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, _xattr, _xkwds_pop

84 from pygeodesy.fmath import fdot, hypot1, zqrt

85 from pygeodesy.fsums import Fsum, _sum

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

87 _invokation_, _negative_, _SPACE_

88 from pygeodesy.karney import _K_2_0, _norm180, _signBit, _sincos2, \

89 _ALL_LAZY

90# from pygeodesy.lazily import _ALL_LAZY # from .karney

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__ = '23.08.31'

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 _CIs(object):

110 '''(INTERAL) Complete integrals cache.

111 '''

112 def __init__(self, **kwds):

113 self.__dict__ = kwds

114

115

116 class _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

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 html/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=NN):

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

144

145 @kwarg name: Optional name (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

156 if name:

157 self.name = name

158

159 @Property_RO

160 def alpha2(self):

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

162 '''

163 return self._alpha2

164

165 @Property_RO

166 def alphap2(self):

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

168 '''

169 return self._alphap2

170

171 @Property_RO

172 def cD(self):

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

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

175 '''

176 return self._reset_cDEKEeps.cD

177

178 @Property_RO

179 def cE(self):

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

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

182 '''

183 return self._reset_cDEKEeps.cE

184

185 @Property_RO

186 def cG(self):

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

188 C{G(α^2, k)} (C{float}).

189 '''

190 return self._reset_cGHPi.cG

191

192 @Property_RO

193 def cH(self):

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

195 C{H(α^2, k)} (C{float}).

196 '''

197 return self._reset_cGHPi.cH

198

199 @Property_RO

200 def cK(self):

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

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

203 '''

204 return self._reset_cDEKEeps.cK

205

206 @Property_RO

207 def cKE(self):

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

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

210 '''

211 return self._reset_cDEKEeps.cKE

212

213 @Property_RO

214 def cPi(self):

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

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

217 '''

218 return self._reset_cGHPi.cPi

219

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

221 '''The periodic Jahnke's incomplete elliptic integral.

222

223 @arg sn: sin(φ).

224 @arg cn: cos(φ).

225 @arg dn: sqrt(1 − k2 * sin(2φ)).

226

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

228

229 @raise EllipticError: Invalid invokation or no convergence.

230 '''

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

232

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

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

235

236 @arg sn: sin(φ).

237 @arg cn: cos(φ).

238 @arg dn: sqrt(1 − k2 * sin(2φ)).

239

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

241

242 @raise EllipticError: Invalid invokation or no convergence.

243 '''

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

245

246 def deltaEinv(self, stau, ctau):

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

248

249 @arg stau: sin(τ)

250 @arg ctau: cos(τ)

251

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

253

254 @raise EllipticError: No convergence.

255 '''

256 try:

257 if _signBit(ctau): # pi periodic

258 stau, ctau = neg_(stau, ctau)

259 t = atan2(stau, ctau)

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

261

262 except Exception as e:

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

264

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

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

267

268 @arg sn: sin(φ).

269 @arg cn: cos(φ).

270 @arg dn: sqrt(1 − k2 * sin(2φ)).

271

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

273

274 @raise EllipticError: Invalid invokation or no convergence.

275 '''

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

277

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

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

280

281 @arg sn: sin(φ).

282 @arg cn: cos(φ).

283 @arg dn: sqrt(1 − k2 * sin(2φ)).

284

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

286

287 @raise EllipticError: Invalid invokation or no convergence.

288 '''

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

290

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

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

293

294 @arg sn: sin(φ).

295 @arg cn: cos(φ).

296 @arg dn: sqrt(1 − k2 * sin(2φ)).

297

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

299

300 @raise EllipticError: Invalid invokation or no convergence.

301 '''

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

303

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

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

306

307 @arg sn: sin(φ).

308 @arg cn: cos(φ).

309 @arg dn: sqrt(1 − k2 * sin(2φ)).

310

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

312 (C{float}).

313

314 @raise EllipticError: Invalid invokation or no convergence.

315 '''

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

317

318 def _Einv(self, x):

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

320 '''

321 E2 = self.cE * _2_0

322 n = floor(x / E2 + _0_5)

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

324 # linear approximation

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

326 Phi = Fsum(phi)

327 # first order correction

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

329 # For kp2 close to zero use asin(r / cE) or J. P. Boyd,

330 # Applied Math. and Computation 218, 7005-7013 (2012)

331 # <https://DOI.org/10.1016/j.amc.2011.12.021>

332 _Phi2, self._iteration = Phi.fsum2_, 0 # aggregate

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

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

335 if dn:

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

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

338 else: # PYCHOK no cover

339 d = _0_0 # XXX continue?

340 if fabs(d) < _TolJAC: # 3-4 trips

341 _iterations(self, i)

342 break

343 else: # PYCHOK no cover

344 raise _convergenceError(d, _TolJAC)

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

346

347 @Property_RO

348 def eps(self):

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

350 '''

351 return self._reset_cDEKEeps.eps

352

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

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

355 Jacobi elliptic functions.

356

357 @arg phi_or_sn: φ or sin(φ).

358 @kwarg cn: C{None} or cos(φ).

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

360

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

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

363

364 @raise EllipticError: Invalid invokation or no convergence.

365 '''

366 def _fD(sn, cn, dn):

367 r = fabs(sn)**3

368 if r:

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

370 return r

371

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

373 self.deltaD, _fD)

374

375 def fDelta(self, sn, cn):

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

377

378 @arg sn: sin(φ).

379 @arg cn: cos(φ).

380

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

382 '''

383 try:

384 k2 = self.k2

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

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

387 return sqrt(s) if s else _0_0

388

389 except Exception as e:

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

391

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

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

394 Jacobi elliptic functions.

395

396 @arg phi_or_sn: φ or sin(φ).

397 @kwarg cn: C{None} or cos(φ).

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

399

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

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

402

403 @raise EllipticError: Invalid invokation or no convergence.

404 '''

405 def _fE(sn, cn, dn):

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

407 '''

408 if sn:

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

410 kp2, k2 = self.kp2, self.k2

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

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

413 if k2:

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

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

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

417 if kp2:

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

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

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

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

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

423 Ei *= fabs(sn)

424 ei = float(Ei)

425 else: # PYCHOK no cover

426 ei = _0_0

427 return ei

428

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

430 self.deltaE, _fE)

431

432 def fEd(self, deg):

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

434 the argument given in C{degrees}.

435

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

437

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

439

440 @raise EllipticError: No convergence.

441 '''

442 if _K_2_0:

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

444 elif fabs(deg) < _180_0:

445 e = _0_0

446 else:

447 e = ceil(deg / _360_0 - _0_5)

448 deg -= e * _360_0

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

450

451 def fEinv(self, x):

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

453

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

455

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

457 (C{float}).

458

459 @raise EllipticError: No convergence.

460 '''

461 try:

462 return self._Einv(x)

463 except Exception as e:

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

465

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

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

468 Jacobi elliptic functions.

469

470 @arg phi_or_sn: φ or sin(φ).

471 @kwarg cn: C{None} or cos(φ).

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

473

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

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

476

477 @raise EllipticError: Invalid invokation or no convergence.

478 '''

479 def _fF(sn, cn, dn):

480 r = fabs(sn)

481 if r:

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

483 return r

484

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

486 self.deltaF, _fF)

487

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

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

490 Jacobi elliptic functions.

491

492 @arg phi_or_sn: φ or sin(φ).

493 @kwarg cn: C{None} or cos(φ).

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

495

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

497

498 @raise EllipticError: Invalid invokation or no convergence.

499

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

501 geodesic in terms of this combination of elliptic

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

503 (1811), p 181<https://Books.Google.com/books?id=

504 riIOAAAAQAAJ&pg=PA181>}.

505

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

507 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}

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

509 '''

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

511 self.cG, self.deltaG)

512

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

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

515 Jacobi elliptic functions.

516

517 @arg phi_or_sn: φ or sin(φ).

518 @kwarg cn: C{None} or cos(φ).

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

520

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

522

523 @raise EllipticError: Invalid invokation or no convergence.

524

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

526 on the geodesic in terms of this combination of

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

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

529

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

531 GeographicLib.SourceForge.io/html/geodesic.html#geodellip>}

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

533 '''

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

535 self.cH, self.deltaH)

536

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

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

539 Jacobi elliptic functions.

540

541 @arg phi_or_sn: φ or sin(φ).

542 @kwarg cn: C{None} or cos(φ).

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

544

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

546

547 @raise EllipticError: Invalid invokation or no convergence.

548 '''

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

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

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

552 self.cPi, self.deltaPi)

553

554 def _fXa(self, phi_or_sn, cn, dn, aX, cX, _deltaX):

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

556 '''

557 def _fX(sn, cn, dn):

558 if sn:

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

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

561 if aX:

562 sn2 = sn**2

563 p = sn2 * self.alphap2 + cn2

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

565 r = float(R.fmul(fabs(sn)))

566 else: # PYCHOK no cover

567 r = _0_0

568 return r

569

570 return self._fXf(phi_or_sn, cn, dn, cX, _deltaX, _fX)

571

572 def _fXf(self, phi_or_sn, cn, dn, cX, _deltaX, _fX):

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

574 '''

575 self._iteration = 0 # aggregate

576 phi = sn = phi_or_sn

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

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

579 if fabs(phi) >= PI:

580 if cX:

581 cX *= (_deltaX(sn, cn, dn) + phi) / PI_2

582 return cX

583 # fall through

584 elif cn is None or dn is None:

585 n = NN(_f_, _deltaX.__name__[5:])

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

587

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

589 xi = cX * _2_0 - _fX(sn, cn, dn)

590 elif cn > 0:

591 xi = _fX(sn, cn, dn)

592 else:

593 xi = cX

594 return copysign0(xi, sn)

595

596 @Property_RO

597 def k2(self):

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

599 '''

600 return self._k2

601

602 @Property_RO

603 def kp2(self):

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

605 '''

606 return self._kp2

607

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

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

610

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

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

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

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

615

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

617 or B{C{alphap2}}.

618

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

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

621 that these conditions are met to enable accuracy to be

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

623 '''

624 if self.__dict__:

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

626

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

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

629

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

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

632 Error=EllipticError)

633

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

635 # K E D

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

637 # k = 1: inf 1 inf

638 # Pi G H

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

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

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

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

643 #

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

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

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

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

648 # Pi(alpha2, 1) = inf

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

650

651 @Property_RO

652 def _reset_cDEKEeps(self):

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

654 '''

655 k2, kp2 = self.k2, self.kp2

656 if k2:

657 if kp2:

658 try:

659 self._iteration = 0

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

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

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

663 cD = float(D)

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

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

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

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

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

669 cK = _rF2(self, kp2, _1_0)

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

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

672

673 except Exception as e:

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

675 else:

676 cD = cK = cKE = INF

677 cE = _1_0

678 eps = k2

679 else:

680 cD = PI_4

681 cE = cK = PI_2

682 cKE = _0_0 # k2 * cD

683 eps = EPS

684

685 return _CIs(cD=cD, cE=cE, cK=cK, cKE=cKE, eps=eps)

686

687 @Property_RO

688 def _reset_cGHPi(self):

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

690 '''

691 alpha2, alphap2, kp2 = self.alpha2, self.alphap2, self.kp2

692 try:

693 self._iteration = 0

694 if alpha2:

695 if alphap2:

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

697 cK = self.cK

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

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

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

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

702 else:

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

704 cPi = INF # XXX or NAN?

705 else:

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

707 else:

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

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

710 # So write (for alpha2 = 0)

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

712 # = 1 / sqrt(1-k2) * int(sin(phi)**2 / sqrt(1-k2/kp2 * sin(phi)**2,...)

713 # = 1 / kp * D(i * k/kp)

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

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

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

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

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

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

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

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

722

723 except Exception as e:

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

725 alphap2=alphap2, cause=e)

726 return _CIs(cG=cG, cH=cH, cPi=cPi)

727

728 def sncndn(self, x):

729 '''The Jacobi elliptic function.

730

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

732

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

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

735

736 @raise EllipticError: No convergence.

737 '''

738 self._iteration = 0 # reset

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

740 if self.kp2:

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

742 dn = _1_0

743 sn, cn = _sincos2(x * cd)

744 if sn:

745 a = cn / sn

746 c *= a

747 for m, n in reversed(mn):

748 a *= c

749 c *= dn

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

751 a = c / m

752 a = _1_0 / hypot1(c)

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

754 cn = c * sn

755 if d and _signBit(self.kp2):

756 cn, dn = dn, cn

757 sn = sn / d # /= chokes PyChecker

758 else:

759 sn = tanh(x)

760 cn = dn = _1_0 / cosh(x)

761

762 except Exception as e:

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

764

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

766

767 def _sncndn3(self, phi):

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

769 '''

770 sn, cn = _sincos2(phi)

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

772

773 @Property_RO

774 def _sncndn4(self):

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

776 '''

777 # Bulirsch's sncndn routine, p 89.

778 d, mc = 0, self.kp2

779 if _signBit(mc):

780 d = _1_0 - mc

781 mc = neg(mc / d)

782 d = sqrt(d)

783

784 mn, a = [], _1_0

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

786 mc = sqrt(mc)

787 mn.append((a, mc))

788 c = (a + mc) * _0_5

789 r = fabs(mc - a)

790 t = _TolJAC * a

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

792 _iterations(self, i)

793 break

794 mc *= a

795 a = c

796 else: # PYCHOK no cover

797 raise _convergenceError(r, t)

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

799 return c, d, cd, mn

800

801 @staticmethod

802 def fRC(x, y):

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

804

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

806

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

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

809 '''

810 return _rC(None, x, y)

811

812 @staticmethod

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

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

815

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

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

818

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

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

821 '''

822 try:

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

824 except Exception as e:

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

826

827 @staticmethod

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

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

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

831

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

833

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

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

836 '''

837 try:

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

839 except Exception as e:

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

841

842 @staticmethod

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

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

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

846

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

848

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

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

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

852 EllipticFunction_8cpp_source.html#l00096>} version 2.3.

853 '''

854 try:

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

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

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

858

859 except Exception as e:

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

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

862

863 @staticmethod

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

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

866

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

868

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

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

871 '''

872 try:

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

874 except Exception as e:

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

876

877_allPropertiesOf_n(15, Elliptic) # # PYCHOK assert, see Elliptic.reset

878

879

880 class EllipticError(_ValueError):

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

882 '''

883 pass

884

885

886 class Elliptic3Tuple(_NamedTuple):

887 '''3-Tuple C{(sn, cn, dn)} all C{scalar}.

888 '''

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

890 _Units_ = ( Scalar, Scalar, Scalar)

891

892

893 class _L(list):

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

895 '''

896 _a0 = None

897# _xyzp = ()

898

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

900 list.__init__(self, xyzp)

901 self._xyzp = xyzp

902

903 def a0(self, n):

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

905 '''

906 t = tuple(self)

907 m = n - len(t)

908 if m > 0:

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

910 try:

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

912 except ValueError: # Fsum(NAN) exception

913 a = _sum(t) / n

914 self._a0 = a

915 return a

916

917 def amrs4(self, inst, n, Tol):

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

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

920 '''

921 L = self

922 a = L.a0(n)

923 m = 1

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

925 for i in range(_TRIPS):

926 d = fabs(a * m)

927 if d > t: # 5-6 trips

928 _iterations(inst, i)

929 break

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

931 try:

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

933 except ValueError: # Fsum(NAN) exception

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

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

936 a = (r + a) * _0_25

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

938 m *= 4

939 else: # PYCHOK no cover

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

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

942

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

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

945 '''

946 # assert am

947 a0 = self._a0

948 for x in xs:

949 yield (a0 - x) / am

950

951

952 def _ab2(inst, x, y):

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

954 '''

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

956 if b > a:

957 a, b = b, a

958 yield a, b # initial x0, y0

959 for i in range(1, _TRIPS):

960 d = fabs(a - b)

961 t = _TolRG0 * a

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

963 _iterations(inst, i - 1)

964 break

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

966 yield a, b # xn, yn

967 else: # PYCHOK no cover

968 raise _convergenceError(d, t)

969

970

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

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

973 '''

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

975 return ValueError(t) # txt only

976

977

978 def _deltaX(sn, cn, dn, cX, _fX):

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

980 '''

981 try:

982 if cn is None or dn is None:

983 raise ValueError(_invalid_)

984

985 if _signBit(cn):

986 sn, cn = neg_(sn, cn)

987 r = _fX(sn, cn, dn) * PI_2 / cX

988 return r - atan2(sn, cn)

989

990 except Exception as e:

991 n = NN(_delta_, _fX.__name__[1:])

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

993

994

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

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

997 '''

998 c = _xkwds_pop(kwds_cause_txt, cause=None)

999 t = _xkwds_pop(kwds_cause_txt, txt=NN)

1000 n = _xattr(where, __name__=where)

1001 n = _DOT_(Elliptic.__name__, n)

1002 n = _SPACE_(_invokation_, n)

1003 u = unstr(n, *args, **kwds_cause_txt)

1004 return EllipticError(u, cause=c, txt=t)

1005

1006

1007 def _Horner(S, e1, E2, E3, E4, E5, *over):

1008 '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below.

1009 '''

1010 E22 = E2**2

1011 # Polynomial is <https://DLMF.NIST.gov/19.36.E2>

1012 # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52

1013 # + 3*E5/26 - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20

1014 # + 45*E2**2*E3/272 - 9*(E3*E4+E2*E5)/68)

1015 # converted to Horner-like form ...

1016 F = Fsum

1017 e = e1 * 4084080

1018 S *= e

1019 S += F(E2 * -540540, 471240).fmul(E5)

1020 S += F(E2 * 612612, E3 * -540540, -556920).fmul(E4)

1021 S += F(E2 * -706860, E22 * 675675, E3 * 306306, 680680).fmul(E3)

1022 S += F(E2 * 417690, E22 * -255255, -875160).fmul(E2)

1023 S += 4084080

1024 return S.fdiv((over[0] * e) if over else e) # Fsum

1025

1026

1027 def _iterations(inst, i):

1028 '''(INTERNAL) Aggregate iterations B{C{i}}.

1029 '''

1030 if inst and i > 0:

1031 inst._iteration += i

1032

1033

1034 def _3over(a, b):

1035 '''(INTERNAL) Return C{3 / (a * b)}.

1036 '''

1037 return _over(_3_0, a * b)

1038

1039

1040 def _rC(unused, x, y):

1041 '''(INTERNAL) Defined only for C{y != 0} and C{x >= 0}.

1042 '''

1043 d = x - y

1044 if d < 0: # catch NaN

1045 # <https://DLMF.NIST.gov/19.2.E18>

1046 d = -d

1047 r = atan(sqrt(d / x)) if x > 0 else PI_2

1048 elif d == 0: # XXX d < EPS0? or EPS02 or _EPSmin

1049 d, r = y, _1_0

1050 elif y > 0: # <https://DLMF.NIST.gov/19.2.E19>

1051 r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x))

1052 elif y < 0: # <https://DLMF.NIST.gov/19.2.E20>

1053 r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y)))

1054 else: # PYCHOK no cover

1055 raise _ellipticError(Elliptic.fRC, x, y)

1056 return r / sqrt(d) # float

1057

1058

1059 def _RD(inst, x, y, z, *over):

1060 '''(INTERNAL) Carlson, eqs 2.28 - 2.34.

1061 '''

1062 L = _L(x, y, z)

1063 S = _D()

1064 for a, m, r, s in L.amrs4(inst, 5, _TolRF):

1065 if s:

1066 S += _over(_3_0, (r + z) * s[2] * m)

1067 z = L[2] # s[2] = sqrt(z)

1068 x, y = L.rescale(-a * m, x, y)

1069 xy = x * y

1070 z = (x + y) / _3_0

1071 z2 = z**2

1072 return _Horner(S(_1_0), sqrt(a) * a * m,

1073 xy - _6_0 * z2,

1074 (xy * _3_0 - _8_0 * z2) * z,

1075 (xy - z2) * _3_0 * z2,

1076 xy * z2 * z, *over) # Fsum

1077

1078

1079 def _rF2(inst, x, y): # 2-arg version, z=0

1080 '''(INTERNAL) Carlson, eqs 2.36 - 2.38.

1081 '''

1082 for a, b in _ab2(inst, x, y): # PYCHOK yield

1083 pass

1084 return _over(PI, a + b) # float

1085

1086

1087 def _RF3(inst, x, y, z): # 3-arg version

1088 '''(INTERNAL) Carlson, eqs 2.2 - 2.7.

1089 '''

1090 L = _L(x, y, z)

1091 for a, m, _, _ in L.amrs4(inst, 3, _TolRF):

1092 pass

1093 x, y = L.rescale(a * m, x, y)

1094 z = neg(x + y)

1095 xy = x * y

1096 e2 = xy - z**2

1097 e3 = xy * z

1098 e4 = e2**2

1099 # Polynomial is <https://DLMF.NIST.gov/19.36.E1>

1100 # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44

1101 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16)

1102 # converted to Horner-like form ...

1103 S = Fsum(e4 * 15015, e3 * 6930, e2 * -16380, 17160).fmul(e3)

1104 S += Fsum(e4 * -5775, e2 * 10010, -24024).fmul(e2)

1105 S += 240240

1106 return S.fdiv(sqrt(a) * 240240) # Fsum

1107

1108

1109 def _rG2(inst, x, y, PI_=PI_4): # 2-args

1110 '''(INTERNAL) Carlson, eqs 2.36 - 2.39.

1111 '''

1112 m = -1 # neg!

1113 S = _D()

1114 # assert not S

1115 for a, b in _ab2(inst, x, y): # PYCHOK yield

1116 if S:

1117 S += (a - b)**2 * m

1118 m *= 2

1119 else: # initial

1120 S += (a + b)**2 * _0_5

1121 return S(PI_).fover(a + b)

1122

1123

1124 def _rG3(inst, x, y, z): # 3-arg version

1125 '''(INTERNAL) C{x}, C{y} and C{z} all non-zero, see C{.fRG}.

1126 '''

1127 R = _RF3(inst, x, y, z) * z

1128 rd = (x - z) * (z - y) # - (y - z)

1129 if rd: # Carlson, eq 1.7

1130 R += _RD(inst, x, y, z, _3_0 / rd)

1131 R += sqrt(x * y / z)

1132 return R.fover(_2_0)

1133

1134

1135 def _RJ(inst, x, y, z, p, *over):

1136 '''(INTERNAL) Carlson, eqs 2.17 - 2.25.

1137 '''

1138 def _xyzp(x, y, z, p):

1139 return (x + p) * (y + p) * (z + p)

1140

1141 L = _L(x, y, z, p)

1142 n = neg(_xyzp(x, y, z, -p))

1143 S = _D()

1144 for a, m, _, s in L.amrs4(inst, 5, _TolRD):

1145 if s:

1146 d = _xyzp(*s)

1147 if d:

1148 if n:

1149 rc = _rC(inst, _1_0, n / d**2 + _1_0)

1150 n *= _1_64th # /= chokes PyChecker

1151 else:

1152 rc = _1_0 # == _rC(None, _1_0, _1_0)

1153 S += rc / (d * m)

1154 else: # PYCHOK no cover

1155 return NAN

1156 x, y, z = L.rescale(a * m, x, y, z)

1157 xyz = x * y * z

1158 p = Fsum(x, y, z).fover(_N_2_0)

1159 p2 = p**2

1160 p3 = p2 * p

1161 E2 = Fsum(x * y, x * z, y * z, -p2 * _3_0)

1162 E2p = E2 * p

1163 return _Horner(S(_6_0), sqrt(a) * a * m, E2,

1164 Fsum(p3 * _4_0, xyz, E2p * _2_0),

1165 Fsum(p3 * _3_0, E2p, xyz * _2_0).fmul(p),

1166 xyz * p2, *over) # Fsum

1167

1168# **) MIT License

1169#

1170# Copyright (C) 2016-2023 -- mrJean1 at Gmail -- All Rights Reserved.

1171#

1172# Permission is hereby granted, free of charge, to any person obtaining a

1173# copy of this software and associated documentation files (the "Software"),

1174# to deal in the Software without restriction, including without limitation

1175# the rights to use, copy, modify, merge, publish, distribute, sublicense,

1176# and/or sell copies of the Software, and to permit persons to whom the

1177# Software is furnished to do so, subject to the following conditions:

1178#

1179# The above copyright notice and this permission notice shall be included

1180# in all copies or substantial portions of the Software.

1181#

1182# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS

1183# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

1184# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL

1185# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR

1186# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,

1187# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR

1188# OTHER DEALINGS IN THE SOFTWARE.