# Coverage for pygeodesy/elliptic.py : 97%

## 340 statements 330 run 10 missing 21 excluded

Hot-keys on this page

r m x p toggle line displays

j k next/prev highlighted chunk

0 (zero) top of page

1 (one) first highlighted chunk

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 | # -*- coding: utf-8 -*- u'''Elliptic integrals and functions transcribed from I{Charles Karney's} C++ class U{EllipticFunction <https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1EllipticFunction.html>} into pure Python class L{Elliptic}. Python method names follow the C++ member functions, except: - member functions I{without arguments} are mapped to Python properties prefixed with C{"c"}, for example C{E()} is property C{cE}, - member functions with 1 or 3 arguments are renamed to Python methods starting with an C{"f"}, example C{E(psi)} to C{fE(psi)} and C{E(sn, cn, dn)} to C{fE(sn, cn, dn)}, - other Python method names conventionally start with a lower-case letter or an underscore if private. Following is a copy of Karney's U{EllipticFunction.hpp <https://GeographicLib.SourceForge.io/html/EllipticFunction_8hpp_source.html>} file C{Header}. Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2008-2017) and licensed under the MIT/X11 License. For more information, see the U{GeographicLib<https://GeographicLib.SourceForge.io>} documentation. B{Elliptic integrals and functions.} This provides the elliptic functions and integrals needed for C{Ellipsoid}, C{GeodesicExact}, and C{TransverseMercatorExact}. Two categories of function are provided: - functions to compute U{symmetric elliptic integrals <https://DLMF.NIST.gov/19.16.i>} - methods to compute U{Legrendre's elliptic integrals <https://DLMF.NIST.gov/19.2.ii>} and U{Jacobi elliptic functions<https://DLMF.NIST.gov/22.2>}. In the latter case, an object is constructed giving the modulus C{k} (and optionally the parameter C{alpha}). The modulus (and parameter) are always passed as squares which allows C{k} to be pure imaginary. (Confusingly, Abramowitz and Stegun call C{m = k**2} the "parameter" and C{n = alpha**2} the "characteristic".) In geodesic applications, it is convenient to separate the incomplete integrals into secular and periodic components, e.g. I{C{E(phi, k) = (2 E(k) / pi) [ phi + delta E(phi, k) ]}} where I{C{delta E(phi, k)}} is an odd periodic function with period I{C{pi}}. The computation of the elliptic integrals uses the algorithms given in U{B. C. Carlson, Computation of real or complex elliptic integrals <https://DOI.org/10.1007/BF02198293>}, Numerical Algorithms 10, 13--26 (1995) with the additional optimizations given U{here <https://DLMF.NIST.gov/19.36.i>}. The computation of the Jacobi elliptic functions uses the algorithm given in U{R. Bulirsch, Numerical Calculation of Elliptic Integrals and Elliptic Functions <https://DOI.org/10.1007/BF01397975>}, Numerische Mathematik 7, 78--90 (1965). The notation follows U{NIST Digital Library of Mathematical Functions <https://DLMF.NIST.gov>} chapters U{19<https://DLMF.NIST.gov/19>} and U{22<https://DLMF.NIST.gov/22>}. ''' # make sure int/int division yields float quotient from __future__ import division division = 1 / 2 # double check int division, see .datum.py, .utily.py if not division: raise ImportError('%s 1/2 == %d' % ('division', division)) del division from pygeodesy.basics import EPS, INF, map2, property_RO from pygeodesy.fmath import fdot, Fsum, fsum_, hypot1 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import _Named, _NamedTuple from pygeodesy.utily import PI, PI_2, PI_4, sincos2, sincos2d from math import asinh, atan, atan2, ceil, copysign, cosh, \ floor, sin, sqrt, tanh __all__ = _ALL_LAZY.elliptic + _ALL_DOCS('Elliptic3Tuple') __version__ = '20.03.23' _TolJAC = sqrt(EPS * 0.01) _TolRD = pow(EPS * 0.002, 0.125) _TolRF = pow(EPS * 0.030, 0.125) _TolRG0 = _TolJAC * 2.7 _TRIPS = 13 # Max depth for sncndn, etc, 5-7 might be enough class EllipticError(ValueError): '''Elliptic integral, function, convergence or other L{Elliptic} issue. ''' pass class Elliptic(_Named): '''Elliptic integrals and functions. @see: Karney's U{Detailed Description<https://GeographicLib.SourceForge.io/ html/classGeographicLib_1_1EllipticFunction.html#details>}. ''' _alpha2 = 0 _alphap2 = 0 _eps = EPS _k2 = 0 _kp2 = 0 _trips_ = _TRIPS _cD = INF _cE = 1.0 _cG = INF _cH = INF _cK = INF _cKE = INF _cPi = INF def __init__(self, k2=0, alpha2=0, kp2=None, alphap2=None): '''Constructor, specifying the C{modulus} and C{parameter}. @kwarg k2: Modulus squared (C{float}, 0 <= k^2 <= 1). @kwarg alpha2: Parameter squared (C{float}, 0 <= α^2 <= 1). @kwarg kp2: Complementary modulus squared (C{float}, k'^2 >= 0). @kwarg alphap2: Complementary parameter squared (C{float}, α'^2 >= 0). @see: Method L{reset} for further details. @note: If only elliptic integrals of the first and second kinds are needed, then set B{C{alpha2}} = 0 (the default value). In that case, we have Π(φ, 0, k) = F(φ, k), G(φ, 0, k) = E(φ, k), and H(φ, 0, k) = F(φ, k) - D(φ, k). ''' self.reset(k2=k2, alpha2=alpha2, kp2=kp2, alphap2=alphap2) @property_RO def alpha2(self): '''Get α^2, the square of the parameter (C{float}). ''' return self._alpha2 @property_RO def alphap2(self): '''Get α'^2, the square of the complementary parameter (C{float}). ''' return self._alphap2 @property_RO def cD(self): '''Get Jahnke's complete integral C{D(k)} (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E6>}. ''' return self._cD @property_RO def cE(self): '''Get the complete integral of the second kind C{E(k)} (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}. ''' return self._cE @property_RO def cG(self): '''Get Legendre's complete geodesic longitude integral C{G(α^2, k)} (C{float}). ''' return self._cG @property_RO def cH(self): '''Get Cayley's complete geodesic longitude difference integral C{H(α^2, k)} (C{float}). ''' return self._cH @property_RO def cK(self): '''Get the complete integral of the first kind C{K(k)} (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}. ''' return self._cK @property_RO def cKE(self): '''Get the difference between the complete integrals of the first and second kinds, C{K(k) − E(k)} (C{float}). ''' return self._cKE @property_RO def cPi(self): '''Get the complete integral of the third kind C{Pi(α^2, k)} (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E7>}. ''' return self._cPi def deltaD(self, sn, cn, dn): '''The periodic Jahnke's incomplete elliptic integral. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π D(φ, k) / (2 D(k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cD, self.fD) def deltaE(self, sn, cn, dn): '''The periodic incomplete integral of the second kind. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π E(φ, k) / (2 E(k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cE, self.fE) def deltaEinv(self, stau, ctau): '''The periodic inverse of the incomplete integral of the second kind. @arg stau: sin(τ) @arg ctau: cos(τ) @return: Periodic function E^−1(τ (2 E(k)/π), k) - τ (C{float}). @raise EllipticError: No convergence. ''' # Function is periodic with period pi t = atan2(-stau, -ctau) if ctau < 0 else atan2(stau, ctau) return self.fEinv(t * self._cE / PI_2) - t def deltaF(self, sn, cn, dn): '''The periodic incomplete integral of the first kind. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π F(φ, k) / (2 K(k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cK, self.fF) def deltaG(self, sn, cn, dn): '''Legendre's periodic geodesic longitude integral. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π G(φ, k) / (2 G(k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cG, self.fG) def deltaH(self, sn, cn, dn): '''Cayley's periodic geodesic longitude difference integral. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π H(φ, k) / (2 H(k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cH, self.fH) def deltaPi(self, sn, cn, dn): '''The periodic incomplete integral of the third kind. @arg sn: sin(φ). @arg cn: cos(φ). @arg dn: sqrt(1 − k2 sin(2φ)). @return: Periodic function π Π(φ, α2, k) / (2 Π(α2, k)) - φ (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._deltaX(sn, cn, dn, self.cPi, self.fPi) def _deltaX(self, sn, cn, dn, cX, fX): '''(INTERNAL) Helper for C{.deltaD} thru C{.deltaPi}. ''' if None in (cn, dn): t = self.classname + '.delta' + fX.__name__[1:] raise EllipticError('%s invalid: %s%r' % ('invokation', t, (sn, cn, dn))) if cn < 0: cn, sn = -cn, -sn return fX(sn, cn, dn) * PI_2 / cX - atan2(sn, cn) @property_RO def eps(self): '''Get epsilon (C{float}). ''' return self._eps def fD(self, phi_or_sn, cn=None, dn=None): '''Jahnke's incomplete elliptic integral in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: D(φ, k) as though φ ∈ (−π, π] (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}. @raise EllipticError: Invalid invokation or no convergence. ''' def _fD(sn, cn, dn): return abs(sn) * sn**2 * _RD_3(cn**2, dn**2, 1) return self._fXf(phi_or_sn, cn, dn, self.cD, self.deltaD, _fD) def fDelta(self, sn, cn): '''The C{Delta} amplitude function. @arg sn: sin(φ). @arg cn: cos(φ). @return: C{sqrt(1 − k2 C{sin}(2φ))} (C{float}). ''' k2 = self.k2 return sqrt((1 - k2 * sn**2) if k2 < 0 else (k2 * cn**2 + self.kp2)) def fE(self, phi_or_sn, cn=None, dn=None): '''The incomplete integral of the second kind in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: E(φ, k) as though φ ∈ (−π, π] (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E5>}. @raise EllipticError: Invalid invokation or no convergence. ''' def _fE(sn, cn, dn): sn2, cn2, dn2 = sn**2, cn**2, dn**2 kp2, k2 = self.kp2, self.k2 if k2 <= 0: # Carlson, eq. 4.6, <https://DLMF.NIST.gov/19.25.E9> ei = _RF(cn2, dn2, 1) - k2 * sn2 * _RD_3(cn2, dn2, 1) elif kp2 >= 0: # <https://DLMF.NIST.gov/19.25.E10> ei = fsum_(kp2 * _RF(cn2, dn2, 1), kp2 * k2 * sn2 * _RD_3(cn2, 1, dn2), k2 * abs(cn) / dn) else: # <https://DLMF.NIST.gov/19.25.E11> ei = dn / abs(cn) - kp2 * sn2 * _RD_3(dn2, 1, cn2) return ei * abs(sn) return self._fXf(phi_or_sn, cn, dn, self.cE, self.deltaE, _fE) def fEd(self, deg): '''The incomplete integral of the second kind with the argument given in degrees. @arg deg: Angle (C{degrees}). @return: E(π B{C{deg}}/180, k) (C{float}). @raise EllipticError: No convergence. ''' n = ceil(deg / 360.0 - 0.5) sn, cn = sincos2d(deg - n * 360.0) return self.fE(sn, cn, self.fDelta(sn, cn)) + 4 * self.cE * n def fEinv(self, x): '''The inverse of the incomplete integral of the second kind. @arg x: Argument (C{float}). @return: φ = 1 / E(B{C{x}}, k), such that E(φ, k) = B{C{x}} (C{float}). @raise EllipticError: No convergence. ''' E2 = self.cE * 2.0 n = floor(x / E2 + 0.5) x -= E2 * n # x now in [-ec, ec) # linear approximation phi = PI * x / E2 # phi in [-pi/2, pi/2) Phi = Fsum(phi) # first order correction phi = Phi.fsum_(-self.eps * sin(2 * phi) * 0.5) # For kp2 close to zero use asin(x/.cE) or J. P. Boyd, # Applied Math. and Computation 218, 7005-7013 (2012) # <https://DOI.org/10.1016/j.amc.2011.12.021> for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC sn, cn, dn = self._sncndn3(phi) phi, e = Phi.fsum2_((x - self.fE(sn, cn, dn)) / dn) if abs(e) < _TolJAC: return n * PI + phi raise _convergenceError(self.fEinv, x) def fF(self, phi_or_sn, cn=None, dn=None): '''The incomplete integral of the first kind in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: F(φ, k) as though φ ∈ (−π, π] (C{float}), U{defined<https://DLMF.NIST.gov/19.2.E4>}. @raise EllipticError: Invalid invokation or no convergence. ''' def _fF(sn, cn, dn): return abs(sn) * _RF(cn**2, dn**2, 1) return self._fXf(phi_or_sn, cn, dn, self.cK, self.deltaF, _fF) def fG(self, phi_or_sn, cn=None, dn=None): '''Legendre's geodesic longitude integral in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: G(φ, k) as though φ ∈ (−π, π] (C{float}). @raise EllipticError: Invalid invokation or no convergence. @note: Legendre expresses the longitude of a point on the geodesic in terms of this combination of elliptic integrals in U{Exercices de Calcul Intégral, Vol 1 (1811), p 181<https://Books. Google.com/books?id=riIOAAAAQAAJ&pg=PA181>}. @see: U{Geodesics in terms of elliptic integrals<https:// GeographicLib.SourceForge.io/html/geodesic.html#geodellip>} for the expression for the longitude in terms of this function. ''' return self._fXa(phi_or_sn, cn, dn, self.alpha2 - self.k2, self.cG, self.deltaG) def fH(self, phi_or_sn, cn=None, dn=None): '''Cayley's geodesic longitude difference integral in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: H(φ, k) as though φ ∈ (−π, π] (C{float}). @raise EllipticError: Invalid invokation or no convergence. @note: Cayley expresses the longitude difference of a point on the geodesic in terms of this combination of elliptic integrals in U{Phil. Mag. B{40} (1870), p 333 <https://Books.Google.com/books?id=Zk0wAAAAIAAJ&pg=PA333>}. @see: U{Geodesics in terms of elliptic integrals<https:// GeographicLib.SourceForge.io/html/geodesic.html#geodellip>} for the expression for the longitude in terms of this function. ''' return self._fXa(phi_or_sn, cn, dn, self.alphap2, self.cH, self.deltaH) def fPi(self, phi_or_sn, cn=None, dn=None): '''The incomplete integral of the third kind in terms of Jacobi elliptic functions. @arg phi_or_sn: φ or sin(φ). @kwarg cn: C{None} or cos(φ). @kwarg dn: C{None} or sqrt(1 − k2 sin(2φ)). @return: Π(φ, α2, k) as though φ ∈ (−π, π] (C{float}). @raise EllipticError: Invalid invokation or no convergence. ''' return self._fXa(phi_or_sn, cn, dn, self.alpha2, self.cPi, self.deltaPi) def _fXa(self, phi_or_sn, cn, dn, aX, cX, deltaX): '''(INTERNAL) Helper for C{.fG}, C{.fH} and C{.fPi}. ''' def _fX(sn, cn, dn): cn2, sn2, dn2 = cn**2, sn**2, dn**2 return abs(sn) * (_RF(cn2, dn2, 1) + aX * sn2 * _RJ_3(cn2, dn2, 1, cn2 + self.alphap2 * sn2)) return self._fXf(phi_or_sn, cn, dn, cX, deltaX, _fX) def _fXf(self, phi_or_sn, cn, dn, cX, deltaX, fX): '''(INTERNAL) Helper for C{f.D}, C{.fE}, C{.fF} and C{._fXa}. ''' phi = sn = phi_or_sn if cn is dn is None: # fX(phi) call sn, cn, dn = self._sncndn3(phi) if abs(phi) >= PI: return (deltaX(sn, cn, dn) + phi) * cX / PI_2 # fall through elif None in (cn, dn): t = self.classname + '.f' + deltaX.__name__[5:] raise EllipticError('%s invalid: %s%r' % ('invokation', t, (sn, cn, dn))) if cn > 0: # enforce usual trig-like symmetries xi = fX(sn, cn, dn) elif cn < 0: xi = 2 * cX - fX(sn, cn, dn) else: xi = cX return copysign(xi, sn) @property_RO def k2(self): '''Get k^2, the square of the modulus (C{float}). ''' return self._k2 @property_RO def kp2(self): '''Get k'^2, the square of the complementary modulus (C{float}). ''' return self._kp2 def reset(self, k2=0, alpha2=0, kp2=None, alphap2=None): # MCCABE 13 '''Reset the modulus and parameter. @kwarg k2: modulus squared (C{float}, 0 <= k^2<= 1). @kwarg alpha2: parameter (C{float}, 0 <= α^2 <= 1). @kwarg kp2: complementary modulus squared (C{float}, k'^2 >= 0). @kwarg alphap2: complementary parameter squared (C{float}, α'^2 >= 0). @raise EllipticError: Invalid B{C{k2}}, B{C{alpha2}}, B{C{kp2}} or B{C{alphap2}} or no convergence. @note: The arguments must satisfy I{B{C{k2}} + B{C{kp2}} = 1} and I{B{C{alpha2}} + B{C{alphap2}} = 1}. No checking is done that these conditions are met to enable accuracy to be maintained, e.g., when C{k} is very close to unity. ''' if 0 > k2 or k2 > 1: raise EllipticError('%s invalid: %r' % ('k2', k2)) self._k2 = k2 = float(k2) if 0 > alpha2 or alpha2 > 1: raise EllipticError('%s invalid: %r' % ('alpha2', alpha2)) self._alpha2 = alpha2 = float(alpha2) if kp2 is None: kp2 = 1 - k2 elif kp2 < 0: raise EllipticError('%s invalid: %r' % ('kp2', kp2)) self._kp2 = kp2 = float(kp2) if alphap2 is None: alphap2 = 1 - alpha2 elif alphap2 < 0: raise EllipticError('%s invalid: %r' % ('alphap2', alphap2)) self._alphap2 = alphap2 = float(alphap2) # Values of complete elliptic integrals for k = 0,1 and alpha = 0,1 # K E D # k = 0: pi/2 pi/2 pi/4 # k = 1: inf 1 inf # Pi G H # k = 0, alpha = 0: pi/2 pi/2 pi/4 # k = 1, alpha = 0: inf 1 1 # k = 0, alpha = 1: inf inf pi/2 # k = 1, alpha = 1: inf inf inf # # G(0, k) = Pi(0, k) = H(1, k) = E(k) # H(0, k) = K(k) - D(k) # Pi(alpha2, 0) = G(alpha2, 0) = pi / (2 * sqrt(1 - alpha2)) # H( alpha2, 0) = pi / (2 * (sqrt(1 - alpha2) + 1)) # Pi(alpha2, 1) = inf # G( alpha2, 1) = H(alpha2, 1) = RC(1, alphap2) if k2: if kp2: self._eps = k2 / (sqrt(kp2) + 1)**2 # D(k) = (K(k) - E(k))/k2, Carlson eq.4.3 # <https://DLMF.NIST.gov/19.25.E1> self._cD = _RD_3(0, kp2, 1) self._cKE = k2 * self.cD # Complete elliptic integral E(k), Carlson eq. 4.2 # <https://DLMF.NIST.gov/19.25.E1> self._cE = _RG_(kp2, 1) * 2 # Complete elliptic integral K(k), Carlson eq. 4.1 # <https://DLMF.NIST.gov/19.25.E1> self._cK = _RF_(kp2, 1) else: self._eps = k2 self._cD = self._cK = self._cKE = INF self._cE = 1.0 else: self._eps = EPS # delattr(self, '_eps') self._cD = PI_4 self._cE = self._cK = PI_2 self._cKE = 0.0 # k2 * self._cD if alpha2: if kp2: # <https://DLMF.NIST.gov/19.25.E2> if alphap2: rj_3 = _RJ_3(0, kp2, 1, alphap2) # G(alpha2, k) self._cG = self.cK + (alpha2 - k2) * rj_3 # H(alpha2, k) self._cH = self.cK - alphap2 * rj_3 # Pi(alpha2, k) self._cPi = self.cK + alpha2 * rj_3 else: self._cG = self._cH = self._cPi = INF # XXX or NAN? elif alphap2: self._cG = self._cH = _RC(1, alphap2) self._cPi = INF # XXX or NAN? else: self._cG = self.cE self._cPi = self.cK # cH = cK - cD but this involves large cancellations # if k2 is close to 1. So write (for alpha2 = 0) # cH = int(cos(phi)**2/sqrt(1-k2*sin(phi)**2),phi,0,pi/2) # = 1/sqrt(1-k2) * int(sin(phi)**2/sqrt(1-k2/kp2*sin(phi)**2,...) # = 1/kp * D(i*k/kp) # and use D(k) = RD(0, kp2, 1) / 3 # so cH = 1/kp * RD(0, 1/kp2, 1) / 3 # = kp2 * RD(0, 1, kp2) / 3 # using <https://DLMF.NIST.gov/19.20.E18> # Equivalently # RF(x, 1) - RD(0, x, 1)/3 = x * RD(0, 1, x)/3 for x > 0 # For k2 = 1 and alpha2 = 0, we have # cH = int(cos(phi),...) = 1 self._cH = kp2 * _RD_3(0, 1, kp2) if kp2 else 1.0 def sncndn(self, x): # PYCHOK x not used? '''The Jacobi elliptic function. @arg x: The argument (C{float}). @return: An L{Elliptic3Tuple}C{(sn, cn, dn)} with C{*n}C{(}B{C{x}}C{, k}C{)}. @raise EllipticError: No convergence. ''' # Bulirsch's sncndn routine, p 89. mc = self.kp2 if mc: # never negative ... if mc < 0: # PYCHOK no cover d = 1.0 - mc mc = -mc / d # /= -d chokes PyChecker d = sqrt(d) x *= d else: d = 0 a, c, mn = 1.0, 0, [] for _ in range(self._trips_): # GEOGRAPHICLIB_PANIC # This converges quadratically, max 6 trips mc = sqrt(mc) mn.append((a, mc)) c = (a + mc) * 0.5 if abs(a - mc) <= (_TolJAC * a): break mc *= a a = c else: raise _convergenceError(self.sncndn, x) x *= c dn = 1 sn, cn = sincos2(x) if sn: a = cn / sn c *= a while mn: m, n = mn.pop() a *= c c *= dn dn = (n + a) / (m + a) a = c / m sn = copysign(1.0 / hypot1(c), sn) cn = c * sn if d: # PYCHOK no cover cn, dn = dn, cn sn = sn / d # /= d chokes PyChecker else: sn = tanh(x) cn = dn = 1.0 / cosh(x) return Elliptic3Tuple(sn, cn, dn) def _sncndn3(self, phi): '''(INTERNAL) Helper for C{.fEinv} and C{._fXf}. ''' sn, cn = sincos2(phi) return Elliptic3Tuple(sn, cn, self.fDelta(sn, cn)) class Elliptic3Tuple(_NamedTuple): # .elliptic.py '''3-Tuple C{(sn, cn, dn)} all C{float}. ''' _Names_ = ('sn', 'cn', 'dn') def _convergenceError(where, *args): '''(INTERNAL) Return an L{EllipticError}. ''' return EllipticError('no %s%r convergence' % (where.__name__, args)) def _Horner(e0, e1, e2, e3, e4, e5): '''(INTERNAL) Horner form for C{_RD} and C{_RJ} below. ''' # Polynomial is <https://DLMF.NIST.gov/19.36.E2> # (1 - 3*E2/14 + E3/6 + 9*E2**2/88 - 3*E4/22 - 9*E2*E3/52 + 3*E5/26 # - E2**3/16 + 3*E3**2/40 + 3*E2*E4/20 + 45*E2**2*E3/272 # - 9*(E3*E4+E2*E5)/68) # converted to Horner form ... H = Fsum(471240, -540540 * e2) * e5 H += Fsum(612612 * e2, -540540 * e3, -556920) * e4 H += Fsum(306306 * e3, 675675 * e2**2, -706860 * e2, 680680) * e3 H += Fsum(417690 * e2, -255255 * e2**2, -875160) * e2 e = 4084080 * e1 return H.fsum_(4084080, e * e0) / e def _Q(A, T, tol): '''(INTERNAL) Helper for C{_RD}, C{_RF} and C{_RJ}. ''' return max(abs(A - t) for t in T[1:]) / tol def _RC(x, y): '''Degenerate symmetric integral of the first kind C{_RC}. @return: C{_RC(x, y) = _RF(x, y, y)}. @see: U{C{_RC} definition<https://DLMF.NIST.gov/19.2.E17>}. ''' # Defined only for y != 0 and x >= 0. d = x - y if d < 0: # catch _NaN # <https://DLMF.NIST.gov/19.2.E18> d = -d r = atan(sqrt(d / x)) if x > 0 else PI_2 elif d == 0: # d < EPS? r, d = 1.0, y elif y > 0: # <https://DLMF.NIST.gov/19.2.E19> r = asinh(sqrt(d / y)) # atanh(sqrt((x - y) / x)) elif y < 0: # <https://DLMF.NIST.gov/19.2.E20> r = asinh(sqrt(-x / y)) # atanh(sqrt(x / (x - y))) else: raise EllipticError('%s invalid: %s%r' % ('y', _RC.__name__, (x, y))) return r / sqrt(d) def _RD_3(x, y, z): '''Degenerate symmetric integral of the third kind C{_RD} / 3. ''' return _RD(x, y, z) / 3.0 def _RD(x, y, z): '''Degenerate symmetric integral of the third kind C{_RD}. @return: C{_RD(x, y, z) = _RJ(x, y, z, z)}. @see: U{C{_RD} definition<https://DLMF.NIST.gov/19.16.E5>}. ''' # Carlson, eqs 2.28 - 2.34 m = 1.0 S = Fsum() A = fsum_(x, y, z, z, z) * 0.2 T = (A, x, y, z) Q = _Q(A, T, _TolRD) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 7 trips break t = T[3] # z0 r, s, T = _rsT(T) S += 1.0 / (m * s[2] * (t + r)) m *= 4 else: raise _convergenceError(_RD, x, y, z) m *= T[0] # An x = (x - A) / m y = (y - A) / m z = (x + y) / 3.0 z2 = z**2 xy = x * y return _Horner(3 * S.fsum(), m * sqrt(T[0]), xy - 6 * z2, (xy * 3 - 8 * z2) * z, (xy - z2) * 3 * z2, xy * z2 * z) def _RF_(x, y): '''Symmetric integral of the first kind C{_RF}. @return: C{_RF(x, y)}. @see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>}. ''' # Carlson, eqs 2.36 - 2.38 a, b = sqrt(x), sqrt(y) if a < b: a, b = b, a while abs(a - b) > (_TolRG0 * a): # max 4 trips b, a = sqrt(a * b), (a + b) * 0.5 return PI / (a + b) def _RF(x, y, z): '''Symmetric integral of the first kind C{_RF}. @return: C{_RF(x, y, z)}. @see: U{C{_RF} definition<https://DLMF.NIST.gov/19.16.E1>}. ''' # Carlson, eqs 2.2 - 2.7 m = 1.0 A = fsum_(x, y, z) / 3.0 T = (A, x, y, z) Q = _Q(A, T, _TolRF) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 6 trips break _, _, T = _rsT(T) m *= 4 else: raise _convergenceError(_RF, x, y, z) m *= T[0] # An x = (A - x) / m y = (A - y) / m z = -(x + y) e2 = x * y - z**2 e3 = x * y * z # Polynomial is <https://DLMF.NIST.gov/19.36.E1> # (1 - E2/10 + E3/14 + E2**2/24 - 3*E2*E3/44 # - 5*E2**3/208 + 3*E3**2/104 + E2**2*E3/16) # converted to Horner form ... H = Fsum( 6930 * e3, 15015 * e2**2, -16380 * e2, 17160) * e3 H += Fsum(10010 * e2, -5775 * e2**2, -24024) * e2 return H.fsum_(240240) / (240240 * sqrt(T[0])) def _RG_(x, y): '''Symmetric integral of the second kind C{_RG}. @return: C{_RG(x, y)}. @see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and in Carlson, eq. 1.5. ''' # Carlson, eqs 2.36 - 2.39 a, b = sqrt(x), sqrt(y) if a < b: a, b = b, a m = 0.25 S = Fsum(m * (a + b)**2) while abs(a - b) > (_TolRG0 * a): # max 4 trips b, a = sqrt(a * b), (a + b) * 0.5 m *= 2 S -= m * (a - b)**2 return S.fsum() * PI_2 / (a + b) def _RG(x, y, z): # used by testElleiptic.py '''Symmetric integral of the second kind C{_RG}. @return: C{_RG(x, y, z)}. @see: U{C{_RG} definition<https://DLMF.NIST.gov/19.16.E3>} and in Carlson, eq. 1.5. ''' if not z: y, z = z, y # Carlson, eq 1.7 return fsum_(_RF(x, y, z) * z, _RD_3(x, y, z) * (x - z) * (z - y), sqrt(x * y / z)) * 0.5 def _RJ_3(x, y, z, p): '''Symmetric integral of the third kind C{_RJ} / 3. ''' return _RJ(x, y, z, p) / 3.0 def _RJ(x, y, z, p): '''Symmetric integral of the third kind C{_RJ}. @return: C{_RJ(x, y, z, p)}. @see: U{C{_RJ} definition<https://DLMF.NIST.gov/19.16.E2>}. ''' def _xyzp(x, y, z, p): return (x + p) * (y + p) * (z + p) # Carlson, eqs 2.17 - 2.25 m = m3 = 1.0 S = Fsum() D = -_xyzp(x, y, z, -p) A = fsum_(x, y, z, 2 * p) * 0.2 T = (A, x, y, z, p) Q = _Q(A, T, _TolRD) for _ in range(_TRIPS): if Q < abs(m * T[0]): # max 7 trips break _, s, T = _rsT(T) d = _xyzp(*s) e = D / (m3 * d**2) S += _RC(1, 1 + e) / (m * d) m *= 4 m3 *= 64 else: raise _convergenceError(_RJ, x, y, z, p) m *= T[0] # An x = (A - x) / m y = (A - y) / m z = (A - z) / m xyz = x * y * z p = -(x + y + z) * 0.5 p2 = p**2 e2 = fsum_(x * y, x * z, y * z, -3 * p2) return _Horner(6 * S.fsum(), m * sqrt(T[0]), e2, fsum_(xyz, 2 * p * e2, 4 * p * p2), fsum_(xyz * 2, p * e2, 3 * p * p2) * p, p2 * xyz) def _rsT(T): '''(INTERNAL) Helper for C{_RD}, C{_RF} and C{_RJ}. ''' s = map2(sqrt, T[1:]) r = fdot(s[:3], s[1], s[2], s[0]) T = tuple((t + r) * 0.25 for t in T) return r, s, T # **) MIT License # # Copyright (C) 2016-2020 -- mrJean1 at Gmail -- All Rights Reserved. # # Permission is hereby granted, free of charge, to any person obtaining a # copy of this software and associated documentation files (the "Software"), # to deal in the Software without restriction, including without limitation # the rights to use, copy, modify, merge, publish, distribute, sublicense, # and/or sell copies of the Software, and to permit persons to whom the # Software is furnished to do so, subject to the following conditions: # # The above copyright notice and this permission notice shall be included # in all copies or substantial portions of the Software. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS # OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, # FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL # THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE.