Coverage for pygeodesy/elliptic.py: 99%
453 statements
« prev ^ index » next coverage.py v7.2.2, created at 2023-04-12 11:45 -0400
« prev ^ index » next coverage.py v7.2.2, created at 2023-04-12 11:45 -0400
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:Charles@Karney.com>} (2008-2022)
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
78from pygeodesy.constants import EPS, _EPStol as _TolJAC, INF, PI, PI_2, PI_4, \
79 _0_0, _0_125, _0_25, _0_5, _1_0, _1_64th, _2_0, \
80 _N_2_0, _3_0, _4_0, _6_0, _8_0, _180_0, _360_0
81from pygeodesy.errors import _ValueError, _xkwds_pop
82from pygeodesy.fmath import fdot, Fsum, hypot1
83# from pygeodesy.fsums import Fsum # from .fmath
84from pygeodesy.interns import NN, _DOT_, _f_, _SPACE_
85from pygeodesy.karney import _ALL_LAZY, _signBit
86# from pygeodesy.lazily import _ALL_LAZY # from .karney
87from pygeodesy.named import _Named, _NamedTuple, _NotImplemented
88from pygeodesy.props import _allPropertiesOf_n, Property_RO, property_RO, \
89 _update_all
90from pygeodesy.streprs import Fmt, unstr
91from pygeodesy.units import Scalar, Scalar_
92from pygeodesy.utily import sincos2, sincos2d
94from math import asinh, atan, atan2, ceil, cosh, fabs, floor, sin, sqrt, tanh
96__all__ = _ALL_LAZY.elliptic
97__version__ = '23.03.19'
99_delta_ = 'delta'
100_invokation_ = 'invokation'
101_TolRD = pow(EPS * 0.002, _0_125) # 8th root: quadquadratic?, octic?, ocrt?
102_TolRF = pow(EPS * 0.030, _0_125) # 4th root: biquadratic, quartic, qurt?
103_TolRG0 = _TolJAC * 2.7
104_TRIPS = 31 # Max depth, 7 might be sufficient
107class _Complete(object):
108 '''(INTERAL) Hold complete integrals.
109 '''
110 def __init__(self, **kwds):
111 self.__dict__ = kwds
114class _Deferred(list):
115 '''(INTERNAL) Collector for L{_Deferred_Fsum}.
116 '''
117 def __init__(self, *xs):
118 if xs:
119 list.__init__(self, xs)
121 def __add__(self, other): # PYCHOK no cover
122 return _NotImplemented(self, other)
124 def __iadd__(self, x): # overide C{list} += C{list}
125 # assert isscalar(x)
126 self.append(float(x))
127 return self
129 def __imul__(self, other): # PYCHOK no cover
130 return _NotImplemented(self, other)
132 def __isub__(self, x): # PYCHOK no cover
133 # assert isscalar(x)
134 self.append(-float(x))
135 return self
137 def __mul__(self, other): # PYCHOK no cover
138 return _NotImplemented(self, other)
140 def __radd__(self, other): # PYCHOK no cover
141 return _NotImplemented(self, other)
143 def __rsub__(self, other): # PYCHOK no cover
144 return _NotImplemented(self, other)
146 def __sub__(self, other): # PYCHOK no cover
147 return _NotImplemented(self, other)
149 @property_RO
150 def Fsum(self):
151 # get a L{_Deferred_Fsum} instance, pre-named
152 return _Deferred_Fsum()._facc(self) # known C{float}s
155class _Deferred_Fsum(Fsum):
156 '''(INTERNAL) Deferred L{Fsum}.
157 '''
158 name = NN # pre-named, overridden below
160 def _update(self, **other): # PYCHOK don't ...
161 # ... waste time zapping non-existing Property/_ROs
162 if other or len(self.__dict__) > 2:
163 Fsum._update(self, **other)
165_Deferred_Fsum.name = _Deferred_Fsum.__name__ # PYCHOK once
168class Elliptic(_Named):
169 '''Elliptic integrals and functions.
171 @see: I{Karney}'s U{Detailed Description<https://GeographicLib.SourceForge.io/
172 html/classGeographicLib_1_1EllipticFunction.html#details>}.
173 '''
174 _alpha2 = 0
175 _alphap2 = 0
176 _eps = EPS
177 _k2 = 0
178 _kp2 = 0
180 def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None, name=NN):
181 '''Constructor, specifying the C{modulus} and C{parameter}.
183 @kwarg name: Optional name (C{str}).
185 @see: Method L{Elliptic.reset} for further details.
187 @note: If only elliptic integrals of the first and second kinds
188 are needed, use C{B{alpha2}=0}, the default value. In
189 that case, we have C{Π(φ, 0, k) = F(φ, k), G(φ, 0, k) =
190 E(φ, k)} and C{H(φ, 0, k) = F(φ, k) - D(φ, k)}.
191 '''
192 self.reset(k2=k2, alpha2=alpha2, kp2=kp2, alphap2=alphap2)
194 if name:
195 self.name = name
197 @Property_RO
198 def alpha2(self):
199 '''Get α^2, the square of the parameter (C{float}).
200 '''
201 return self._alpha2
203 @Property_RO
204 def alphap2(self):
205 '''Get α'^2, the square of the complementary parameter (C{float}).
206 '''
207 return self._alphap2
209 @Property_RO
210 def cD(self):
211 '''Get Jahnke's complete integral C{D(k)} (C{float}),
212 U{defined<https://DLMF.NIST.gov/19.2.E6>}.
213 '''
214 return self._reset_cDcEcKcKE_eps.cD
216 @Property_RO
217 def cE(self):
218 '''Get the complete integral of the second kind C{E(k)}
219 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}.
220 '''
221 return self._reset_cDcEcKcKE_eps.cE
223 @Property_RO
224 def cG(self):
225 '''Get Legendre's complete geodesic longitude integral
226 C{G(α^2, k)} (C{float}).
227 '''
228 return self._reset_cGcHcPi.cG
230 @Property_RO
231 def cH(self):
232 '''Get Cayley's complete geodesic longitude difference integral
233 C{H(α^2, k)} (C{float}).
234 '''
235 return self._reset_cGcHcPi.cH
237 @Property_RO
238 def cK(self):
239 '''Get the complete integral of the first kind C{K(k)}
240 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}.
241 '''
242 return self._reset_cDcEcKcKE_eps.cK
244 @Property_RO
245 def cKE(self):
246 '''Get the difference between the complete integrals of the
247 first and second kinds, C{K(k) − E(k)} (C{float}).
248 '''
249 return self._reset_cDcEcKcKE_eps.cKE
251 @Property_RO
252 def cPi(self):
253 '''Get the complete integral of the third kind C{Pi(α^2, k)}
254 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}.
255 '''
256 return self._reset_cGcHcPi.cPi
258 def deltaD(self, sn, cn, dn):
259 '''The periodic Jahnke's incomplete elliptic integral.
261 @arg sn: sin(φ).
262 @arg cn: cos(φ).
263 @arg dn: sqrt(1 − k2 * sin(2φ)).
265 @return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}).
267 @raise EllipticError: Invalid invokation or no convergence.
268 '''
269 return self._deltaX(sn, cn, dn, self.cD, self.fD)
271 def deltaE(self, sn, cn, dn):
272 '''The periodic incomplete integral of the second kind.
274 @arg sn: sin(φ).
275 @arg cn: cos(φ).
276 @arg dn: sqrt(1 − k2 * sin(2φ)).
278 @return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}).
280 @raise EllipticError: Invalid invokation or no convergence.
281 '''
282 return self._deltaX(sn, cn, dn, self.cE, self.fE)
284 def deltaEinv(self, stau, ctau):
285 '''The periodic inverse of the incomplete integral of the second kind.
287 @arg stau: sin(τ)
288 @arg ctau: cos(τ)
290 @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}).
292 @raise EllipticError: No convergence.
293 '''
294 # Function is periodic with period pi
295 t = atan2(-stau, -ctau) if _signBit(ctau) else atan2(stau, ctau)
296 return self.fEinv(t * self.cE / PI_2) - t
298 def deltaF(self, sn, cn, dn):
299 '''The periodic incomplete integral of the first kind.
301 @arg sn: sin(φ).
302 @arg cn: cos(φ).
303 @arg dn: sqrt(1 − k2 * sin(2φ)).
305 @return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}).
307 @raise EllipticError: Invalid invokation or no convergence.
308 '''
309 return self._deltaX(sn, cn, dn, self.cK, self.fF)
311 def deltaG(self, sn, cn, dn):
312 '''Legendre's periodic geodesic longitude integral.
314 @arg sn: sin(φ).
315 @arg cn: cos(φ).
316 @arg dn: sqrt(1 − k2 * sin(2φ)).
318 @return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}).
320 @raise EllipticError: Invalid invokation or no convergence.
321 '''
322 return self._deltaX(sn, cn, dn, self.cG, self.fG)
324 def deltaH(self, sn, cn, dn):
325 '''Cayley's periodic geodesic longitude difference integral.
327 @arg sn: sin(φ).
328 @arg cn: cos(φ).
329 @arg dn: sqrt(1 − k2 * sin(2φ)).
331 @return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}).
333 @raise EllipticError: Invalid invokation or no convergence.
334 '''
335 return self._deltaX(sn, cn, dn, self.cH, self.fH)
337 def deltaPi(self, sn, cn, dn):
338 '''The periodic incomplete integral of the third kind.
340 @arg sn: sin(φ).
341 @arg cn: cos(φ).
342 @arg dn: sqrt(1 − k2 * sin(2φ)).
344 @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ
345 (C{float}).
347 @raise EllipticError: Invalid invokation or no convergence.
348 '''
349 return self._deltaX(sn, cn, dn, self.cPi, self.fPi)
351 def _deltaX(self, sn, cn, dn, cX, fX):
352 '''(INTERNAL) Helper for C{.deltaD} thru C{.deltaPi}.
353 '''
354 if cn is None or dn is None:
355 n = NN(_delta_, fX.__name__[1:])
356 raise _invokationError(n, sn, cn, dn)
358 if _signBit(cn):
359 cn, sn = -cn, -sn
360 return fX(sn, cn, dn) * PI_2 / cX - atan2(sn, cn)
362 @Property_RO
363 def eps(self):
364 '''Get epsilon (C{float}).
365 '''
366 return self._reset_cDcEcKcKE_eps.eps
368 def fD(self, phi_or_sn, cn=None, dn=None):
369 '''Jahnke's incomplete elliptic integral in terms of
370 Jacobi elliptic functions.
372 @arg phi_or_sn: φ or sin(φ).
373 @kwarg cn: C{None} or cos(φ).
374 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
376 @return: D(φ, k) as though φ ∈ (−π, π] (C{float}),
377 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
379 @raise EllipticError: Invalid invokation or no convergence.
380 '''
381 def _fD(sn, cn, dn):
382 r = fabs(sn)**3
383 if r:
384 r = _RD(self, cn**2, dn**2, _1_0, _3_0 / r)
385 return r
387 return self._fXf(phi_or_sn, cn, dn, self.cD,
388 self.deltaD, _fD)
390 def fDelta(self, sn, cn):
391 '''The C{Delta} amplitude function.
393 @arg sn: sin(φ).
394 @arg cn: cos(φ).
396 @return: sqrt(1 − k2 * sin(2φ)) (C{float}).
397 '''
398 k2 = self.k2
399 s = (_1_0 - k2 * sn**2) if k2 < 0 else (self.kp2
400 + ((k2 * cn**2) if k2 > 0 else _0_0))
401 return sqrt(s) if s else _0_0
403 def fE(self, phi_or_sn, cn=None, dn=None):
404 '''The incomplete integral of the second kind in terms of
405 Jacobi elliptic functions.
407 @arg phi_or_sn: φ or sin(φ).
408 @kwarg cn: C{None} or cos(φ).
409 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
411 @return: E(φ, k) as though φ ∈ (−π, π] (C{float}),
412 U{defined<https://DLMF.NIST.gov/19.2.E5>}.
414 @raise EllipticError: Invalid invokation or no convergence.
415 '''
416 def _fE(sn, cn, dn):
417 '''(INTERNAL) Core of C{.fE}.
418 '''
419 if sn:
420 sn2, cn2, dn2 = sn**2, cn**2, dn**2
421 kp2, k2 = self.kp2, self.k2
422 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9>
423 ei = _RF3(self, cn2, dn2, _1_0)
424 if k2:
425 ei -= _RD(self, cn2, dn2, _1_0, _3over(k2, sn2))
426 elif kp2 >= 0: # <https://DLMF.NIST.gov/19.25.E10>
427 ei = k2 * fabs(cn) / dn
428 if kp2:
429 ei += (_RD( self, cn2, _1_0, dn2, _3over(k2, sn2)) +
430 _RF3(self, cn2, dn2, _1_0)) * kp2
431 else: # <https://DLMF.NIST.gov/19.25.E11>
432 ei = dn / fabs(cn)
433 ei -= _RD(self, dn2, _1_0, cn2, _3over(kp2, sn2))
434 ei *= fabs(sn)
435 else: # PYCHOK no cover
436 ei = _0_0
437 return ei
439 return self._fXf(phi_or_sn, cn, dn, self.cE,
440 self.deltaE, _fE)
442 def fEd(self, deg):
443 '''The incomplete integral of the second kind with
444 the argument given in degrees.
446 @arg deg: Angle (C{degrees}).
448 @return: E(π B{C{deg}}/180, k) (C{float}).
450 @raise EllipticError: No convergence.
451 '''
452 if fabs(deg) < _180_0:
453 e = _0_0
454 else: # PYCHOK no cover
455 e = ceil(deg / _360_0 - _0_5)
456 deg -= e * _360_0
457 e *= self.cE * _4_0
458 sn, cn = sincos2d(deg)
459 return self.fE(sn, cn, self.fDelta(sn, cn)) + e
461 def fEinv(self, x):
462 '''The inverse of the incomplete integral of the second kind.
464 @arg x: Argument (C{float}).
466 @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}
467 (C{float}).
469 @raise EllipticError: No convergence.
470 '''
471 E2 = self.cE * _2_0
472 n = floor(x / E2 + _0_5)
473 y = x - E2 * n # y now in [-ec, ec)
474 # linear approximation
475 phi = PI * y / E2 # phi in [-pi/2, pi/2)
476 Phi = Fsum(phi)
477 # first order correction
478 phi = Phi.fsum_(self.eps * sin(phi * _2_0) / _N_2_0)
479 # For kp2 close to zero use asin(x/.cE) or J. P. Boyd,
480 # Applied Math. and Computation 218, 7005-7013 (2012)
481 # <https://DOI.org/10.1016/j.amc.2011.12.021>
482 fE = self.fE
483 _sncndnPhi = self._sncndnPhi
484 _Phi2 = Phi.fsum2_
485 self._iteration = 0 # aggregate
486 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
487 sn, cn, dn = _sncndnPhi(phi)
488 phi, e = _Phi2((y - fE(sn, cn, dn)) / dn)
489 if fabs(e) < _TolJAC:
490 self._iteration += i
491 break
492 else: # PYCHOK no cover
493 raise _no_convergenceError(e, _TolJAC, self.fEinv, x)
494 return Phi.fsum_(n * PI) if n else phi
496 def fF(self, phi_or_sn, cn=None, dn=None):
497 '''The incomplete integral of the first kind in terms of
498 Jacobi elliptic functions.
500 @arg phi_or_sn: φ or sin(φ).
501 @kwarg cn: C{None} or cos(φ).
502 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
504 @return: F(φ, k) as though φ ∈ (−π, π] (C{float}),
505 U{defined<https://DLMF.NIST.gov/19.2.E4>}.
507 @raise EllipticError: Invalid invokation or no convergence.
508 '''
509 def _fF(sn, cn, dn):
510 r = fabs(sn)
511 if r:
512 r *= _RF3(self, cn**2, dn**2, _1_0)
513 return r
515 return self._fXf(phi_or_sn, cn, dn, self.cK,
516 self.deltaF, _fF)
518 def fG(self, phi_or_sn, cn=None, dn=None):
519 '''Legendre's geodesic longitude integral in terms of
520 Jacobi elliptic functions.
522 @arg phi_or_sn: φ or sin(φ).
523 @kwarg cn: C{None} or cos(φ).
524 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).
526 @return: G(φ, k) as though φ ∈ (−π, π] (C{float}).
528 @raise EllipticError: Invalid invokation or no convergence.
530 @note: Legendre expresses the longitude of a point on the
531 geodesic in terms of this combination of elliptic
532 integrals in U{Exercices de Calcul Intégral, Vol 1
533 (1811), p 181<https://Books.Google.com/books?id=
534 riIOAAAAQAAJ&pg=PA181>}.
536 @see: U{Geodesics in terms of elliptic integrals<https://
537 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
538 for the expression for the longitude in terms of this function.
539 '''
540 return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2,
541 self.cG, self.deltaG)
543 def fH(self, phi_or_sn, cn=None, dn=None):
544 '''Cayley's geodesic longitude difference integral 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: H(φ, k) as though φ ∈ (−π, π] (C{float}).
553 @raise EllipticError: Invalid invokation or no convergence.
555 @note: Cayley expresses the longitude difference of a point
556 on the geodesic in terms of this combination of
557 elliptic integrals in U{Phil. Mag. B{40} (1870), p 333
558 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}.
560 @see: U{Geodesics in terms of elliptic integrals<https://
561 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}
562 for the expression for the longitude in terms of this function.
563 '''
564 return self._fXa(phi_or_sn, cn, dn, -self.alphap2,
565 self.cH, self.deltaH)
567 def fPi(self, phi_or_sn, cn=None, dn=None):
568 '''The incomplete integral of the third kind 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: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}).
577 @raise EllipticError: Invalid invokation or no convergence.
578 '''
579 if dn is None and cn is not None: # and isscalar(phi_or_sn)
580 dn = self.fDelta(phi_or_sn, cn) # in .triaxial
581 return self._fXa(phi_or_sn, cn, dn, self.alpha2,
582 self.cPi, self.deltaPi)
584 def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX):
585 '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}.
586 '''
587 def _fX(sn, cn, dn):
588 if sn:
589 cn2, sn2, dn2 = cn**2, sn**2, dn**2
590 r = _RF3(self, cn2, dn2, _1_0)
591 if aX:
592 z = cn2 + sn2 * self.alphap2
593 r += _RJ(self, cn2, dn2, _1_0, z, _3over(aX, sn2))
594 r *= fabs(sn)
595 else: # PYCHOK no cover
596 r = _0_0
597 return r
599 return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX)
601 def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX):
602 '''(INTERNAL) Helper for C{f.D}, C{.fE}, C{.fF} and C{._fXa}.
603 '''
604 self._iteration = 0 # aggregate
605 phi = sn = phi_or_sn
606 if cn is dn is None: # fX(phi) call
607 sn, cn, dn = self._sncndnPhi(phi)
608 if fabs(phi) >= PI: # PYCHOK no cover
609 return (deltaX(sn, cn, dn) + phi) * cX / PI_2
610 # fall through
611 elif cn is None or dn is None:
612 n = NN(_f_, deltaX.__name__[5:])
613 raise _invokationError(n, sn, cn, dn)
615 if _signBit(cn): # enforce usual trig-like symmetries
616 xi = _2_0 * cX - fX(sn, cn, dn)
617 elif cn > 0:
618 xi = fX(sn, cn, dn)
619 else:
620 xi = cX
621 return copysign0(xi, sn)
623 @Property_RO
624 def k2(self):
625 '''Get k^2, the square of the modulus (C{float}).
626 '''
627 return self._k2
629 @Property_RO
630 def kp2(self):
631 '''Get k'^2, the square of the complementary modulus (C{float}).
632 '''
633 return self._kp2
635 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13
636 '''Reset the modulus, parameter and the complementaries.
638 @kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1).
639 @kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1).
640 @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0).
641 @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0).
643 @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}}
644 or B{C{alphap2}}.
646 @note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and
647 C{B{alpha2} + B{alphap2} = 1}. No checking is done
648 that these conditions are met to enable accuracy to be
649 maintained, e.g., when C{k} is very close to unity.
650 '''
651 _update_all(self, _Named.iteration._uname, Base=Property_RO)
653 self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0)
654 self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError)
656 self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0)
657 self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2),
658 Error=EllipticError)
660 # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1
661 # K E D
662 # k = 0: pi/2 pi/2 pi/4
663 # k = 1: inf 1 inf
664 # Pi G H
665 # k = 0, alpha = 0: pi/2 pi/2 pi/4
666 # k = 1, alpha = 0: inf 1 1
667 # k = 0, alpha = 1: inf inf pi/2
668 # k = 1, alpha = 1: inf inf inf
669 #
670 # G(0, k) = Pi(0, k) = H(1, k) = E(k)
671 # H(0, k) = K(k) - D(k)
672 # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2))
673 # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1))
674 # Pi(alpha2, 1) = inf
675 # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)
677 @Property_RO
678 def _reset_cDcEcKcKE_eps(self):
679 '''(INTERNAL) Get the complete integrals D, E, K and KE plus C{eps}.
680 '''
681 k2 = self.k2
682 if k2:
683 kp2 = self.kp2
684 if kp2:
685 self._iteration = 0
686 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3
687 # <https://DLMF.NIST.gov/19.25.E1>
688 cD = _RD(self, _0_0, kp2, _1_0, _3_0)
689 # Complete elliptic integral E(k), Carlson eq. 4.2
690 # <https://DLMF.NIST.gov/19.25.E1>
691 cE = _RG2(self, kp2, _1_0)
692 # Complete elliptic integral K(k), Carlson eq. 4.1
693 # <https://DLMF.NIST.gov/19.25.E1>
694 cK = _RF2(self, kp2, _1_0)
695 cKE = k2 * cD
696 eps = k2 / (sqrt(kp2) + _1_0)**2
697 else: # PYCHOK no cover
698 cD = cK = cKE = INF
699 cE = _1_0
700 eps = k2
701 else: # PYCHOK no cover
702 cD = PI_4
703 cE = cK = PI_2
704 cKE = _0_0 # k2 * cD
705 eps = EPS
707 return _Complete(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps)
709 @Property_RO
710 def _reset_cGcHcPi(self):
711 '''(INTERNAL) Get the complete integrals G, H and Pi.
712 '''
713 self._iteration = 0
714 alpha2 = self.alpha2
715 if alpha2:
716 alphap2 = self.alphap2
717 if alphap2:
718 kp2 = self.kp2
719 if kp2: # <https://DLMF.NIST.gov/19.25.E2>
720 rj = _RJ(self, _0_0, kp2, _1_0, alphap2, _3_0)
721 cPi = cH = cG = self.cK
722 cG += (alpha2 - self.k2) * rj # G(alpha2, k)
723 cH -= alphap2 * rj # H(alpha2, k)
724 cPi += alpha2 * rj # Pi(alpha2, k)
725 else: # PYCHOK no cover
726 cG = cH = _RC(self, _1_0, alphap2)
727 cPi = INF # XXX or NAN?
728 else: # PYCHOK no cover
729 cG = cH = cPi = INF # XXX or NAN?
730 else:
731 cG, cPi, kp2 = self.cE, self.cK, self.kp2
732 # H = K - D but this involves large cancellations if k2 is near 1.
733 # So write (for alpha2 = 0)
734 # H = int(cos(phi)**2/sqrt(1-k2*sin(phi)**2),phi,0,pi/2)
735 # = 1/sqrt(1-k2) * int(sin(phi)**2/sqrt(1-k2/kp2*sin(phi)**2,...)
736 # = 1/kp * D(i*k/kp)
737 # and use D(k) = RD(0, kp2, 1) / 3
738 # so H = 1/kp * RD(0, 1/kp2, 1) / 3
739 # = kp2 * RD(0, 1, kp2) / 3
740 # using <https://DLMF.NIST.gov/19.20.E18>. Equivalently
741 # RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0
742 # For k2 = 1 and alpha2 = 0, we have
743 # H = int(cos(phi),...) = 1
744 cH = _RD(self, _0_0, _1_0, kp2, _3_0 / kp2) if kp2 else _1_0
746 return _Complete(cG=cG, cH=cH, cPi=cPi)
748 def sncndn(self, x):
749 '''The Jacobi elliptic function.
751 @arg x: The argument (C{float}).
753 @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with
754 C{*n(B{x}, k)}.
756 @raise EllipticError: No convergence.
757 '''
758 self._iteration = 0 # reset
759 # Bulirsch's sncndn routine, p 89.
760 if self.kp2:
761 c, d, cd, mn_ = self._sncndnBulirsch4
762 dn = _1_0
763 sn, cn = sincos2(x * cd)
764 if sn:
765 a = cn / sn
766 c *= a
767 for m, n in mn_:
768 a *= c
769 c *= dn
770 dn = (n + a) / (m + a)
771 a = c / m
772 sn = copysign0(_1_0 / hypot1(c), sn) # _signBit(sn)
773 cn = c * sn
774 if d and _signBit(self.kp2): # PYCHOK no cover
775 cn, dn = dn, cn
776 sn = sn / d # /= chokes PyChecker
777 else:
778 sn = tanh(x)
779 cn = dn = _1_0 / cosh(x)
781 return Elliptic3Tuple(sn, cn, dn, iteration=self._iteration)
783 @Property_RO
784 def _sncndnBulirsch4(self):
785 '''(INTERNAL) Get Bulirsch' 4-tuple C{(c, d, cd, mn_)}.
786 '''
787 # Bulirsch's sncndn routine, p 89.
788 d, mc = 0, self.kp2
789 if _signBit(mc): # PYCHOK no cover
790 d = _1_0 - mc
791 mc = neg(mc / d)
792 d = sqrt(d)
794 mn, a = [], _1_0
795 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC
796 # This converges quadratically, max 6 trips
797 mc = sqrt(mc)
798 mn.append((a, mc))
799 c = (a + mc) * _0_5
800 t = _TolJAC * a
801 if fabs(a - mc) <= t:
802 self._iteration += i # accumulate
803 break
804 mc *= a
805 a = c
806 else: # PYCHOK no cover
807 raise _no_convergenceError(a - mc, t, None, kp=self.kp, kp2=self.kp2)
808 cd = (c * d) if d else c
809 return c, d, cd, tuple(reversed(mn)) # mn reversed!
811 def _sncndnPhi(self, phi):
812 '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}.
813 '''
814 sn, cn = sincos2(phi)
815 return Elliptic3Tuple(sn, cn, self.fDelta(sn, cn))
817 @staticmethod
818 def fRC(x, y):
819 '''Degenerate symmetric integral of the first kind C{RC(x, y)}.
821 @return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}.
823 @see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and
824 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
825 '''
826 return _RC(None, x, y)
828 @staticmethod
829 def fRD(x, y, z):
830 '''Degenerate symmetric integral of the third kind C{RD(x, y, z)}.
832 @return: C{RD(x, y, z)}, equivalent to C{RJ(x, y, z, z)}.
834 @see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and
835 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
836 '''
837 return _RD(None, x, y, z)
839 @staticmethod
840 def fRF(x, y, *z):
841 '''Symmetric or complete symmetric integral of the first kind
842 C{RF(x, y, z)} respectively C{RF(x, y)}.
844 @return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}.
846 @see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and
847 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
848 '''
849 return _RF3(None, x, y, *z) if z and z[0] else _RF2(None, x, y)
851 @staticmethod
852 def fRG(x, y, *z):
853 '''Symmetric or complete symmetric integral of the second kind
854 C{RG(x, y, z)} respectively C{RG(x, y)}.
856 @return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}.
858 @see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>} and
859 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
860 '''
861 return _RG3(None, x, y, *z) if z and z[0] else (
862 _RG2(None, x, y) * _0_5)
864 @staticmethod
865 def fRJ(x, y, z, p):
866 '''Symmetric integral of the third kind C{RJ(x, y, z, p)}.
868 @return: C{RJ(x, y, z, p)}.
870 @see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and
871 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.
872 '''
873 return _RJ(None, x, y, z, p)
875_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset
878class EllipticError(_ValueError):
879 '''Elliptic integral, function, convergence or other L{Elliptic} issue.
880 '''
881 pass
884class Elliptic3Tuple(_NamedTuple):
885 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.
886 '''
887 _Names_ = ('sn', 'cn', 'dn')
888 _Units_ = ( Scalar, Scalar, Scalar)
891class _Lxyz(list):
892 '''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}.
893 '''
894 _a = None
895 _a0 = None
897 def __init__(self, *xyz_): # x, y, z [, p]
898 list.__init__(self, xyz_)
900 def a0(self, n):
901 '''Compute the initial C{a}.
902 '''
903 t = tuple(self)
904 m = n - len(t)
905 if m > 0:
906 t += t[-1:] * m
907 self._a0 = self._a = Fsum(*t).fover(n)
908 return self._a0
910 def asr3(self, a):
911 '''Compute next C{a}, C{sqrt(xyz_)} and C{fdot(sqrt(xyz))}.
912 '''
913 L = self
914 # assert a is L._a
915 s = map2(sqrt, L) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)]
916 r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ...
917 L[:] = [(x + r) * _0_25 for x in L]
918 # assert L is self
919 L._a = a = (a + r) * _0_25
920 return a, s, r
922 def rescale(self, am, *xy_):
923 '''Rescale C{x}, C{y}, ...
924 '''
925 for x in xy_:
926 yield (self._a0 - x) / am
928 def thresh(self, Tol):
929 '''Return the convergence threshold.
930 '''
931 return max(fabs(self._a0 - x) for x in self) / Tol
934def _horner(S, e1, E2, E3, E4, E5, *over):
935 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.
936 '''
937 E22 = E2**2
938 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>
939 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52
940 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20
941 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)
942 # converted to Horner-like form ...
943 e = e1 * 4084080
944 S *= e
945 S += Fsum( E2 * -540540, 471240).fmul(E5)
946 S += Fsum( E3 * -540540, E2 * 612612, -556920).fmul(E4)
947 S += Fsum(E22 * 675675, E3 * 306306, E2 * -706860, 680680).fmul(E3)
948 S += Fsum(E22 * -255255, E2 * 417690, -875160).fmul(E2)
949 S += 4084080
950 return S.fover((e * over[0]) if over else e)
953def _invokationError(name, *args): # PYCHOK no cover
954 '''(INTERNAL) Return an L{EllipticError}.
955 '''
956 n = _DOT_(Elliptic.__name__, name)
957 n = _SPACE_(_invokation_, n)
958 return EllipticError(NN(n, repr(args))) # unstr
961def _iterations(inst, i):
962 '''(INTERNAL) Aggregate iterations B{C{i}}.
963 '''
964 if inst:
965 inst._iteration += i
968def _no_convergenceError(d, tol, where, *args, **kwds_thresh): # PYCHOK no cover
969 '''(INTERNAL) Return an L{EllipticError}.
970 '''
971 n = Elliptic.__name__
972 if where:
973 n = _DOT_(n, where.__name__)
974 if kwds_thresh:
975 q = _xkwds_pop(kwds_thresh, thresh=False)
976 t = unstr(n, *args, **kwds_thresh)
977 else:
978 q = False
979 t = unstr(n, *args)
980 return EllipticError(Fmt.no_convergence(d, tol, thresh=q), txt=t)
983def _3over(a, b):
984 '''(INTERNAL) Return C{3 / (a * b)}.
985 '''
986 return _3_0 / (a * b)
989def _RC(unused, x, y):
990 '''(INTERNAL) Defined only for y != 0 and x >= 0.
991 '''
992 d = x - y
993 if d < 0: # catch _NaN
994 # <https://DLMF.NIST.gov/19.2.E18>
995 d = -d
996 r = atan(sqrt(d / x)) if x > 0 else PI_2
997 elif d == _0_0: # XXX d < EPS0? or EPS02 or _EPSmin
998 d, r = y, _1_0
999 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>
1000 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))
1001 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>
1002 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))
1003 else:
1004 raise _invokationError(Elliptic.fRC.__name__, x, y)
1005 return r / sqrt(d)
1008def _RD(inst, x, y, z, *over):
1009 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.
1010 '''
1011 L = _Lxyz(x, y, z)
1012 a = L.a0(5)
1013 q = L.thresh(_TolRF)
1014 S = _Deferred()
1015 am, m = a, 1
1016 for i in range(_TRIPS):
1017 if fabs(am) > q: # max 7 trips
1018 _iterations(inst, i)
1019 break
1020 t = L[2] # z0...n
1021 a, s, r = L.asr3(a)
1022 S += _1_0 / ((t + r) * s[2] * m)
1023 m *= 4
1024 am = a * m
1025 else: # PYCHOK no cover
1026 raise _no_convergenceError(am, q, Elliptic.fRD, x, y, z, *over,
1027 thresh=True)
1028 x, y = L.rescale(-am, x, y)
1029 xy = x * y
1030 z = (x + y) / _3_0
1031 z2 = z**2
1032 S = S.Fsum.fmul(_3_0)
1033 return _horner(S, am * sqrt(a),
1034 xy - _6_0 * z2,
1035 (xy * _3_0 - _8_0 * z2) * z,
1036 (xy - z2) * _3_0 * z2,
1037 xy * z2 * z, *over)
1040def _RF2(inst, x, y): # 2-arg version, z=0
1041 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.
1042 '''
1043 a, b = sqrt(x), sqrt(y)
1044 if a < b:
1045 a, b = b, a
1046 for i in range(_TRIPS):
1047 t = _TolRG0 * a
1048 if fabs(a - b) <= t: # max 4 trips
1049 _iterations(inst, i)
1050 return (PI / (a + b))
1051 a, b = ((a + b) * _0_5), sqrt(a * b)
1052 else: # PYCHOK no cover
1053 raise _no_convergenceError(a - b, t, Elliptic.fRF, x, y)
1056def _RF3(inst, x, y, z): # 3-arg version
1057 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.
1058 '''
1059 L = _Lxyz(x, y, z)
1060 a = L.a0(3)
1061 q = L.thresh(_TolRF)
1062 am, m = a, 1
1063 for i in range(_TRIPS):
1064 if fabs(am) > q: # max 6 trips
1065 _iterations(inst, i)
1066 break
1067 a, _, _ = L.asr3(a)
1068 m *= 4
1069 am = a * m
1070 else: # PYCHOK no cover
1071 raise _no_convergenceError(am, q, Elliptic.fRF, x, y, z,
1072 thresh=True)
1073 x, y = L.rescale(am, x, y)
1074 z = neg(x + y)
1075 xy = x * y
1076 e2 = xy - z**2
1077 e3 = xy * z
1078 e4 = e2**2
1079 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>
1080 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44
1081 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)
1082 # converted to Horner-like form ...
1083 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)
1084 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)
1085 S += Fsum(240240)
1086 return S.fover(sqrt(a) * 240240)
1089def _RG2(inst, x, y): # 2-args and I{doubled}
1090 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.
1091 '''
1092 a, b = sqrt(x), sqrt(y)
1093 if a < b:
1094 a, b = b, a
1095 ab = a - b # fabs(a - b)
1096 S = _Deferred(_0_5 * (a + b)**2)
1097 m = -1
1098 for i in range(_TRIPS): # max 4 trips
1099 t = _TolRG0 * a
1100 if ab <= t:
1101 _iterations(inst, i)
1102 return S.Fsum.fover((a + b) / PI_2)
1103 a, b = ((a + b) * _0_5), sqrt(a * b)
1104 ab = fabs(a - b)
1105 S += ab**2 * m
1106 m *= 2
1107 else: # PYCHOK no cover
1108 raise _no_convergenceError(ab, t, Elliptic.fRG, x, y)
1111def _RG3(inst, x, y, z): # 3-arg version
1112 '''(INTERNAL) Never called with zero B{C{z}}, see C{.fRG}.
1113 '''
1114# if not z:
1115# y, z = z, y
1116 rd = (x - z) * (z - y) # - (y - z)
1117 if rd: # Carlson, eq 1.7
1118 rd = _RD(inst, x, y, z, _3_0 * z / rd)
1119 xyz = x * y
1120 if xyz:
1121 xyz = sqrt(xyz / z**3)
1122 return Fsum(_RF3(inst, x, y, z), rd, xyz).fover(_2_0 / z)
1125def _RJ(inst, x, y, z, p, *over):
1126 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.
1127 '''
1128 def _xyzp(x, y, z, p):
1129 return (x + p) * (y + p) * (z + p)
1131 L = _Lxyz(x, y, z, p)
1132 a = L.a0(5)
1133 q = L.thresh(_TolRD)
1134 S = _Deferred()
1135 n = neg(_xyzp(x, y, z, -p))
1136 am, m = a, 1
1137 for i in range(_TRIPS):
1138 if fabs(am) > q: # max 7 trips
1139 _iterations(inst, i)
1140 break
1141 a, s, _ = L.asr3(a)
1142 d = _xyzp(*s)
1143 if n:
1144 rc = _RC(inst, _1_0, n / d**2 + _1_0)
1145 n *= _1_64th # /= chokes PyChecker
1146 else:
1147 rc = _1_0 # == _RC(None, _1_0, _1_0)
1148 S += rc / (d * m)
1149 m *= 4
1150 am = a * m
1151 else: # PYCHOK no cover
1152 raise _no_convergenceError(am, q, Elliptic.fRJ, x, y, z, p,
1153 thresh=True)
1154 x, y, z = L.rescale(am, x, y, z)
1155 xyz = x * y * z
1156 p = Fsum(x, y, z).fover(_N_2_0)
1157 p2 = p**2
1158 p3 = p2*p
1159 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)
1160 E2p = E2 * p
1161 S = S.Fsum.fmul(_6_0)
1162 return _horner(S, am * sqrt(a), E2,
1163 Fsum(p3 * _4_0, xyz, E2p * _2_0),
1164 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),
1165 xyz * p2, *over)
1167# **) MIT License
1168#
1169# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.
1170#
1171# Permission is hereby granted, free of charge, to any person obtaining a
1172# copy of this software and associated documentation files (the "Software"),
1173# to deal in the Software without restriction, including without limitation
1174# the rights to use, copy, modify, merge, publish, distribute, sublicense,
1175# and/or sell copies of the Software, and to permit persons to whom the
1176# Software is furnished to do so, subject to the following conditions:
1177#
1178# The above copyright notice and this permission notice shall be included
1179# in all copies or substantial portions of the Software.
1180#
1181# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
1182# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
1183# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
1184# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
1185# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
1186# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
1187# OTHER DEALINGS IN THE SOFTWARE.