation or no convergence.
602 @note: Cayley expresses the longitude difference of a point
603 on the geodesic in terms of this combination of
604 elliptic integrals in U{Phil. Mag. B{40} (1870), p 333
605 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}.
607 @see: U{Geodesics in terms of elliptic integrals<https://
608 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
609 for the expression for the longitude in terms of this function.
610 '''
611 return self._fXa(phi_or_sn, cn, dn, -self.alphap2,
612 self.cH, self.deltaH)
614 def fPi(self, phi_or_sn, cn=None, dn=None):
615 '''The incomplete integral of the third kind in terms of
616 Jacobi elliptic functions.
618 @arg phi_or_sn: φ or sin(φ).
619 @kwarg cn: C{None} or cos(φ).
620 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
622 @return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}).
624 @raise EllipticError: Invalid invokation or no convergence.
625 '''
626 if dn is None and cn is not None: # and isscalar(phi_or_sn)
627 dn = self.fDelta(phi_or_sn, cn) # in .triaxial
628 return self._fXa(phi_or_sn, cn, dn, self.alpha2,
629 self.cPi, self.deltaPi)
631 def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX):
632 '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}.
633 '''
634 def _fX(sn, cn, dn):
635 if sn:
636 cn2, dn2 = cn**2, dn**2
637 R = _RF3(self, cn2, dn2, _1_0)
638 if aX:
639 sn2 = sn**2
640 p = sn2 * self.alphap2 + cn2
641 R += _RJ(self, cn2, dn2, _1_0, p, _3over(aX, sn2))
642 R *= fabs(sn)
643 r = float(R)
644 else: # PYCHOK no cover
645 r = _0_0
646 return r
648 return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX)
650 def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX):
651 '''(INTERNAL) Helper for C{.fD}, C{.fE}, C{.fF} and C{._fXa}.
652 '''
653 self._iteration = 0 # aggregate
654 phi = sn = phi_or_sn
655 if cn is dn is None: # fX(phi) call
656 sn, cn, dn = self._sncndn3(phi)
657 if fabs(phi) >= PI:
658 return (deltaX(sn, cn, dn) + phi) * cX / PI_2
659 # fall through
660 elif cn is None or dn is None:
661 n = NN(_f_, deltaX.__name__[5:])
662 raise _ellipticError(n, sn, cn, dn)
664 if _signBit(cn): # enforce usual trig-like symmetries
665 xi = cX * _2_0 - fX(sn, cn, dn)
666 else:
667 xi = fX(sn, cn, dn) if cn > 0 else cX
668 return copysign0(xi, sn)
670 @Property_RO
671 def k2(self):
672 '''Get k^2, the square of the modulus (C{float}).
673 '''
674 return self._k2
676 @Property_RO
677 def kp2(self):
678 '''Get k'^2, the square of the complementary modulus (C{float}).
679 '''
680 return self._kp2
682 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13
683 '''Reset the modulus, parameter and the complementaries.
685 @kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1).
686 @kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1).
687 @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0).
688 @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0).
690 @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}}
691 or B{C{alphap2}}.
693 @note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and
694 C{B{alpha2} + B{alphap2} = 1}. No checking is done
695 that these conditions are met to enable accuracy to be
696 maintained, e.g., when C{k} is very close to unity.
697 '''
698 if self.__dict__:
699 _update_all(self, _Named.iteration._uname, Base=Property_RO)
701 self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0)
702 self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError)
704 self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0)
705 self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2),
706 Error=EllipticError)
708 # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
709 # K E D
710 # k = 0: pi/2 pi/2 pi/4
711 # k = 1: inf 1 inf
712 # Pi G H
713 # k = 0, alpha = 0: pi/2 pi/2 pi/4
714 # k = 1, alpha = 0: inf 1 1
715 # k = 0, alpha = 1: inf inf pi/2
716 # k = 1, alpha = 1: inf inf inf
717 #
718 # G(0, k) = Pi(0, k) = H(1, k) = E(k)
719 # H(0, k) = K(k) - D(k)
720 # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2))
721 # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1))
722 # Pi(alpha2, 1) = inf
723 # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
725 def sncndn(self, x):
726 '''The Jacobi elliptic function.
728 @arg x: The argument (C{float}).
730 @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with
731 C{*n(B{x}, k)}.
733 @raise EllipticError: No convergence.
734 '''
735 self._iteration = 0 # reset
736 try: # Bulirsch's sncndn routine, p 89.
737 if self.kp2:
738 c, d, cd, mn = self._sncndn4
739 dn = _1_0
740 sn, cn = _sincos2(x * cd)
741 if sn:
742 a = cn / sn
743 c *= a
744 for m, n in reversed(mn):
745 a *= c
746 c *= dn
747 dn = (n + a) / (m + a)
748 a = c / m
749 a = _1_0 / hypot1(c)
750 sn = neg(a) if _signBit(sn) else a
751 cn = c * sn
752 if d and _signBit(self.kp2):
753 cn, dn = dn, cn
754 sn = sn / d # /= chokes PyChecker
755 else:
756 sn = tanh(x)
757 cn = dn = _1_0 / cosh(x)
759 except Exception as e:
760 raise _ellipticError(self.sncndn, x, kp2=self.kp2, cause=e)
762 return Elliptic3Tuple(sn, cn, dn, iteration=self._iteration)
764 def _sncndn3(self, phi):
765 '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}.
766 '''
767 sn, cn = _sincos2(phi)
768 return sn, cn, self.fDelta(sn, cn)
770 @Property_RO
771 def _sncndn4(self):
772 '''(INTERNAL) Get Bulirsch' 4-tuple C{(c, d, cd, mn)}.
773 '''
774 # Bulirsch's sncndn routine, p 89.
775 d, mc = 0, self.kp2
776 if _signBit(mc):
777 d = _1_0 - mc
778 mc = neg(mc / d)
779 d = sqrt(d)
781 mn, a = [], _1_0
782 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
783 mc = sqrt(mc)
784 mn.append((a, mc))
785 c = (a + mc) * _0_5
786 r = fabs(mc - a)
787 t = _TolJAC * a
788 if r <= t: # 6 trips, quadratic
789 _iterations(self, i)
790 break
791 mc *= a
792 a = c
793 else: # PYCHOK no cover
794 raise _convergenceError(r, t)
795 cd = (c * d) if d else c
796 return c, d, cd, mn
798 @staticmethod
799 def fRC(x, y):
800 '''Degenerate symmetric integral of the first kind C{RC(x, y)}.
802 @return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}.
804 @see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and
805 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
806 '''
807 return _rC(None, x, y)
809 @staticmethod
810 def fRD(x, y, z, *over):
811 '''Degenerate symmetric integral of the third kind C{RD(x, y, z)}.
813 @return: C{RD(x, y, z) / over}, equivalent to C{RJ(x, y, z, z)
814 / over} with C{over} typically 3.
816 @see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and
817 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
818 '''
819 try:
820 return float(_RD(None, x, y, z, *over))
821 except Exception as e:
822 raise _ellipticError(Elliptic.fRD, x, y, z, *over, cause=e)
824 @staticmethod
825 def fRF(x, y, z=0):
826 '''Symmetric or complete symmetric integral of the first kind
827 C{RF(x, y, z)} respectively C{RF(x, y)}.
829 @return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}.
831 @see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and
832 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
833 '''
834 try:
835 return float(_RF3(None, x, y, z)) if z else _rF2(None, x, y)
836 except Exception as e:
837 raise _ellipticError(Elliptic.fRF, x, y, z, cause=e)
839 @staticmethod
840 def fRG(x, y, z=0):
841 '''Symmetric or complete symmetric integral of the second kind
842 C{RG(x, y, z)} respectively C{RG(x, y)}.
844 @return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}.
846 @see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>},
847 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>} and
848 U{RG<https://GeographicLib.SourceForge.io/C++/doc/
849 EllipticFunction_8cpp_source.html#l00096>} version 2.3.
850 '''
851 try:
852 return _rG2(None, x, y) if z == 0 else (
853 _rG2(None, z, x) if y == 0 else (
854 _rG2(None, y, z) if x == 0 else _rG3(None, x, y, z)))
855 except Exception as e:
856 t = _negative_ if min(x, y, z) < 0 else NN
857 raise _ellipticError(Elliptic.fRG, x, y, z, cause=e, txt=t)
859 @staticmethod
860 def fRJ(x, y, z, p):
861 '''Symmetric integral of the third kind C{RJ(x, y, z, p)}.
863 @return: C{RJ(x, y, z, p)}.
865 @see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and
866 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
867 '''
868 try:
869 return float(_RJ(None, x, y, z, p))
870 except Exception as e:
871 raise _ellipticError(Elliptic.fRJ, x, y, z, p, cause=e)
873 @staticmethod
874 def _RFRD(x, y, z, m):
875 # in .auxilats.AuxDLat.DE, .auxilats.AuxLat.Rectifying
876 try: # float(RF(x, y, z) - RD(x, y, z, 3 / m))
877 R = _RF3(None, x, y, z)
878 if m:
879 R -= _RD(None, x, y, z, _3_0 / m)
880 except Exception as e:
881 raise _ellipticError(Elliptic._RFRD, x, y, z, m, cause=e)
882 return float(R)
884_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset
887class EllipticError(_ValueError):
888 '''Elliptic function, integral, convergence or other L{Elliptic} issue.
889 '''
890 pass
893class Elliptic3Tuple(_NamedTuple):
894 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.
895 '''
896 _Names_ = ('sn', 'cn', 'dn')
897 _Units_ = ( Scalar, Scalar, Scalar)
900class _L(list):
901 '''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}.
902 '''
903 _a0 = None
904# _xyzp = ()
906 def __init__(self, *xyzp): # x, y, z [, p]
907 list.__init__(self, xyzp)
908 self._xyzp = xyzp
910 def a0(self, n):
911 '''Compute the initial C{a}.
912 '''
913 t = tuple(self)
914 m = n - len(t)
915 if m > 0:
916 t += t[-1:] * m
917 try:
918 a = Fsum(*t).fover(n)
919 except ValueError: # Fsum(NAN) exception
920 a = _sum(t) / n
921 self._a0 = a
922 return a
924 def amrs4(self, inst, y, Tol):
925 '''Yield Carlson 4-tuples C{(An, mul, lam, s)} plus sentinel, with
926 C{lam = fdot(sqrt(x), ... (z))} and C{s = (sqrt(x), ... (p))}.
927 '''
928 L = self
929 a = L.a0(5 if y else 3)
930 m = 1
931 t = max(fabs(a - _) for _ in L) / Tol
932 for i in range(_TRIPS):
933 d = fabs(a * m)
934 if d > t: # 3-6 trips
935 _iterations(inst, i)
936 break
937 s = map2(sqrt, L) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)]
938 try:
939 r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ...
940 except ValueError: # Fsum(NAN) exception
941 r = _sum(s[i] * s[(i + 1) % 3] for i in range(3))
942 L[:] = [(r + _) * _0_25 for _ in L]
943 a = (r + a) * _0_25
944 if y: # yield only if used
945 yield a, m, r, s # L[2] is next z
946 m *= 4
947 else: # PYCHOK no cover
948 raise _convergenceError(d, t, thresh=True)
949 yield a, m, None, () # sentinel: same a, next m, no r and s
951 def rescale(self, am, *xs):
952 '''Rescale C{x}, C{y}, ...
953 '''
954 # assert am
955 a0 = self._a0
956 for x in xs:
957 yield (a0 - x) / am
960def _ab2(inst, x, y):
961 '''(INTERNAL) Yield Carlson 2-tuples C{(xn, yn)}.
962 '''
963 a, b = sqrt(x), sqrt(y)
964 if b > a:
965 a, b = b, a
966 yield a, b # initial x0, y0
967 for i in range(_TRIPS):
968 d = fabs(a - b)
969 t = _TolRG0 * a
970 if d <= t: # 3-4 trips
971 _iterations(inst, i)
972 break
973 a, b = ((a + b) * _0_5), sqrt(a * b)
974 yield a, b # xn, yn
975 else: # PYCHOK no cover
976 raise _convergenceError(d, t)
979def _convergenceError(d, tol, **thresh):
980 '''(INTERNAL) Format a no-convergence Error.
981 '''
982 t = Fmt.no_convergence(d, tol, **thresh)
983 return ValueError(t) # txt only
986def _deltaX(sn, cn, dn, cX, fX):
987 '''(INTERNAL) Helper for C{Elliptic.deltaD} thru C{.deltaPi}.
988 '''
989 try:
990 if cn is None or dn is None:
991 raise ValueError(_invalid_)
993 if _signBit(cn):
994 sn, cn = neg_(sn, cn)
995 r = fX(sn, cn, dn) * PI_2 / cX
996 return r - atan2(sn, cn)
998 except Exception as e:
999 n = NN(_delta_, fX.__name__[1:])
1000 raise _ellipticError(n, sn, cn, dn, cause=e)
1003def _ellipticError(where, *args, **kwds_cause_txt):
1004 '''(INTERNAL) Format an L{EllipticError}.
1005 '''
1006 c = _xkwds_pop(kwds_cause_txt, cause=None)
1007 t = _xkwds_pop(kwds_cause_txt, txt=NN)
1008 n = _xattr(where, __name__=where) # _dunder_nameof(where, where)
1009 n = _DOT_(Elliptic.__name__, n)
1010 n = _SPACE_(_invokation_, n)
1011 u = unstr(n, *args, **kwds_cause_txt)
1012 return EllipticError(u, cause=c, txt=t)
1015def _Horner(S, e1, E2, E3, E4, E5, *over):
1016 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.
1017 '''
1018 E22 = E2**2
1019 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>
1020 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52
1021 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20
1022 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)
1023 # converted to Horner-like form ...
1024 F = Fsum
1025 e = e1 * 4084080
1026 S *= e
1027 S += F(E2 * -540540, 471240).fmul(E5)
1028 S += F(E2 * 612612, E3 * -540540, -556920).fmul(E4)
1029 S += F(E2 * -706860, E22 * 675675, E3 * 306306, 680680).fmul(E3)
1030 S += F(E2 * 417690, E22 * -255255, -875160).fmul(E2)
1031 S += 4084080
1032 return S.fdiv((over[0] * e) if over else e) # Fsum
1035def _iterations(inst, i):
1036 '''(INTERNAL) Aggregate iterations B{C{i}}.
1037 '''
1038 if inst and i > 0:
1039 inst._iteration += i
1042def _3over(a, b):
1043 '''(INTERNAL) Return C{3 / (a * b)}.
1044 '''
1045 return _over(_3_0, a * b)
1048def _rC(unused, x, y):
1049 '''(INTERNAL) Defined only for C{y != 0} and C{x >= 0}.
1050 '''
1051 d = x - y
1052 if d < 0: # catch NaN
1053 # <https://DLMF.NIST.gov/19.2.E18>
1054 d = -d
1055 r = atan(sqrt(d / x)) if x > 0 else PI_2
1056 elif d == 0: # XXX d < EPS0? or EPS02 or _EPSmin
1057 d, r = y, _1_0
1058 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>
1059 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))
1060 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>
1061 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))
1062 else: # PYCHOK no cover
1063 raise _ellipticError(Elliptic.fRC, x, y)
1064 return r / sqrt(d) # float
1067def _RD(inst, x, y, z, *over):
1068 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.
1069 '''
1070 L = _L(x, y, z)
1071 S = _D()
1072 for a, m, r, s in L.amrs4(inst, True, _TolRF):
1073 if s:
1074 S += _over(_3_0, (r + z) * s[2] * m)
1075 z = L[2] # s[2] = sqrt(z)
1076 x, y = L.rescale(-a * m, x, y)
1077 xy = x * y
1078 z = (x + y) / _3_0
1079 z2 = z**2
1080 return _Horner(S(_1_0), sqrt(a) * a * m,
1081 xy - _6_0 * z2,
1082 (xy * _3_0 - _8_0 * z2) * z,
1083 (xy - z2) * _3_0 * z2,
1084 xy * z2 * z, *over) # Fsum
1087def _rF2(inst, x, y): # 2-arg version, z=0
1088 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.
1089 '''
1090 for a, b in _ab2(inst, x, y): # PYCHOK yield
1091 pass
1092 return _over(PI, a + b) # float
1095def _RF3(inst, x, y, z): # 3-arg version
1096 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.
1097 '''
1098 L = _L(x, y, z)
1099 for a, m, _, _ in L.amrs4(inst, False, _TolRF):
1100 pass
1101 x, y = L.rescale(a * m, x, y)
1102 z = neg(x + y)
1103 xy = x * y
1104 e2 = xy - z**2
1105 e3 = xy * z
1106 e4 = e2**2
1107 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>
1108 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44
1109 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)
1110 # converted to Horner-like form ...
1111 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)
1112 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)
1113 S += 240240
1114 return S.fdiv(sqrt(a) * 240240) # Fsum
1117def _rG2(inst, x, y, PI_=PI_4): # 2-args
1118 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.
1119 '''
1120 m = -1 # neg!
1121 S = _D()
1122 # assert not S
1123 for a, b in _ab2(inst, x, y): # PYCHOK yield
1124 if S:
1125 S += (a - b)**2 * m
1126 m *= 2
1127 else: # initial
1128 S += (a + b)**2 * _0_5
1129 return S(PI_).fover(a + b)
1132def _rG3(inst, x, y, z): # 3-arg version
1133 '''(INTERNAL) C{x}, C{y} and C{z} all non-zero, see C{.fRG}.
1134 '''
1135 R = _RF3(inst, x, y, z) * z
1136 rd = (x - z) * (z - y) # - (y - z)
1137 if rd: # Carlson, eq 1.7
1138 R += _RD(inst, x, y, z, _3_0 / rd)
1139 R += sqrt(x * y / z)
1140 return R.fover(_2_0)
1143def _RJ(inst, x, y, z, p, *over):
1144 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.
1145 '''
1146 def _xyzp(x, y, z, p):
1147 return (x + p) * (y + p) * (z + p)
1149 L = _L(x, y, z, p)
1150 n = neg(_xyzp(x, y, z, -p))
1151 S = _D()
1152 for a, m, _, s in L.amrs4(inst, True, _TolRD):
1153 if s:
1154 d = _xyzp(*s)
1155 if d:
1156 if n:
1157 rc = _rC(inst, _1_0, n / d**2 + _1_0)
1158 n *= _1_64th # /= chokes PyChecker
1159 else:
1160 rc = _1_0 # == _rC(None, _1_0, _1_0)
1161 S += rc / (d * m)
1162 else: # PYCHOK no cover
1163 return NAN
1164 x, y, z = L.rescale(a * m, x, y, z)
1165 p = Fsum(x, y, z).fover(_N_2_0)
1166 p2 = p**2
1167 p3 = p2 * p
1168 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)
1169 E2p = E2 * p
1170 xyz = x * y * z
1171 return _Horner(S(_6_0), sqrt(a) * a * m, E2,
1172 Fsum(p3 * _4_0, xyz, E2p * _2_0),
1173 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),
1174 xyz * p2, *over) # Fsum
1176# **) MIT License
1177#
1178# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.
1179#
1180# Permission is hereby granted, free of charge, to any person obtaining a
1181# copy of this software and associated documentation files (the "Software"),
1182# to deal in the Software without restriction, including without limitation
1183# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1184# and/or sell copies of the Software, and to permit persons to whom the
1185# Software is furnished to do so, subject to the following conditions:
1186#
1187# The above copyright notice and this permission notice shall be included
1188# in all copies or substantial portions of the Software.
1189#
1190# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1191# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1192# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1193# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1194# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1195# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1196# OTHER DEALINGS IN THE SOFTWARE.