ceph/src/boost/boost/geometry/util/series_expansion.hpp

   1 // Boost.Geometry
   2
   3 // Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.
   4
   5 // Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.
   6
   7 // This file was modified by Oracle on 2019.
   8 // Modifications copyright (c) 2019 Oracle and/or its affiliates.
   9
  10 // Contributed and/or modified by Adam Wulkiewicz, on behalf of Oracle
  11
  12 // Use, modification and distribution is subject to the Boost Software License,
  13 // Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
  14 // http://www.boost.org/LICENSE_1_0.txt)
  15
  16 // This file is converted from GeographicLib, https://geographiclib.sourceforge.io
  17 // GeographicLib is originally written by Charles Karney.
  18
  19 // Author: Charles Karney (2008-2017)
  20
  21 // Last updated version of GeographicLib: 1.49
  22
  23 // Original copyright notice:
  24
  25 // Copyright (c) Charles Karney (2008-2017) <charles@karney.com> and licensed
  26 // under the MIT/X11 License. For more information, see
  27 // https://geographiclib.sourceforge.io
  28
  29 #ifndef BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP
  30 #define BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP
  31
  32 #include <boost/geometry/core/assert.hpp>
  33 #include <boost/geometry/util/math.hpp>
  34
  35 namespace boost { namespace geometry { namespace series_expansion {
  36
  37     /*
  38      Generate and evaluate the series expansion of the following integral
  39
  40      I1 = integrate( sqrt(1+k2*sin(sigma1)^2), sigma1, 0, sigma )
  41
  42      which is valid for k2 small. We substitute k2 = 4 * eps / (1 - eps)^2
  43      and expand (1 - eps) * I1 retaining terms up to order eps^maxpow
  44      in A1 and C1[l].
  45
  46      The resulting series is of the form
  47
  48      A1 * ( sigma + sum(C1[l] * sin(2*l*sigma), l, 1, maxpow) ).
  49
  50      The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
  51
  52      The expansion above is performed in Maxima, a Computer Algebra System.
  53      The C++ code (that yields the function evaluate_A1 below) is
  54      generated by the following Maxima script:
  55      geometry/doc/other/maxima/geod.mac
  56
  57      To replace each number x by CT(x) the following
  58      script can be used:
  59        sed -e 's/[0-9]\+/CT(&)/g; s/\[CT/\[/g; s/)\]/\]/g;
  60                s/case\sCT(/case /g; s/):/:/g; s/epsCT(2)/eps2/g;'
  61     */
  62     template <size_t SeriesOrder, typename CT>
  63     inline CT evaluate_A1(CT eps)
  64     {
  65         CT eps2 = math::sqr(eps);
  66         CT t;
  67         switch (SeriesOrder/2) {
  68         case 0:
  69             t = CT(0);
  70             break;
  71         case 1:
  72             t = eps2/CT(4);
  73             break;
  74         case 2:
  75             t = eps2*(eps2+CT(16))/CT(64);
  76             break;
  77         case 3:
  78             t = eps2*(eps2*(eps2+CT(4))+CT(64))/CT(256);
  79             break;
  80         case 4:
  81             t = eps2*(eps2*(eps2*(CT(25)*eps2+CT(64))+CT(256))+CT(4096))/CT(16384);
  82             break;
  83         }
  84         return (t + eps) / (CT(1) - eps);
  85     }
  86
  87     /*
  88      Generate and evaluate the series expansion of the following integral
  89
  90      I2 = integrate( 1/sqrt(1+k2*sin(sigma1)^2), sigma1, 0, sigma )
  91
  92      which is valid for k2 small. We substitute k2 = 4 * eps / (1 - eps)^2
  93      and expand (1 - eps) * I2 retaining terms up to order eps^maxpow
  94      in A2 and C2[l].
  95
  96      The resulting series is of the form
  97
  98      A2 * ( sigma + sum(C2[l] * sin(2*l*sigma), l, 1, maxpow) )
  99
 100      The scale factor A2-1 = mean value of (d/dsigma)2 - 1
 101
 102      The expansion above is performed in Maxima, a Computer Algebra System.
 103      The C++ code (that yields the function evaluate_A2 below) is
 104      generated by the following Maxima script:
 105      geometry/doc/other/maxima/geod.mac
 106     */
 107     template <size_t SeriesOrder, typename CT>
 108     inline CT evaluate_A2(CT const& eps)
 109     {
 110         CT const eps2 = math::sqr(eps);
 111         CT t;
 112         switch (SeriesOrder/2) {
 113         case 0:
 114             t = CT(0);
 115             break;
 116         case 1:
 117             t = -CT(3)*eps2/CT(4);
 118             break;
 119         case 2:
 120             t = (-CT(7)*eps2-CT(48))*eps2/CT(64);
 121             break;
 122         case 3:
 123             t = eps2*((-CT(11)*eps2-CT(28))*eps2-CT(192))/CT(256);
 124             break;
 125         default:
 126             t = eps2*(eps2*((-CT(375)*eps2-CT(704))*eps2-CT(1792))-CT(12288))/CT(16384);
 127             break;
 128         }
 129         return (t - eps) / (CT(1) + eps);
 130     }
 131
 132     /*
 133      Express
 134
 135         I3 = integrate( (2-f)/(1+(1-f)*sqrt(1+k2*sin(sigma1)^2)), sigma1, 0, sigma )
 136
 137      as a series
 138
 139         A3 * ( sigma + sum(C3[l] * sin(2*l*sigma), l, 1, maxpow-1) )
 140
 141      valid for f and k2 small.  It is convenient to write k2 = 4 * eps / (1 -
 142      eps)^2 and f = 2*n/(1+n) and expand in eps and n.  This procedure leads
 143      to a series where the coefficients of eps^j are terminating series in n.
 144
 145      The scale factor A3 = mean value of (d/dsigma)I3
 146
 147      The expansion above is performed in Maxima, a Computer Algebra System.
 148      The C++ code (that yields the function evaluate_coeffs_A3 below) is
 149      generated by the following Maxima script:
 150      geometry/doc/other/maxima/geod.mac
 151     */
 152     template <typename Coeffs, typename CT>
 153     inline void evaluate_coeffs_A3(Coeffs &c, CT const& n)
 154     {
 155         switch (int(Coeffs::static_size)) {
 156         case 0:
 157             break;
 158         case 1:
 159             c[0] = CT(1);
 160             break;
 161         case 2:
 162             c[0] = CT(1);
 163             c[1] = -CT(1)/CT(2);
 164             break;
 165         case 3:
 166             c[0] = CT(1);
 167             c[1] = (n-CT(1))/CT(2);
 168             c[2] = -CT(1)/CT(4);
 169             break;
 170         case 4:
 171             c[0] = CT(1);
 172             c[1] = (n-CT(1))/CT(2);
 173             c[2] = (-n-CT(2))/CT(8);
 174             c[3] = -CT(1)/CT(16);
 175             break;
 176         case 5:
 177             c[0] = CT(1);
 178             c[1] = (n-CT(1))/CT(2);
 179             c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
 180             c[3] = (-CT(3)*n-CT(1))/CT(16);
 181             c[4] = -CT(3)/CT(64);
 182             break;
 183         case 6:
 184             c[0] = CT(1);
 185             c[1] = (n-CT(1))/CT(2);
 186             c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
 187             c[3] = ((-n-CT(3))*n-CT(1))/CT(16);
 188             c[4] = (-CT(2)*n-CT(3))/CT(64);
 189             c[5] = -CT(3)/CT(128);
 190             break;
 191         case 7:
 192             c[0] = CT(1);
 193             c[1] = (n-CT(1))/CT(2);
 194             c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
 195             c[3] = (n*(n*(CT(5)*n-CT(1))-CT(3))-CT(1))/CT(16);
 196             c[4] = ((-CT(10)*n-CT(2))*n-CT(3))/CT(64);
 197             c[5] = (-CT(5)*n-CT(3))/CT(128);
 198             c[6] = -CT(5)/CT(256);
 199             break;
 200         default:
 201             c[0] = CT(1);
 202             c[1] = (n-CT(1))/CT(2);
 203             c[2] = (n*(CT(3)*n-CT(1))-CT(2))/CT(8);
 204             c[3] = (n*(n*(CT(5)*n-CT(1))-CT(3))-CT(1))/CT(16);
 205             c[4] = (n*((-CT(5)*n-CT(20))*n-CT(4))-CT(6))/CT(128);
 206             c[5] = ((-CT(5)*n-CT(10))*n-CT(6))/CT(256);
 207             c[6] = (-CT(15)*n-CT(20))/CT(1024);
 208             c[7] = -CT(25)/CT(2048);
 209             break;
 210         }
 211     }
 212
 213     /*
 214      The coefficients C1[l] in the Fourier expansion of B1.
 215
 216      The expansion below is performed in Maxima, a Computer Algebra System.
 217      The C++ code (that yields the function evaluate_coeffs_C1 below) is
 218      generated by the following Maxima script:
 219      geometry/doc/other/maxima/geod.mac
 220     */
 221     template <typename Coeffs, typename CT>
 222     inline void evaluate_coeffs_C1(Coeffs &c, CT const& eps)
 223     {
 224         CT eps2 = math::sqr(eps);
 225         CT d = eps;
 226         switch (int(Coeffs::static_size) - 1) {
 227         case 0:
 228             break;
 229         case 1:
 230             c[1] = -d/CT(2);
 231             break;
 232         case 2:
 233             c[1] = -d/CT(2);
 234             d *= eps;
 235             c[2] = -d/CT(16);
 236             break;
 237         case 3:
 238             c[1] = d*(CT(3)*eps2-CT(8))/CT(16);
 239             d *= eps;
 240             c[2] = -d/CT(16);
 241             d *= eps;
 242             c[3] = -d/CT(48);
 243             break;
 244         case 4:
 245             c[1] = d*(CT(3)*eps2-CT(8))/CT(16);
 246             d *= eps;
 247             c[2] = d*(eps2-CT(2))/CT(32);
 248             d *= eps;
 249             c[3] = -d/CT(48);
 250             d *= eps;
 251             c[4] = -CT(5)*d/CT(512);
 252             break;
 253         case 5:
 254             c[1] = d*((CT(6)-eps2)*eps2-CT(16))/CT(32);
 255             d *= eps;
 256             c[2] = d*(eps2-CT(2))/CT(32);
 257             d *= eps;
 258             c[3] = d*(CT(9)*eps2-CT(16))/CT(768);
 259             d *= eps;
 260             c[4] = -CT(5)*d/CT(512);
 261             d *= eps;
 262             c[5] = -CT(7)*d/CT(1280);
 263             break;
 264         case 6:
 265             c[1] = d*((CT(6)-eps2)*eps2-CT(16))/CT(32);
 266             d *= eps;
 267             c[2] = d*((CT(64)-CT(9)*eps2)*eps2-CT(128))/CT(2048);
 268             d *= eps;
 269             c[3] = d*(CT(9)*eps2-CT(16))/CT(768);
 270             d *= eps;
 271             c[4] = d*(CT(3)*eps2-CT(5))/CT(512);
 272             d *= eps;
 273             c[5] = -CT(7)*d/CT(1280);
 274             d *= eps;
 275             c[6] = -CT(7)*d/CT(2048);
 276             break;
 277         case 7:
 278             c[1] = d*(eps2*(eps2*(CT(19)*eps2-CT(64))+CT(384))-CT(1024))/CT(2048);
 279             d *= eps;
 280             c[2] = d*((CT(64)-CT(9)*eps2)*eps2-CT(128))/CT(2048);
 281             d *= eps;
 282             c[3] = d*((CT(72)-CT(9)*eps2)*eps2-CT(128))/CT(6144);
 283             d *= eps;
 284             c[4] = d*(CT(3)*eps2-CT(5))/CT(512);
 285             d *= eps;
 286             c[5] = d*(CT(35)*eps2-CT(56))/CT(10240);
 287             d *= eps;
 288             c[6] = -CT(7)*d/CT(2048);
 289             d *= eps;
 290             c[7] = -CT(33)*d/CT(14336);
 291             break;
 292         default:
 293             c[1] = d*(eps2*(eps2*(CT(19)*eps2-CT(64))+CT(384))-CT(1024))/CT(2048);
 294             d *= eps;
 295             c[2] = d*(eps2*(eps2*(CT(7)*eps2-CT(18))+CT(128))-CT(256))/CT(4096);
 296             d *= eps;
 297             c[3] = d*((CT(72)-CT(9)*eps2)*eps2-CT(128))/CT(6144);
 298             d *= eps;
 299             c[4] = d*((CT(96)-CT(11)*eps2)*eps2-CT(160))/CT(16384);
 300             d *= eps;
 301             c[5] = d*(CT(35)*eps2-CT(56))/CT(10240);
 302             d *= eps;
 303             c[6] = d*(CT(9)*eps2-CT(14))/CT(4096);
 304             d *= eps;
 305             c[7] = -CT(33)*d/CT(14336);
 306             d *= eps;
 307             c[8] = -CT(429)*d/CT(262144);
 308             break;
 309         }
 310     }
 311
 312     /*
 313      The coefficients C1p[l] in the Fourier expansion of B1p.
 314
 315      The expansion below is performed in Maxima, a Computer Algebra System.
 316      The C++ code (that yields the function evaluate_coeffs_C1p below) is
 317      generated by the following Maxima script:
 318      geometry/doc/other/maxima/geod.mac
 319     */
 320     template <typename Coeffs, typename CT>
 321     inline void evaluate_coeffs_C1p(Coeffs& c, CT const& eps)
 322     {
 323         CT const eps2 = math::sqr(eps);
 324         CT d = eps;
 325         switch (int(Coeffs::static_size) - 1) {
 326         case 0:
 327             break;
 328         case 1:
 329             c[1] = d/CT(2);
 330             break;
 331         case 2:
 332             c[1] = d/CT(2);
 333             d *= eps;
 334             c[2] = CT(5)*d/CT(16);
 335             break;
 336         case 3:
 337             c[1] = d*(CT(16)-CT(9)*eps2)/CT(32);
 338             d *= eps;
 339             c[2] = CT(5)*d/CT(16);
 340             d *= eps;
 341             c[3] = CT(29)*d/CT(96);
 342             break;
 343         case 4:
 344             c[1] = d*(CT(16)-CT(9)*eps2)/CT(32);
 345             d *= eps;
 346             c[2] = d*(CT(30)-CT(37)*eps2)/CT(96);
 347             d *= eps;
 348             c[3] = CT(29)*d/CT(96);
 349             d *= eps;
 350             c[4] = CT(539)*d/CT(1536);
 351             break;
 352         case 5:
 353             c[1] = d*(eps2*(CT(205)*eps2-CT(432))+CT(768))/CT(1536);
 354             d *= eps;
 355             c[2] = d*(CT(30)-CT(37)*eps2)/CT(96);
 356             d *= eps;
 357             c[3] = d*(CT(116)-CT(225)*eps2)/CT(384);
 358             d *= eps;
 359             c[4] = CT(539)*d/CT(1536);
 360             d *= eps;
 361             c[5] = CT(3467)*d/CT(7680);
 362             break;
 363         case 6:
 364             c[1] = d*(eps2*(CT(205)*eps2-CT(432))+CT(768))/CT(1536);
 365             d *= eps;
 366             c[2] = d*(eps2*(CT(4005)*eps2-CT(4736))+CT(3840))/CT(12288);
 367             d *= eps;
 368             c[3] = d*(CT(116)-CT(225)*eps2)/CT(384);
 369             d *= eps;
 370             c[4] = d*(CT(2695)-CT(7173)*eps2)/CT(7680);
 371             d *= eps;
 372             c[5] = CT(3467)*d/CT(7680);
 373             d *= eps;
 374             c[6] = CT(38081)*d/CT(61440);
 375             break;
 376         case 7:
 377             c[1] = d*(eps2*((CT(9840)-CT(4879)*eps2)*eps2-CT(20736))+CT(36864))/CT(73728);
 378             d *= eps;
 379             c[2] = d*(eps2*(CT(4005)*eps2-CT(4736))+CT(3840))/CT(12288);
 380             d *= eps;
 381             c[3] = d*(eps2*(CT(8703)*eps2-CT(7200))+CT(3712))/CT(12288);
 382             d *= eps;
 383             c[4] = d*(CT(2695)-CT(7173)*eps2)/CT(7680);
 384             d *= eps;
 385             c[5] = d*(CT(41604)-CT(141115)*eps2)/CT(92160);
 386             d *= eps;
 387             c[6] = CT(38081)*d/CT(61440);
 388             d *= eps;
 389             c[7] = CT(459485)*d/CT(516096);
 390             break;
 391         default:
 392             c[1] = d*(eps2*((CT(9840)-CT(4879)*eps2)*eps2-CT(20736))+CT(36864))/CT(73728);
 393             d *= eps;
 394             c[2] = d*(eps2*((CT(120150)-CT(86171)*eps2)*eps2-CT(142080))+CT(115200))/CT(368640);
 395             d *= eps;
 396             c[3] = d*(eps2*(CT(8703)*eps2-CT(7200))+CT(3712))/CT(12288);
 397             d *= eps;
 398             c[4] = d*(eps2*(CT(1082857)*eps2-CT(688608))+CT(258720))/CT(737280);
 399             d *= eps;
 400             c[5] = d*(CT(41604)-CT(141115)*eps2)/CT(92160);
 401             d *= eps;
 402             c[6] = d*(CT(533134)-CT(2200311)*eps2)/CT(860160);
 403             d *= eps;
 404             c[7] = CT(459485)*d/CT(516096);
 405             d *= eps;
 406             c[8] = CT(109167851)*d/CT(82575360);
 407             break;
 408         }
 409     }
 410
 411     /*
 412      The coefficients C2[l] in the Fourier expansion of B2.
 413
 414      The expansion below is performed in Maxima, a Computer Algebra System.
 415      The C++ code (that yields the function evaluate_coeffs_C2 below) is
 416      generated by the following Maxima script:
 417      geometry/doc/other/maxima/geod.mac
 418     */
 419     template <typename Coeffs, typename CT>
 420     inline void evaluate_coeffs_C2(Coeffs& c, CT const& eps)
 421     {
 422         CT const eps2 = math::sqr(eps);
 423         CT d = eps;
 424         switch (int(Coeffs::static_size) - 1) {
 425         case 0:
 426             break;
 427         case 1:
 428             c[1] = d/CT(2);
 429             break;
 430         case 2:
 431             c[1] = d/CT(2);
 432             d *= eps;
 433             c[2] = CT(3)*d/CT(16);
 434             break;
 435         case 3:
 436             c[1] = d*(eps2+CT(8))/CT(16);
 437             d *= eps;
 438             c[2] = CT(3)*d/CT(16);
 439             d *= eps;
 440             c[3] = CT(5)*d/CT(48);
 441             break;
 442         case 4:
 443             c[1] = d*(eps2+CT(8))/CT(16);
 444             d *= eps;
 445             c[2] = d*(eps2+CT(6))/CT(32);
 446             d *= eps;
 447             c[3] = CT(5)*d/CT(48);
 448             d *= eps;
 449             c[4] = CT(35)*d/CT(512);
 450             break;
 451         case 5:
 452             c[1] = d*(eps2*(eps2+CT(2))+CT(16))/CT(32);
 453             d *= eps;
 454             c[2] = d*(eps2+CT(6))/CT(32);
 455             d *= eps;
 456             c[3] = d*(CT(15)*eps2+CT(80))/CT(768);
 457             d *= eps;
 458             c[4] = CT(35)*d/CT(512);
 459             d *= eps;
 460             c[5] = CT(63)*d/CT(1280);
 461             break;
 462         case 6:
 463             c[1] = d*(eps2*(eps2+CT(2))+CT(16))/CT(32);
 464             d *= eps;
 465             c[2] = d*(eps2*(CT(35)*eps2+CT(64))+CT(384))/CT(2048);
 466             d *= eps;
 467             c[3] = d*(CT(15)*eps2+CT(80))/CT(768);
 468             d *= eps;
 469             c[4] = d*(CT(7)*eps2+CT(35))/CT(512);
 470             d *= eps;
 471             c[5] = CT(63)*d/CT(1280);
 472             d *= eps;
 473             c[6] = CT(77)*d/CT(2048);
 474             break;
 475         case 7:
 476             c[1] = d*(eps2*(eps2*(CT(41)*eps2+CT(64))+CT(128))+CT(1024))/CT(2048);
 477             d *= eps;
 478             c[2] = d*(eps2*(CT(35)*eps2+CT(64))+CT(384))/CT(2048);
 479             d *= eps;
 480             c[3] = d*(eps2*(CT(69)*eps2+CT(120))+CT(640))/CT(6144);
 481             d *= eps;
 482             c[4] = d*(CT(7)*eps2+CT(35))/CT(512);
 483             d *= eps;
 484             c[5] = d*(CT(105)*eps2+CT(504))/CT(10240);
 485             d *= eps;
 486             c[6] = CT(77)*d/CT(2048);
 487             d *= eps;
 488             c[7] = CT(429)*d/CT(14336);
 489             break;
 490         default:
 491             c[1] = d*(eps2*(eps2*(CT(41)*eps2+CT(64))+CT(128))+CT(1024))/CT(2048);
 492             d *= eps;
 493             c[2] = d*(eps2*(eps2*(CT(47)*eps2+CT(70))+CT(128))+CT(768))/CT(4096);
 494             d *= eps;
 495             c[3] = d*(eps2*(CT(69)*eps2+CT(120))+CT(640))/CT(6144);
 496             d *= eps;
 497             c[4] = d*(eps2*(CT(133)*eps2+CT(224))+CT(1120))/CT(16384);
 498             d *= eps;
 499             c[5] = d*(CT(105)*eps2+CT(504))/CT(10240);
 500             d *= eps;
 501             c[6] = d*(CT(33)*eps2+CT(154))/CT(4096);
 502             d *= eps;
 503             c[7] = CT(429)*d/CT(14336);
 504             d *= eps;
 505             c[8] = CT(6435)*d/CT(262144);
 506             break;
 507         }
 508     }
 509
 510     /*
 511      The coefficients C3[l] in the Fourier expansion of B3.
 512
 513      The expansion below is performed in Maxima, a Computer Algebra System.
 514      The C++ code (that yields the function evaluate_coeffs_C3 below) is
 515      generated by the following Maxima script:
 516      geometry/doc/other/maxima/geod.mac
 517     */
 518     template <size_t SeriesOrder, typename Coeffs, typename CT>
 519     inline void evaluate_coeffs_C3x(Coeffs &c, CT const& n) {
 520         size_t const coeff_size = Coeffs::static_size;
 521         size_t const expected_size = (SeriesOrder * (SeriesOrder - 1)) / 2;
 522         BOOST_GEOMETRY_ASSERT((coeff_size == expected_size));
 523
 524         const CT n2 = math::sqr(n);
 525         switch (SeriesOrder) {
 526         case 0:
 527             break;
 528         case 1:
 529             break;
 530         case 2:
 531             c[0] = (CT(1)-n)/CT(4);
 532             break;
 533         case 3:
 534             c[0] = (CT(1)-n)/CT(4);
 535             c[1] = (CT(1)-n2)/CT(8);
 536             c[2] = ((n-CT(3))*n+CT(2))/CT(32);
 537             break;
 538         case 4:
 539             c[0] = (CT(1)-n)/CT(4);
 540             c[1] = (CT(1)-n2)/CT(8);
 541             c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
 542             c[3] = ((n-CT(3))*n+CT(2))/CT(32);
 543             c[4] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
 544             c[5] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
 545             break;
 546         case 5:
 547             c[0] = (CT(1)-n)/CT(4);
 548             c[1] = (CT(1)-n2)/CT(8);
 549             c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
 550             c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
 551             c[4] = ((n-CT(3))*n+CT(2))/CT(32);
 552             c[5] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
 553             c[6] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
 554             c[7] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
 555             c[8] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
 556             c[9] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
 557             break;
 558         case 6:
 559             c[0] = (CT(1)-n)/CT(4);
 560             c[1] = (CT(1)-n2)/CT(8);
 561             c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
 562             c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
 563             c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
 564             c[5] = ((n-CT(3))*n+CT(2))/CT(32);
 565             c[6] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
 566             c[7] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
 567             c[8] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
 568             c[9] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
 569             c[10] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
 570             c[11] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
 571             c[12] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
 572             c[13] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
 573             c[14] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
 574             break;
 575         case 7:
 576             c[0] = (CT(1)-n)/CT(4);
 577             c[1] = (CT(1)-n2)/CT(8);
 578             c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
 579             c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
 580             c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
 581             c[5] = (CT(10)*n+CT(21))/CT(1024);
 582             c[6] = ((n-CT(3))*n+CT(2))/CT(32);
 583             c[7] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
 584             c[8] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
 585             c[9] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
 586             c[10] = (CT(69)*n+CT(108))/CT(8192);
 587             c[11] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
 588             c[12] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
 589             c[13] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
 590             c[14] = (CT(12)-n)/CT(1024);
 591             c[15] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
 592             c[16] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
 593             c[17] = (CT(72)-CT(43)*n)/CT(8192);
 594             c[18] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
 595             c[19] = (CT(9)-CT(15)*n)/CT(1024);
 596             c[20] = (CT(44)-CT(99)*n)/CT(8192);
 597             break;
 598         default:
 599             c[0] = (CT(1)-n)/CT(4);
 600             c[1] = (CT(1)-n2)/CT(8);
 601             c[2] = (n*((-CT(5)*n-CT(1))*n+CT(3))+CT(3))/CT(64);
 602             c[3] = (n*((CT(2)-CT(2)*n)*n+CT(2))+CT(5))/CT(128);
 603             c[4] = (n*(CT(3)*n+CT(11))+CT(12))/CT(512);
 604             c[5] = (CT(10)*n+CT(21))/CT(1024);
 605             c[6] = CT(243)/CT(16384);
 606             c[7] = ((n-CT(3))*n+CT(2))/CT(32);
 607             c[8] = (n*(n*(CT(2)*n-CT(3))-CT(2))+CT(3))/CT(64);
 608             c[9] = (n*((-CT(6)*n-CT(9))*n+CT(2))+CT(6))/CT(256);
 609             c[10] = ((CT(1)-CT(2)*n)*n+CT(5))/CT(256);
 610             c[11] = (CT(69)*n+CT(108))/CT(8192);
 611             c[12] = CT(187)/CT(16384);
 612             c[13] = (n*((CT(5)-n)*n-CT(9))+CT(5))/CT(192);
 613             c[14] = (n*(n*(CT(10)*n-CT(6))-CT(10))+CT(9))/CT(384);
 614             c[15] = ((-CT(77)*n-CT(8))*n+CT(42))/CT(3072);
 615             c[16] = (CT(12)-n)/CT(1024);
 616             c[17] = CT(139)/CT(16384);
 617             c[18] = (n*((CT(20)-CT(7)*n)*n-CT(28))+CT(14))/CT(1024);
 618             c[19] = ((-CT(7)*n-CT(40))*n+CT(28))/CT(2048);
 619             c[20] = (CT(72)-CT(43)*n)/CT(8192);
 620             c[21] = CT(127)/CT(16384);
 621             c[22] = (n*(CT(75)*n-CT(90))+CT(42))/CT(5120);
 622             c[23] = (CT(9)-CT(15)*n)/CT(1024);
 623             c[24] = CT(99)/CT(16384);
 624             c[25] = (CT(44)-CT(99)*n)/CT(8192);
 625             c[26] = CT(99)/CT(16384);
 626             c[27] = CT(429)/CT(114688);
 627             break;
 628         }
 629     }
 630
 631     /*
 632     \brief Given the set of coefficients coeffs2[] evaluate on
 633       C3 and return the set of coefficients coeffs1[].
 634
 635       Elements coeffs1[1] through coeffs1[SeriesOrder - 1] are set.
 636     */
 637     template <typename Coeffs1, typename Coeffs2, typename CT>
 638     inline void evaluate_coeffs_C3(Coeffs1 &coeffs1, Coeffs2 &coeffs2, CT const& eps)
 639     {
 640         CT mult = 1;
 641         size_t offset = 0;
 642
 643         // i is the index of C3[i].
 644         for (size_t i = 1; i < Coeffs1::static_size; ++i)
 645         {
 646             // Order of polynomial in eps.
 647             size_t m = Coeffs1::static_size - i;
 648             mult *= eps;
 649
 650             coeffs1[i] = mult * math::horner_evaluate(eps, coeffs2.begin() + offset,
 651                                                            coeffs2.begin() + offset + m);
 652
 653             offset += m;
 654         }
 655         // Post condition: offset == coeffs_C3_size
 656     }
 657
 658     /*
 659     \brief Evaluate the following:
 660
 661      y = sum(c[i] * sin(2*i * x), i, 1, n)
 662
 663      using Clenshaw summation.
 664     */
 665     template <typename CT, typename Coeffs>
 666     inline CT sin_cos_series(CT const& sinx, CT const& cosx, Coeffs const& coeffs)
 667     {
 668         size_t n = Coeffs::static_size - 1;
 669         size_t index = 0;
 670
 671         // Point to one beyond last element.
 672         index += (n + 1);
 673         CT ar = 2 * (cosx - sinx) * (cosx + sinx);
 674
 675         // If n is odd, get the last element.
 676         CT k0 = n & 1 ? coeffs[--index] : 0;
 677         CT k1 = 0;
 678
 679         // Make n even.
 680         n /= 2;
 681         while (n--) {
 682             // Unroll loop x 2, so accumulators return to their original role.
 683             k1 = ar * k0 - k1 + coeffs[--index];
 684             k0 = ar * k1 - k0 + coeffs[--index];
 685         }
 686
 687         return 2 * sinx * cosx * k0;
 688     }
 689
 690     /*
 691      The coefficient containers for the series expansions.
 692      These structs allow the caller to only know the series order.
 693     */
 694     template <size_t SeriesOrder, typename CT>
 695     struct coeffs_C1 : boost::array<CT, SeriesOrder + 1>
 696     {
 697         coeffs_C1(CT const& epsilon)
 698         {
 699             evaluate_coeffs_C1(*this, epsilon);
 700         }
 701     };
 702
 703     template <size_t SeriesOrder, typename CT>
 704     struct coeffs_C1p : boost::array<CT, SeriesOrder + 1>
 705     {
 706         coeffs_C1p(CT const& epsilon)
 707         {
 708             evaluate_coeffs_C1p(*this, epsilon);
 709         }
 710     };
 711
 712     template <size_t SeriesOrder, typename CT>
 713     struct coeffs_C2 : boost::array<CT, SeriesOrder + 1>
 714     {
 715         coeffs_C2(CT const& epsilon)
 716         {
 717             evaluate_coeffs_C2(*this, epsilon);
 718         }
 719     };
 720
 721     template <size_t SeriesOrder, typename CT>
 722     struct coeffs_C3x : boost::array<CT, (SeriesOrder * (SeriesOrder - 1)) / 2>
 723     {
 724         coeffs_C3x(CT const& n)
 725         {
 726             evaluate_coeffs_C3x<SeriesOrder>(*this, n);
 727         }
 728     };
 729
 730     template <size_t SeriesOrder, typename CT>
 731     struct coeffs_C3 : boost::array<CT, SeriesOrder>
 732     {
 733         coeffs_C3(CT const& n, CT const& epsilon)
 734         {
 735             coeffs_C3x<SeriesOrder, CT> coeffs_C3x(n);
 736
 737             evaluate_coeffs_C3(*this, coeffs_C3x, epsilon);
 738         }
 739     };
 740
 741     template <size_t SeriesOrder, typename CT>
 742     struct coeffs_A3 : boost::array<CT, SeriesOrder>
 743     {
 744         coeffs_A3(CT const& n)
 745         {
 746             evaluate_coeffs_A3(*this, n);
 747         }
 748     };
 749
 750 }}} // namespace boost::geometry::series_expansion
 751
 752 #endif // BOOST_GEOMETRY_UTIL_SERIES_EXPANSION_HPP