1

2# -*- coding: utf-8 -*-

3

4 u'''I{Karney}'s elliptic functions and integrals.

5

6 Class L{Elliptic} transcoded from I{Charles Karney}'s C++ class U{EllipticFunction

7<https://GeographicLib.SourceForge.io/C++/doc/classGeographicLib_1_1EllipticFunction.html>}

8 to pure Python, including symmetric integrals L{Elliptic.fRC}, L{Elliptic.fRD},

9 L{Elliptic.fRF}, L{Elliptic.fRG} and L{Elliptic.fRJ} as C{static methods}.

10

11 Python method names follow the C++ member functions, I{except}:

12

13 - member functions I{without arguments} are mapped to Python properties

14 prefixed with C{"c"}, for example C{E()} is property C{cE},

15

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)},

19

20 - other Python method names conventionally start with a lower-case

21 letter or an underscore if private.

22

23 Following is a copy of I{Karney}'s U{EllipticFunction.hpp

24<https://GeographicLib.SourceForge.io/C++/doc/EllipticFunction_8hpp_source.html>}

25 file C{Header}.

26

27 Copyright (C) U{Charles Karney<mailto:Karney@Alum.MIT.edu>} (2008-2023)

28 and licensed under the MIT/X11 License. For more information, see the

29 U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation.

30

31 B{Elliptic integrals and functions.}

32

33 This provides the elliptic functions and integrals needed for

34 C{Ellipsoid}, C{GeodesicExact}, and C{TransverseMercatorExact}. Two

35 categories of function are provided:

36

37 - functions to compute U{symmetric elliptic integrals

38 <https://DLMF.NIST.gov/19.16.i>}

39

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>}.

43

44 In the latter case, an object is constructed giving the modulus

45 C{k} (and optionally the parameter C{alpha}). The modulus (and

46 parameter) are always passed as squares which allows C{k} to be

47 pure imaginary. (Confusingly, Abramowitz and Stegun call C{m = k**2}

48 the "parameter" and C{n = alpha**2} the "characteristic".)

49

50 In geodesic applications, it is convenient to separate the incomplete

51 integrals into secular and periodic components, e.g.

52

53 I{C{E(phi, k) = (2 E(k) / pi) [ phi + delta E(phi, k) ]}}

54

55 where I{C{delta E(phi, k)}} is an odd periodic function with

56 period I{C{pi}}.

57

58 The computation of the elliptic integrals uses the algorithms given

59 in 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,

62 13--26 (1995) with the additional optimizations given U{here

63<https://DLMF.NIST.gov/19.36.i>}.

64

65 The computation of the Jacobi elliptic functions uses the algorithm

66 given in U{R. Bulirsch, Numerical Calculation of Elliptic Integrals

67 and Elliptic Functions<https://DOI.org/10.1007/BF01397975>},

68 Numerische Mathematik 7, 78--90 (1965).

69

70 The notation follows U{NIST Digital Library of Mathematical Functions

71<https://DLMF.NIST.gov>} chapters U{19<https://DLMF.NIST.gov/19>} and

72 U{22<https://DLMF.NIST.gov/22>}.

73'''

74# make sure int/int division yields float quotient, see .basics

75 from __future__ import division as _; del _ # PYCHOK semicolon

76

77 from pygeodesy.basics import copysign0, map2, neg, neg_

78 from 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

84 from pygeodesy.fmath import fdot, hypot1, zqrt, _ValueError

85 from pygeodesy.fsums import Fsum, _sum

86# from pygeodesy.internals import _dunder_nameof # from .lazily

87 from pygeodesy.interns import NN, _delta_, _DOT_, _f_, _invalid_, \

88 _invokation_, _negative_, _SPACE_

89 from pygeodesy.karney import _K_2_0, _norm180, _signBit, _sincos2

90 from pygeodesy.lazily import _ALL_LAZY, _dunder_nameof

91 from pygeodesy.named import _Named, _NamedTuple, Fmt, unstr

92 from pygeodesy.props import _allPropertiesOf_n, Property_RO, _update_all

93# from pygeodesy.streprs import Fmt, unstr # from .named

94 from pygeodesy.units import Scalar, Scalar_

95# from pygeodesy.utily import sincos2 as _sincos2 # from .karney

96

97 from math import asinh, atan, atan2, ceil, cosh, fabs, floor, \

98 radians, sin, sqrt, tanh

99

100__all__ = _ALL_LAZY.elliptic

101__version__ = '24.05.18'

102

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

107

108

109 class _Cs(object):

110 '''(INTERAL) Complete integrals cache.

111 '''

112 def __init__(self, **kwds):

113 self.__dict__ = kwds

114

115

116 class _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

124

125 def __iadd__(self, x):

126 list.append(self, x)

127 return self

128

129

130 class Elliptic(_Named):

131 '''Elliptic integrals and functions.

132

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

141

142 def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None, **name):

143 '''Constructor, specifying the C{modulus} and C{parameter}.

144

145 @kwarg name: Optional C{B{name}=NN} (C{str}).

146

147 @see: Method L{Elliptic.reset} for further details.

148

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)

155 if name:

156 self.name = name

157

158 @Property_RO

159 def alpha2(self):

160 '''Get α^2, the square of the parameter (C{float}).

161 '''

162 return self._alpha2

163

164 @Property_RO

165 def alphap2(self):

166 '''Get α'^2, the square of the complementary parameter (C{float}).

167 '''

168 return self._alphap2

169

170 @Property_RO

171 def cD(self):

172 '''Get Jahnke's complete integral C{D(k)} (C{float}),

173 U{defined<https://DLMF.NIST.gov/19.2.E6>}.

174 '''

175 return self._cDEKEeps.cD

176

177 @Property_RO

178 def _cDEKEeps(self):

179 '''(INTERNAL) Get the complete integrals D, E, K and KE plus C{eps}.

180 '''

181 k2, kp2 = self.k2, self.kp2

182 if k2:

183 if kp2:

184 try:

185 self._iteration = 0

186 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3

187 # <https://DLMF.NIST.gov/19.25.E1>

188 D = _RD(self, _0_0, kp2, _1_0, _3_0)

189 cD = float(D)

190 # Complete elliptic integral E(k), Carlson eq. 4.2

191 # <https://DLMF.NIST.gov/19.25.E1>

192 cE = _rG2(self, kp2, _1_0, PI_=PI_2)

193 # Complete elliptic integral K(k), Carlson eq. 4.1

194 # <https://DLMF.NIST.gov/19.25.E1>

195 cK = _rF2(self, kp2, _1_0)

196 cKE = float(D.fmul(k2))

197 eps = k2 / (sqrt(kp2) + _1_0)**2

198

199 except Exception as e:

200 raise _ellipticError(self.reset, k2=k2, kp2=kp2, cause=e)

201 else:

202 cD = cK = cKE = INF

203 cE = _1_0

204 eps = k2

205 else:

206 cD = PI_4

207 cE = cK = PI_2

208 cKE = _0_0 # k2 * cD

209 eps = EPS

210

211 return _Cs(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps)

212

213 @Property_RO

214 def cE(self):

215 '''Get the complete integral of the second kind C{E(k)}

216 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}.

217 '''

218 return self._cDEKEeps.cE

219

220 @Property_RO

221 def cG(self):

222 '''Get Legendre's complete geodesic longitude integral

223 C{G(α^2, k)} (C{float}).

224 '''

225 return self._cGHPi.cG

226

227 @Property_RO

228 def _cGHPi(self):

229 '''(INTERNAL) Get the complete integrals G, H and Pi.

230 '''

231 alpha2, alphap2, kp2 = self.alpha2, self.alphap2, self.kp2

232 try:

233 self._iteration = 0

234 if alpha2:

235 if alphap2:

236 if kp2: # <https://DLMF.NIST.gov/19.25.E2>

237 cK = self.cK

238 Rj = _RJ(self, _0_0, kp2, _1_0, alphap2, _3_0)

239 cG = float(Rj * (alpha2 - self.k2) + cK) # G(alpha2, k)

240 cH = -float(Rj * alphap2 - cK) # H(alpha2, k)

241 cPi = float(Rj * alpha2 + cK) # Pi(alpha2, k)

242 else:

243 cG = cH = _rC(self, _1_0, alphap2)

244 cPi = INF # XXX or NAN?

245 else:

246 cG = cH = cPi = INF # XXX or NAN?

247 else:

248 cG, cPi = self.cE, self.cK

249 # H = K - D but this involves large cancellations if k2 is near 1.

250 # So write (for alpha2 = 0)

251 # H = int(cos(phi)**2 / sqrt(1-k2 * sin(phi)**2), phi, 0, pi/2)

252 # = 1 / sqrt(1-k2) * int(sin(phi)**2 / sqrt(1-k2/kp2 * sin(phi)**2,...)

253 # = 1 / kp * D(i * k/kp)

254 # and use D(k) = RD(0, kp2, 1) / 3, so

255 # H = 1/kp * RD(0, 1/kp2, 1) / 3

256 # = kp2 * RD(0, 1, kp2) / 3

257 # using <https://DLMF.NIST.gov/19.20.E18>. Equivalently

258 # RF(x, 1) - RD(0, x, 1) / 3 = x * RD(0, 1, x) / 3 for x > 0

259 # For k2 = 1 and alpha2 = 0, we have

260 # H = int(cos(phi),...) = 1

261 cH = float(_RD(self, _0_0, _1_0, kp2, _3_0 / kp2)) if kp2 else _1_0

262

263 except Exception as e:

264 raise _ellipticError(self.reset, kp2=kp2, alpha2 =alpha2,

265 alphap2=alphap2, cause=e)

266 return _Cs(cG=cG, cH=cH, cPi=cPi)

267

268 @Property_RO

269 def cH(self):

270 '''Get Cayley's complete geodesic longitude difference integral

271 C{H(α^2, k)} (C{float}).

272 '''

273 return self._cGHPi.cH

274

275 @Property_RO

276 def cK(self):

277 '''Get the complete integral of the first kind C{K(k)}

278 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}.

279 '''

280 return self._cDEKEeps.cK

281

282 @Property_RO

283 def cKE(self):

284 '''Get the difference between the complete integrals of the

285 first and second kinds, C{K(k) − E(k)} (C{float}).

286 '''

287 return self._cDEKEeps.cKE

288

289 @Property_RO

290 def cPi(self):

291 '''Get the complete integral of the third kind C{Pi(α^2, k)}

292 (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}.

293 '''

294 return self._cGHPi.cPi

295

296 def deltaD(self, sn, cn, dn):

297 '''Jahnke's periodic incomplete elliptic integral.

298

299 @arg sn: sin(φ).

300 @arg cn: cos(φ).

301 @arg dn: sqrt(1 − k2 * sin(2φ)).

302

303 @return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}).

304

305 @raise EllipticError: Invalid invokation or no convergence.

306 '''

307 return _deltaX(sn, cn, dn, self.cD, self.fD)

308

309 def deltaE(self, sn, cn, dn):

310 '''The periodic incomplete integral of the second kind.

311

312 @arg sn: sin(φ).

313 @arg cn: cos(φ).

314 @arg dn: sqrt(1 − k2 * sin(2φ)).

315

316 @return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}).

317

318 @raise EllipticError: Invalid invokation or no convergence.

319 '''

320 return _deltaX(sn, cn, dn, self.cE, self.fE)

321

322 def deltaEinv(self, stau, ctau):

323 '''The periodic inverse of the incomplete integral of the second kind.

324

325 @arg stau: sin(τ)

326 @arg ctau: cos(τ)

327

328 @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}).

329

330 @raise EllipticError: No convergence.

331 '''

332 try:

333 if _signBit(ctau): # pi periodic

334 stau, ctau = neg_(stau, ctau)

335 t = atan2(stau, ctau)

336 return self._Einv(t * self.cE / PI_2) - t

337

338 except Exception as e:

339 raise _ellipticError(self.deltaEinv, stau, ctau, cause=e)

340

341 def deltaF(self, sn, cn, dn):

342 '''The periodic incomplete integral of the first kind.

343

344 @arg sn: sin(φ).

345 @arg cn: cos(φ).

346 @arg dn: sqrt(1 − k2 * sin(2φ)).

347

348 @return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}).

349

350 @raise EllipticError: Invalid invokation or no convergence.

351 '''

352 return _deltaX(sn, cn, dn, self.cK, self.fF)

353

354 def deltaG(self, sn, cn, dn):

355 '''Legendre's periodic geodesic longitude integral.

356

357 @arg sn: sin(φ).

358 @arg cn: cos(φ).

359 @arg dn: sqrt(1 − k2 * sin(2φ)).

360

361 @return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}).

362

363 @raise EllipticError: Invalid invokation or no convergence.

364 '''

365 return _deltaX(sn, cn, dn, self.cG, self.fG)

366

367 def deltaH(self, sn, cn, dn):

368 '''Cayley's periodic geodesic longitude difference integral.

369

370 @arg sn: sin(φ).

371 @arg cn: cos(φ).

372 @arg dn: sqrt(1 − k2 * sin(2φ)).

373

374 @return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}).

375

376 @raise EllipticError: Invalid invokation or no convergence.

377 '''

378 return _deltaX(sn, cn, dn, self.cH, self.fH)

379

380 def deltaPi(self, sn, cn, dn):

381 '''The periodic incomplete integral of the third kind.

382

383 @arg sn: sin(φ).

384 @arg cn: cos(φ).

385 @arg dn: sqrt(1 − k2 * sin(2φ)).

386

387 @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ

388 (C{float}).

389

390 @raise EllipticError: Invalid invokation or no convergence.

391 '''

392 return _deltaX(sn, cn, dn, self.cPi, self.fPi)

393

394 def _Einv(self, x):

395 '''(INTERNAL) Helper for C{.deltaEinv} and C{.fEinv}.

396 '''

397 E2 = self.cE * _2_0

398 n = floor(x / E2 + _0_5)

399 r = x - E2 * n # r in [-cE, cE)

400 # linear approximation

401 phi = PI * r / E2 # phi in [-PI_2, PI_2)

402 Phi = Fsum(phi)

403 # first order correction

404 phi = Phi.fsum_(self.eps * sin(phi * _2_0) / _N_2_0)

405 self._iteration = 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 = Phi.fsum2f_ # aggregate

410 _abs = fabs

411 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC

412 sn, cn, dn = self._sncndn3(phi)

413 if dn:

414 sn = self.fE(sn, cn, dn)

415 phi, d = _Phi2((r - sn) / dn)

416 else: # PYCHOK no cover

417 d = _0_0 # XXX continue?

418 if _abs(d) < _TolJAC: # 3-4 trips

419 _iterations(self, i)

420 break

421 else: # PYCHOK no cover

422 raise _convergenceError(d, _TolJAC)

423 return Phi.fsum_(n * PI) if n else phi

424

425 @Property_RO

426 def eps(self):

427 '''Get epsilon (C{float}).

428 '''

429 return self._cDEKEeps.eps

430

431 def fD(self, phi_or_sn, cn=None, dn=None):

432 '''Jahnke's incomplete elliptic integral in terms of

433 Jacobi elliptic functions.

434

435 @arg phi_or_sn: φ or sin(φ).

436 @kwarg cn: C{None} or cos(φ).

437 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

438

439 @return: D(φ, k) as though φ ∈ (−π, π] (C{float}),

440 U{defined<https://DLMF.NIST.gov/19.2.E4>}.

441

442 @raise EllipticError: Invalid invokation or no convergence.

443 '''

444 def _fD(sn, cn, dn):

445 r = fabs(sn)**3

446 if r:

447 r = float(_RD(self, cn**2, dn**2, _1_0, _3_0 / r))

448 return r

449

450 return self._fXf(phi_or_sn, cn, dn, self.cD,

451 self.deltaD, _fD)

452

453 def fDelta(self, sn, cn):

454 '''The C{Delta} amplitude function.

455

456 @arg sn: sin(φ).

457 @arg cn: cos(φ).

458

459 @return: sqrt(1 − k2 * sin(2φ)) (C{float}).

460 '''

461 try:

462 k2 = self.k2

463 s = (self.kp2 + cn**2 * k2) if k2 > 0 else (

464 (_1_0 - sn**2 * k2) if k2 < 0 else self.kp2)

465 return sqrt(s) if s else _0_0

466

467 except Exception as e:

468 raise _ellipticError(self.fDelta, sn, cn, k2=k2, cause=e)

469

470 def fE(self, phi_or_sn, cn=None, dn=None):

471 '''The incomplete integral of the second kind in terms of

472 Jacobi elliptic functions.

473

474 @arg phi_or_sn: φ or sin(φ).

475 @kwarg cn: C{None} or cos(φ).

476 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

477

478 @return: E(φ, k) as though φ ∈ (−π, π] (C{float}),

479 U{defined<https://DLMF.NIST.gov/19.2.E5>}.

480

481 @raise EllipticError: Invalid invokation or no convergence.

482 '''

483 def _fE(sn, cn, dn):

484 '''(INTERNAL) Core of C{.fE}.

485 '''

486 if sn:

487 sn2, cn2, dn2 = sn**2, cn**2, dn**2

488 kp2, k2 = self.kp2, self.k2

489 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9>

490 Ei = _RF3(self, cn2, dn2, _1_0)

491 if k2:

492 Ei -= _RD(self, cn2, dn2, _1_0, _3over(k2, sn2))

493 elif kp2 >= 0: # k2 > 0, <https://DLMF.NIST.gov/19.25.E10>

494 Ei = _over(k2 * fabs(cn), dn) # float

495 if kp2:

496 Ei += (_RD( self, cn2, _1_0, dn2, _3over(k2, sn2)) +

497 _RF3(self, cn2, dn2, _1_0)) * kp2

498 else: # kp2 < 0, <https://DLMF.NIST.gov/19.25.E11>

499 Ei = _over(dn, fabs(cn))

500 Ei -= _RD(self, dn2, _1_0, cn2, _3over(kp2, sn2))

501 Ei *= fabs(sn)

502 ei = float(Ei)

503 else: # PYCHOK no cover

504 ei = _0_0

505 return ei

506

507 return self._fXf(phi_or_sn, cn, dn, self.cE,

508 self.deltaE, _fE)

509

510 def fEd(self, deg):

511 '''The incomplete integral of the second kind with

512 the argument given in C{degrees}.

513

514 @arg deg: Angle (C{degrees}).

515

516 @return: E(π B{C{deg}} / 180, k) (C{float}).

517

518 @raise EllipticError: No convergence.

519 '''

520 if _K_2_0:

521 e = round((deg - _norm180(deg)) / _360_0)

522 elif fabs(deg) < _180_0:

523 e = _0_0

524 else:

525 e = ceil(deg / _360_0 - _0_5)

526 deg -= e * _360_0

527 return self.fE(radians(deg)) + e * self.cE * _4_0

528

529 def fEinv(self, x):

530 '''The inverse of the incomplete integral of the second kind.

531

532 @arg x: Argument (C{float}).

533

534 @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}}

535 (C{float}).

536

537 @raise EllipticError: No convergence.

538 '''

539 try:

540 return self._Einv(x)

541 except Exception as e:

542 raise _ellipticError(self.fEinv, x, cause=e)

543

544 def fF(self, phi_or_sn, cn=None, dn=None):

545 '''The incomplete integral of the first kind in terms of

546 Jacobi elliptic functions.

547

548 @arg phi_or_sn: φ or sin(φ).

549 @kwarg cn: C{None} or cos(φ).

550 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

551

552 @return: F(φ, k) as though φ ∈ (−π, π] (C{float}),

553 U{defined<https://DLMF.NIST.gov/19.2.E4>}.

554

555 @raise EllipticError: Invalid invokation or no convergence.

556 '''

557 def _fF(sn, cn, dn):

558 r = fabs(sn)

559 if r:

560 r = float(_RF3(self, cn**2, dn**2, _1_0).fmul(r))

561 return r

562

563 return self._fXf(phi_or_sn, cn, dn, self.cK,

564 self.deltaF, _fF)

565

566 def fG(self, phi_or_sn, cn=None, dn=None):

567 '''Legendre's geodesic longitude integral in terms of

568 Jacobi elliptic functions.

569

570 @arg phi_or_sn: φ or sin(φ).

571 @kwarg cn: C{None} or cos(φ).

572 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

573

574 @return: G(φ, k) as though φ ∈ (−π, π] (C{float}).

575

576 @raise EllipticError: Invalid invokation or no convergence.

577

578 @note: Legendre expresses the longitude of a point on the

579 geodesic in terms of this combination of elliptic

580 integrals in U{Exercices de Calcul Intégral, Vol 1

581 (1811), p 181<https://Books.Google.com/books?id=

582 riIOAAAAQAAJ&pg=PA181>}.

583

584 @see: U{Geodesics in terms of elliptic integrals<https://

585 GeographicLib.SourceForge.io/C++/doc/geodesic.html#geodellip>}

586 for the expression for the longitude in terms of this function.

587 '''

588 return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2,

589 self.cG, self.deltaG)

590

591 def fH(self, phi_or_sn, cn=None, dn=None):

592 '''Cayley's geodesic longitude difference integral in terms of

593 Jacobi elliptic functions.

594

595 @arg phi_or_sn: φ or sin(φ).

596 @kwarg cn: C{None} or cos(φ).

597 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

598

599 @return: H(φ, k) as though φ ∈ (−π, π] (C{float}).

600

601 @raise EllipticError: Invalid invokation or no convergence.

602

603 @note: Cayley expresses the longitude difference of a point

604 on the geodesic in terms of this combination of

605 elliptic integrals in U{Phil. Mag. B{40} (1870), p 333

606 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}.

607

608 @see: U{Geodesics in terms of elliptic integrals<https://

609 GeographicLib.SourceForge.io/C++/doc/geodesic.html#geodellip>}

610 for the expression for the longitude in terms of this function.

611 '''

612 return self._fXa(phi_or_sn, cn, dn, -self.alphap2,

613 self.cH, self.deltaH)

614

615 def fPi(self, phi_or_sn, cn=None, dn=None):

616 '''The incomplete integral of the third kind in terms of

617 Jacobi elliptic functions.

618

619 @arg phi_or_sn: φ or sin(φ).

620 @kwarg cn: C{None} or cos(φ).

621 @kwarg dn: C{None} or sqrt(1 − k2 * sin(2φ)).

622

623 @return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}).

624

625 @raise EllipticError: Invalid invokation or no convergence.

626 '''

627 if dn is None and cn is not None: # and isscalar(phi_or_sn)

628 dn = self.fDelta(phi_or_sn, cn) # in .triaxial

629 return self._fXa(phi_or_sn, cn, dn, self.alpha2,

630 self.cPi, self.deltaPi)

631

632 def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX):

633 '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}.

634 '''

635 def _fX(sn, cn, dn):

636 if sn:

637 cn2, dn2 = cn**2, dn**2

638 R = _RF3(self, cn2, dn2, _1_0)

639 if aX:

640 sn2 = sn**2

641 p = sn2 * self.alphap2 + cn2

642 R += _RJ(self, cn2, dn2, _1_0, p, _3over(aX, sn2))

643 R *= fabs(sn)

644 r = float(R)

645 else: # PYCHOK no cover

646 r = _0_0

647 return r

648

649 return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX)

650

651 def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX):

652 '''(INTERNAL) Helper for C{.fD}, C{.fE}, C{.fF} and C{._fXa}.

653 '''

654 self._iteration = 0 # aggregate

655 phi = sn = phi_or_sn

656 if cn is dn is None: # fX(phi) call

657 sn, cn, dn = self._sncndn3(phi)

658 if fabs(phi) >= PI:

659 return (deltaX(sn, cn, dn) + phi) * cX / PI_2

660 # fall through

661 elif cn is None or dn is None:

662 n = NN(_f_, deltaX.__name__[5:])

663 raise _ellipticError(n, sn, cn, dn)

664

665 if _signBit(cn): # enforce usual trig-like symmetries

666 xi = cX * _2_0 - fX(sn, cn, dn)

667 else:

668 xi = fX(sn, cn, dn) if cn > 0 else cX

669 return copysign0(xi, sn)

670

671 @Property_RO

672 def k2(self):

673 '''Get k^2, the square of the modulus (C{float}).

674 '''

675 return self._k2

676

677 @Property_RO

678 def kp2(self):

679 '''Get k'^2, the square of the complementary modulus (C{float}).

680 '''

681 return self._kp2

682

683 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13

684 '''Reset the modulus, parameter and the complementaries.

685

686 @kwarg k2: Modulus squared (C{float}, NINF <= k^2 <= 1).

687 @kwarg alpha2: Parameter squared (C{float}, NINF <= α^2 <= 1).

688 @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0).

689 @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0).

690

691 @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}}

692 or B{C{alphap2}}.

693

694 @note: The arguments must satisfy C{B{k2} + B{kp2} = 1} and

695 C{B{alpha2} + B{alphap2} = 1}. No checking is done

696 that these conditions are met to enable accuracy to be

697 maintained, e.g., when C{k} is very close to unity.

698 '''

699 if self.__dict__:

700 _update_all(self, _Named.iteration._uname, Base=Property_RO)

701

702 self._k2 = Scalar_(k2=k2, Error=EllipticError, low=None, high=_1_0)

703 self._kp2 = Scalar_(kp2=((_1_0 - k2) if kp2 is None else kp2), Error=EllipticError)

704

705 self._alpha2 = Scalar_(alpha2=alpha2, Error=EllipticError, low=None, high=_1_0)

706 self._alphap2 = Scalar_(alphap2=((_1_0 - alpha2) if alphap2 is None else alphap2),

707 Error=EllipticError)

708

709 # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1

710 # K E D

711 # k = 0: pi/2 pi/2 pi/4

712 # k = 1: inf 1 inf

713 # Pi G H

714 # k = 0, alpha = 0: pi/2 pi/2 pi/4

715 # k = 1, alpha = 0: inf 1 1

716 # k = 0, alpha = 1: inf inf pi/2

717 # k = 1, alpha = 1: inf inf inf

718 #

719 # G(0, k) = Pi(0, k) = H(1, k) = E(k)

720 # H(0, k) = K(k) - D(k)

721 # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2))

722 # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1))

723 # Pi(alpha2, 1) = inf

724 # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2)

725

726 def sncndn(self, x):

727 '''The Jacobi elliptic function.

728

729 @arg x: The argument (C{float}).

730

731 @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with

732 C{*n(B{x}, k)}.

733

734 @raise EllipticError: No convergence.

735 '''

736 self._iteration = 0 # reset

737 try: # Bulirsch's sncndn routine, p 89.

738 if self.kp2:

739 c, d, cd, mn = self._sncndn4

740 dn = _1_0

741 sn, cn = _sincos2(x * cd)

742 if sn:

743 a = cn / sn

744 c *= a

745 for m, n in reversed(mn):

746 a *= c

747 c *= dn

748 dn = (n + a) / (m + a)

749 a = c / m

750 a = _1_0 / hypot1(c)

751 sn = neg(a) if _signBit(sn) else a

752 cn = c * sn

753 if d and _signBit(self.kp2):

754 cn, dn = dn, cn

755 sn = sn / d # /= chokes PyChecker

756 else:

757 sn = tanh(x)

758 cn = dn = _1_0 / cosh(x)

759

760 except Exception as e:

761 raise _ellipticError(self.sncndn, x, kp2=self.kp2, cause=e)

762

763 return Elliptic3Tuple(sn, cn, dn, iteration=self._iteration)

764

765 def _sncndn3(self, phi):

766 '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}.

767 '''

768 sn, cn = _sincos2(phi)

769 return sn, cn, self.fDelta(sn, cn)

770

771 @Property_RO

772 def _sncndn4(self):

773 '''(INTERNAL) Get Bulirsch' 4-tuple C{(c, d, cd, mn)}.

774 '''

775 # Bulirsch's sncndn routine, p 89.

776 d, mc = 0, self.kp2

777 if _signBit(mc):

778 d = _1_0 - mc

779 mc = neg(mc / d)

780 d = sqrt(d)

781

782 mn, a = [], _1_0

783 for i in range(1, _TRIPS): # GEOGRAPHICLIB_PANIC

784 mc = sqrt(mc)

785 mn.append((a, mc))

786 c = (a + mc) * _0_5

787 r = fabs(mc - a)

788 t = _TolJAC * a

789 if r <= t: # 6 trips, quadratic

790 _iterations(self, i)

791 break

792 mc *= a

793 a = c

794 else: # PYCHOK no cover

795 raise _convergenceError(r, t)

796 cd = (c * d) if d else c

797 return c, d, cd, mn

798

799 @staticmethod

800 def fRC(x, y):

801 '''Degenerate symmetric integral of the first kind C{RC(x, y)}.

802

803 @return: C{RC(x, y)}, equivalent to C{RF(x, y, y)}.

804

805 @see: U{C{RC} definition<https://DLMF.NIST.gov/19.2.E17>} and

806 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.

807 '''

808 return _rC(None, x, y)

809

810 @staticmethod

811 def fRD(x, y, z, *over):

812 '''Degenerate symmetric integral of the third kind C{RD(x, y, z)}.

813

814 @return: C{RD(x, y, z) / over}, equivalent to C{RJ(x, y, z, z)

815 / over} with C{over} typically 3.

816

817 @see: U{C{RD} definition<https://DLMF.NIST.gov/19.16.E5>} and

818 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.

819 '''

820 try:

821 return float(_RD(None, x, y, z, *over))

822 except Exception as e:

823 raise _ellipticError(Elliptic.fRD, x, y, z, *over, cause=e)

824

825 @staticmethod

826 def fRF(x, y, z=0):

827 '''Symmetric or complete symmetric integral of the first kind

828 C{RF(x, y, z)} respectively C{RF(x, y)}.

829

830 @return: C{RF(x, y, z)} or C{RF(x, y)} for missing or zero B{C{z}}.

831

832 @see: U{C{RF} definition<https://DLMF.NIST.gov/19.16.E1>} and

833 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.

834 '''

835 try:

836 return float(_RF3(None, x, y, z)) if z else _rF2(None, x, y)

837 except Exception as e:

838 raise _ellipticError(Elliptic.fRF, x, y, z, cause=e)

839

840 @staticmethod

841 def fRG(x, y, z=0):

842 '''Symmetric or complete symmetric integral of the second kind

843 C{RG(x, y, z)} respectively C{RG(x, y)}.

844

845 @return: C{RG(x, y, z)} or C{RG(x, y)} for missing or zero B{C{z}}.

846

847 @see: U{C{RG} definition<https://DLMF.NIST.gov/19.16.E3>},

848 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>} and

849 U{RG<https://GeographicLib.SourceForge.io/C++/doc/

850 EllipticFunction_8cpp_source.html#l00096>} version 2.3.

851 '''

852 try:

853 return _rG2(None, x, y) if z == 0 else (

854 _rG2(None, z, x) if y == 0 else (

855 _rG2(None, y, z) if x == 0 else _rG3(None, x, y, z)))

856 except Exception as e:

857 t = _negative_ if min(x, y, z) < 0 else NN

858 raise _ellipticError(Elliptic.fRG, x, y, z, cause=e, txt=t)

859

860 @staticmethod

861 def fRJ(x, y, z, p):

862 '''Symmetric integral of the third kind C{RJ(x, y, z, p)}.

863

864 @return: C{RJ(x, y, z, p)}.

865

866 @see: U{C{RJ} definition<https://DLMF.NIST.gov/19.16.E2>} and

867 U{Carlson<https://ArXiv.org/pdf/math/9409227.pdf>}.

868 '''

869 try:

870 return float(_RJ(None, x, y, z, p))

871 except Exception as e:

872 raise _ellipticError(Elliptic.fRJ, x, y, z, p, cause=e)

873

874 @staticmethod

875 def _RFRD(x, y, z, m):

876 # in .auxilats.AuxDLat.DE, .auxilats.AuxLat.Rectifying

877 try: # float(RF(x, y, z) - RD(x, y, z, 3 / m))

878 R = _RF3(None, x, y, z)

879 if m:

880 R -= _RD(None, x, y, z, _3_0 / m)

881 except Exception as e:

882 raise _ellipticError(Elliptic._RFRD, x, y, z, m, cause=e)

883 return float(R)

884

885_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset

886

887

888 class EllipticError(_ValueError):

889 '''Elliptic function, integral, convergence or other L{Elliptic} issue.

890 '''

891 pass

892

893

894 class Elliptic3Tuple(_NamedTuple):

895 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.

896 '''

897 _Names_ = ('sn', 'cn', 'dn')

898 _Units_ = ( Scalar, Scalar, Scalar)

899

900

901 class _List(list):

902 '''(INTERNAL) Helper for C{_RD}, C{_RF3} and C{_RJ}.

903 '''

904 _a0 = None

905# _xyzp = ()

906

907 def __init__(self, *xyzp): # x, y, z [, p]

908 list.__init__(self, xyzp)

909 self._xyzp = xyzp

910

911 def a0(self, n):

912 '''Compute the initial C{a}.

913 '''

914 t = tuple(self)

915 m = n - len(t)

916 if m > 0:

917 t += t[-1:] * m

918 try:

919 a = Fsum(*t).fover(n)

920 except ValueError: # Fsum(NAN) exception

921 a = _sum(t) / n

922 self._a0 = a

923 return a

924

925 def amrs4(self, inst, y, Tol):

926 '''Yield Carlson 4-tuples C{(An, mul, lam, s)} plus sentinel, with

927 C{lam = fdot(sqrt(x), ... (z))} and C{s = (sqrt(x), ... (p))}.

928 '''

929 L = self

930 a = L.a0(5 if y else 3)

931 m = 1

932 t = max(fabs(a - _) for _ in L) / Tol

933 for i in range(_TRIPS):

934 d = fabs(a * m)

935 if d > t: # 3-6 trips

936 _iterations(inst, i)

937 break

938 s = map2(sqrt, L) # sqrt(x), srqt(y), sqrt(z) [, sqrt(p)]

939 try:

940 r = fdot(s[:3], s[1], s[2], s[0]) # sqrt(x) * sqrt(y) + ...

941 except ValueError: # Fsum(NAN) exception

942 r = _sum(s[i] * s[(i + 1) % 3] for i in range(3))

943 L[:] = ((r + _) * _0_25 for _ in L)

944 a = (r + a) * _0_25

945 if y: # yield only if used

946 yield a, m, r, s # L[2] is next z

947 m *= 4

948 else: # PYCHOK no cover

949 raise _convergenceError(d, t, thresh=True)

950 yield a, m, None, () # sentinel: same a, next m, no r and s

951

952 def rescale(self, am, *xs):

953 '''Rescale C{x}, C{y}, ...

954 '''

955 # assert am

956 a0 = self._a0

957 _am = _1_0 / am

958 for x in xs:

959 yield (a0 - x) * _am

960

961

962 def _ab2(inst, x, y):

963 '''(INTERNAL) Yield Carlson 2-tuples C{(xn, yn)}.

964 '''

965 a, b = sqrt(x), sqrt(y)

966 if b > a:

967 b, a = a, b

968 for i in range(_TRIPS):

969 yield a, b # xi, yi

970 d = fabs(a - b)

971 t = _TolRG0 * a

972 if d <= t: # 3-4 trips

973 _iterations(inst, i)

974 break

975 a, b = ((a + b) * _0_5), sqrt(a * b)

976 else: # PYCHOK no cover

977 raise _convergenceError(d, t)

978

979

980 def _convergenceError(d, tol, **thresh):

981 '''(INTERNAL) Format a no-convergence Error.

982 '''

983 t = Fmt.no_convergence(d, tol, **thresh)

984 return ValueError(t) # txt only

985

986

987 def _deltaX(sn, cn, dn, cX, fX):

988 '''(INTERNAL) Helper for C{Elliptic.deltaD} thru C{.deltaPi}.

989 '''

990 try:

991 if cn is None or dn is None:

992 raise ValueError(_invalid_)

993

994 if _signBit(cn):

995 sn, cn = neg_(sn, cn)

996 r = fX(sn, cn, dn) * PI_2 / cX

997 return r - atan2(sn, cn)

998

999 except Exception as e:

1000 n = NN(_delta_, fX.__name__[1:])

1001 raise _ellipticError(n, sn, cn, dn, cause=e)

1002

1003

1004 def _ellipticError(where, *args, **kwds_cause_txt):

1005 '''(INTERNAL) Format an L{EllipticError}.

1006 '''

1007 def _x_t_kwds(cause=None, txt=NN, **kwds):

1008 return cause, txt, kwds

1009

1010 x, t, kwds = _x_t_kwds(**kwds_cause_txt)

1011

1012 n = _dunder_nameof(where, where)

1013 n = _DOT_(Elliptic.__name__, n)

1014 n = _SPACE_(_invokation_, n)

1015 u = unstr(n, *args, **kwds)

1016 return EllipticError(u, cause=x, txt=t)

1017

1018

1019 def _Horner(S, e1, E2, E3, E4, E5, *over):

1020 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.

1021 '''

1022 E22 = E2**2

1023 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>

1024 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52

1025 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20

1026 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)

1027 # converted to Horner-like form ...

1028 e = e1 * 4084080

1029 S *= e

1030 S += Fsum(E2 * -540540, 471240).fmul(E5)

1031 S += Fsum(E2 * 612612, E3 * -540540, -556920).fmul(E4)

1032 S += Fsum(E2 * -706860, E22 * 675675, E3 * 306306, 680680).fmul(E3)

1033 S += Fsum(E2 * 417690, E22 * -255255, -875160).fmul(E2)

1034 S += 4084080

1035 if over:

1036 e *= over[0]

1037 return S.fdiv(e) # Fsum

1038

1039

1040 def _iterations(inst, i):

1041 '''(INTERNAL) Aggregate iterations B{C{i}}.

1042 '''

1043 if inst and i > 0:

1044 inst._iteration += i

1045

1046

1047 def _3over(a, b):

1048 '''(INTERNAL) Return C{3 / (a * b)}.

1049 '''

1050 return _over(_3_0, a * b)

1051

1052

1053 def _rC(unused, x, y):

1054 '''(INTERNAL) Defined only for C{y != 0} and C{x >= 0}.

1055 '''

1056 d = x - y

1057 if d < 0: # catch NaN

1058 # <https://DLMF.NIST.gov/19.2.E18>

1059 d = -d

1060 r = atan(sqrt(d / x)) if x > 0 else PI_2

1061 elif d == 0: # XXX d < EPS0? or EPS02 or _EPSmin

1062 d, r = y, _1_0

1063 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>

1064 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))

1065 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>

1066 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))

1067 else: # PYCHOK no cover

1068 raise _ellipticError(Elliptic.fRC, x, y)

1069 return r / sqrt(d) # float

1070

1071

1072 def _RD(inst, x, y, z, *over):

1073 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.

1074 '''

1075 L = _List(x, y, z)

1076 S = _Dsum()

1077 for a, m, r, s in L.amrs4(inst, True, _TolRF):

1078 if s:

1079 S += _over(_3_0, (r + z) * s[2] * m)

1080 z = L[2] # s[2] = sqrt(z)

1081 x, y = L.rescale(-a * m, x, y)

1082 xy = x * y

1083 z = (x + y) / _3_0

1084 z2 = z**2

1085 return _Horner(S(_1_0), sqrt(a) * a * m,

1086 xy - _6_0 * z2,

1087 (xy * _3_0 - _8_0 * z2) * z,

1088 (xy - z2) * _3_0 * z2,

1089 xy * z2 * z, *over) # Fsum

1090

1091

1092 def _rF2(inst, x, y): # 2-arg version, z=0

1093 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.

1094 '''

1095 for a, b in _ab2(inst, x, y): # PYCHOK yield

1096 pass

1097 return _over(PI, a + b) # float

1098

1099

1100 def _RF3(inst, x, y, z): # 3-arg version

1101 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.

1102 '''

1103 L = _List(x, y, z)

1104 for a, m, _, _ in L.amrs4(inst, False, _TolRF):

1105 pass

1106 x, y = L.rescale(a * m, x, y)

1107 z = neg(x + y)

1108 xy = x * y

1109 e2 = xy - z**2

1110 e3 = xy * z

1111 e4 = e2**2

1112 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>

1113 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44

1114 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)

1115 # converted to Horner-like form ...

1116 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)

1117 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)

1118 S += 240240

1119 return S.fdiv(sqrt(a) * 240240) # Fsum

1120

1121

1122 def _rG2(inst, x, y, PI_=PI_4): # 2-args

1123 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.

1124 '''

1125 m = -1 # neg!

1126 S = None

1127 for a, b in _ab2(inst, x, y): # PYCHOK yield

1128 if S is None: # initial

1129 S = _Dsum()

1130 S += (a + b)**2 * _0_5

1131 else:

1132 S += (a - b)**2 * m

1133 m *= 2

1134 return S(PI_).fover(a + b)

1135

1136

1137 def _rG3(inst, x, y, z): # 3-arg version

1138 '''(INTERNAL) C{x}, C{y} and C{z} all non-zero, see C{.fRG}.

1139 '''

1140 R = _RF3(inst, x, y, z) * z

1141 rd = (x - z) * (z - y) # - (y - z)

1142 if rd: # Carlson, eq 1.7

1143 R += _RD(inst, x, y, z, _3_0 / rd)

1144 R += sqrt(x * y / z)

1145 return R.fover(_2_0)

1146

1147

1148 def _RJ(inst, x, y, z, p, *over):

1149 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.

1150 '''

1151 def _xyzp(x, y, z, p):

1152 return (x + p) * (y + p) * (z + p)

1153

1154 L = _List(x, y, z, p)

1155 n = neg(_xyzp(x, y, z, -p))

1156 S = _Dsum()

1157 for a, m, _, s in L.amrs4(inst, True, _TolRD):

1158 if s:

1159 d = _xyzp(*s)

1160 if d:

1161 if n:

1162 rc = _rC(inst, _1_0, n / d**2 + _1_0)

1163 n *= _1_64th # /= chokes PyChecker

1164 else:

1165 rc = _1_0 # == _rC(None, _1_0, _1_0)

1166 S += rc / (d * m)

1167 else: # PYCHOK no cover

1168 return NAN

1169 x, y, z = L.rescale(a * m, x, y, z)

1170 p = Fsum(x, y, z).fover(_N_2_0)

1171 p2 = p**2

1172 p3 = p2 * p

1173 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)

1174 E2p = E2 * p

1175 xyz = x * y * z

1176 return _Horner(S(_6_0), sqrt(a) * a * m, E2,

1177 Fsum(p3 * _4_0, xyz, E2p * _2_0),

1178 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),

1179 xyz * p2, *over) # Fsum

1180

1181# **) MIT License

1182#

1183# Copyright (C) 2016-2024 -- mrJean1 at Gmail -- All Rights Reserved.

1184#

1185# Permission is hereby granted, free of charge, to any person obtaining a

1186# copy of this software and associated documentation files (the "Software"),

1187# to deal in the Software without restriction, including without limitation

1188# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1189# and/or sell copies of the Software, and to permit persons to whom the

1190# Software is furnished to do so, subject to the following conditions:

1191#

1192# The above copyright notice and this permission notice shall be included

1193# in all copies or substantial portions of the Software.

1194#

1195# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1196# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1197# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1198# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1199# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1200# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1201# OTHER DEALINGS IN THE SOFTWARE.