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   4 <title>Comparison of Elliptic Integral Root Finding Algoritghms</title>
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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.root_comparison.elliptic_comparison"></a><a class="link" href="elliptic_comparison.html" title="Comparison of Elliptic Integral Root Finding Algoritghms">Comparison
  28       of Elliptic Integral Root Finding Algoritghms</a>
  29 </h3></div></div></div>
  30 <p>
  31         A second example compares four root finding algorithms for locating the second
  32         radius of an ellipse with first radius 28 and arc length 300, for four floating-point
  33         types, <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
  34         <span class="keyword">double</span></code> and a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
  35         type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>.
  36       </p>
  37 <p>
  38         Which is to say we're solving:
  39       </p>
  40 <pre class="programlisting">4xE(sqrt(1 - 28<sup>2</sup> / x<sup>2</sup>)) - 300 = 0</pre>
  41 <p>
  42         In each case the target accuracy was set using our "recomended"
  43         accuracy limits (or at least limits that make a good starting point - which
  44         is likely to give close to full accuracy without resorting to unnecessary
  45         iterations).
  46       </p>
  47 <div class="informaltable"><table class="table">
  48 <colgroup>
  49 <col>
  50 <col>
  51 </colgroup>
  52 <thead><tr>
  53 <th>
  54                 <p>
  55                   Function
  56                 </p>
  57               </th>
  58 <th>
  59                 <p>
  60                   Precision Requested
  61                 </p>
  62               </th>
  63 </tr></thead>
  64 <tbody>
  65 <tr>
  66 <td>
  67                 <p>
  68                   TOMS748
  69                 </p>
  70               </td>
  71 <td>
  72                 <p>
  73                   numeric_limits&lt;T&gt;::digits - 2
  74                 </p>
  75               </td>
  76 </tr>
  77 <tr>
  78 <td>
  79                 <p>
  80                   Newton
  81                 </p>
  82               </td>
  83 <td>
  84                 <p>
  85                   floor(numeric_limits&lt;T&gt;::digits * 0.6)
  86                 </p>
  87               </td>
  88 </tr>
  89 <tr>
  90 <td>
  91                 <p>
  92                   Halley
  93                 </p>
  94               </td>
  95 <td>
  96                 <p>
  97                   floor(numeric_limits&lt;T&gt;::digits * 0.4)
  98                 </p>
  99               </td>
 100 </tr>
 101 <tr>
 102 <td>
 103                 <p>
 104                   Schr&#246;der
 105                 </p>
 106               </td>
 107 <td>
 108                 <p>
 109                   floor(numeric_limits&lt;T&gt;::digits * 0.4)
 110                 </p>
 111               </td>
 112 </tr>
 113 </tbody>
 114 </table></div>
 115 <p>
 116         Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source
 117         code <a href="../../../../example/root_elliptic_finding.cpp" target="_top">root_elliptic_finding.cpp</a>.
 118       </p>
 119 <p>
 120         The timing uncertainty (especially using MSVC) is at least 5% of normalized
 121         time 'Norm'.
 122       </p>
 123 <p>
 124         To pick out the 'best' and 'worst' algorithms are highlighted in blue and
 125         red. More than one result can be 'best' when normalized times are indistinguishable
 126         within the uncertainty.
 127       </p>
 128 <h4>
 129 <a name="math_toolkit.root_comparison.elliptic_comparison.h0"></a>
 130         <span class="phrase"><a name="math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_fi"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_fi">Program
 131         root_elliptic_finding.cpp,
 132         Microsoft Visual C++ version 14.1, Dinkumware standard library version 650,
 133         Win32 Compiled in optimise mode., _X86_SSE2</a>
 134       </h4>
 135 <div class="table">
 136 <a name="math_toolkit.root_comparison.elliptic_comparison.elliptic"></a><p class="title"><b>Table&#160;10.12.&#160;root with radius 28 and arc length 300) for float, double, long double
 137         and cpp_bin_float_50 types, using _X86_SSE2</b></p>
 138 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
 139         and cpp_bin_float_50 types, using _X86_SSE2">
 140 <colgroup>
 141 <col>
 142 <col>
 143 <col>
 144 <col>
 145 <col>
 146 <col>
 147 <col>
 148 <col>
 149 <col>
 150 <col>
 151 <col>
 152 <col>
 153 <col>
 154 <col>
 155 <col>
 156 <col>
 157 <col>
 158 <col>
 159 <col>
 160 <col>
 161 <col>
 162 </colgroup>
 163 <thead><tr>
 164 <th>
 165               </th>
 166 <th>
 167                 <p>
 168                   float
 169                 </p>
 170               </th>
 171 <th>
 172               </th>
 173 <th>
 174               </th>
 175 <th>
 176               </th>
 177 <th>
 178               </th>
 179 <th>
 180                 <p>
 181                   double
 182                 </p>
 183               </th>
 184 <th>
 185               </th>
 186 <th>
 187               </th>
 188 <th>
 189               </th>
 190 <th>
 191               </th>
 192 <th>
 193                 <p>
 194                   long d
 195                 </p>
 196               </th>
 197 <th>
 198               </th>
 199 <th>
 200               </th>
 201 <th>
 202               </th>
 203 <th>
 204               </th>
 205 <th>
 206                 <p>
 207                   cpp50
 208                 </p>
 209               </th>
 210 <th>
 211               </th>
 212 <th>
 213               </th>
 214 <td class="auto-generated">&#160;</td>
 215 <td class="auto-generated">&#160;</td>
 216 </tr></thead>
 217 <tbody>
 218 <tr>
 219 <td>
 220                 <p>
 221                   Algo
 222                 </p>
 223               </td>
 224 <td>
 225                 <p>
 226                   Its
 227                 </p>
 228               </td>
 229 <td>
 230                 <p>
 231                   Times
 232                 </p>
 233               </td>
 234 <td>
 235                 <p>
 236                   Norm
 237                 </p>
 238               </td>
 239 <td>
 240                 <p>
 241                   Dis
 242                 </p>
 243               </td>
 244 <td>
 245               </td>
 246 <td>
 247                 <p>
 248                   Its
 249                 </p>
 250               </td>
 251 <td>
 252                 <p>
 253                   Times
 254                 </p>
 255               </td>
 256 <td>
 257                 <p>
 258                   Norm
 259                 </p>
 260               </td>
 261 <td>
 262                 <p>
 263                   Dis
 264                 </p>
 265               </td>
 266 <td>
 267               </td>
 268 <td>
 269                 <p>
 270                   Its
 271                 </p>
 272               </td>
 273 <td>
 274                 <p>
 275                   Times
 276                 </p>
 277               </td>
 278 <td>
 279                 <p>
 280                   Norm
 281                 </p>
 282               </td>
 283 <td>
 284                 <p>
 285                   Dis
 286                 </p>
 287               </td>
 288 <td>
 289               </td>
 290 <td>
 291                 <p>
 292                   Its
 293                 </p>
 294               </td>
 295 <td>
 296                 <p>
 297                   Times
 298                 </p>
 299               </td>
 300 <td>
 301                 <p>
 302                   Norm
 303                 </p>
 304               </td>
 305 <td>
 306                 <p>
 307                   Dis
 308                 </p>
 309               </td>
 310 <td>
 311               </td>
 312 </tr>
 313 <tr>
 314 <td>
 315                 <p>
 316                   TOMS748
 317                 </p>
 318               </td>
 319 <td>
 320                 <p>
 321                   6
 322                 </p>
 323               </td>
 324 <td>
 325                 <p>
 326                   906
 327                 </p>
 328               </td>
 329 <td>
 330                 <p>
 331                   2.07
 332                 </p>
 333               </td>
 334 <td>
 335                 <p>
 336                   0
 337                 </p>
 338               </td>
 339 <td>
 340               </td>
 341 <td>
 342                 <p>
 343                   9
 344                 </p>
 345               </td>
 346 <td>
 347                 <p>
 348                   1312
 349                 </p>
 350               </td>
 351 <td>
 352                 <p>
 353                   1.79
 354                 </p>
 355               </td>
 356 <td>
 357                 <p>
 358                   1
 359                 </p>
 360               </td>
 361 <td>
 362               </td>
 363 <td>
 364                 <p>
 365                   9
 366                 </p>
 367               </td>
 368 <td>
 369                 <p>
 370                   1281
 371                 </p>
 372               </td>
 373 <td>
 374                 <p>
 375                   1.75
 376                 </p>
 377               </td>
 378 <td>
 379                 <p>
 380                   1
 381                 </p>
 382               </td>
 383 <td>
 384               </td>
 385 <td>
 386                 <p>
 387                   11
 388                 </p>
 389               </td>
 390 <td>
 391                 <p>
 392                   1690625
 393                 </p>
 394               </td>
 395 <td>
 396                 <p>
 397                   1.52
 398                 </p>
 399               </td>
 400 <td>
 401                 <p>
 402                   -3
 403                 </p>
 404               </td>
 405 <td>
 406               </td>
 407 </tr>
 408 <tr>
 409 <td>
 410                 <p>
 411                   Newton
 412                 </p>
 413               </td>
 414 <td>
 415                 <p>
 416                   3
 417                 </p>
 418               </td>
 419 <td>
 420                 <p>
 421                   640
 422                 </p>
 423               </td>
 424 <td>
 425                 <p>
 426                   1.46
 427                 </p>
 428               </td>
 429 <td>
 430                 <p>
 431                   -1
 432                 </p>
 433               </td>
 434 <td>
 435               </td>
 436 <td>
 437                 <p>
 438                   4
 439                 </p>
 440               </td>
 441 <td>
 442                 <p>
 443                   875
 444                 </p>
 445               </td>
 446 <td>
 447                 <p>
 448                   1.19
 449                 </p>
 450               </td>
 451 <td>
 452                 <p>
 453                   1
 454                 </p>
 455               </td>
 456 <td>
 457               </td>
 458 <td>
 459                 <p>
 460                   4
 461                 </p>
 462               </td>
 463 <td>
 464                 <p>
 465                   843
 466                 </p>
 467               </td>
 468 <td>
 469                 <p>
 470                   1.15
 471                 </p>
 472               </td>
 473 <td>
 474                 <p>
 475                   1
 476                 </p>
 477               </td>
 478 <td>
 479               </td>
 480 <td>
 481                 <p>
 482                   5
 483                 </p>
 484               </td>
 485 <td>
 486                 <p>
 487                   1368750
 488                 </p>
 489               </td>
 490 <td>
 491                 <p>
 492                   1.23
 493                 </p>
 494               </td>
 495 <td>
 496                 <p>
 497                   0
 498                 </p>
 499               </td>
 500 <td>
 501               </td>
 502 </tr>
 503 <tr>
 504 <td>
 505                 <p>
 506                   Halley
 507                 </p>
 508               </td>
 509 <td>
 510                 <p>
 511                   2
 512                 </p>
 513               </td>
 514 <td>
 515                 <p>
 516                   437
 517                 </p>
 518               </td>
 519 <td>
 520                 <p>
 521                   <span class="blue">1.00</span>
 522                 </p>
 523               </td>
 524 <td>
 525                 <p>
 526                   0
 527                 </p>
 528               </td>
 529 <td>
 530               </td>
 531 <td>
 532                 <p>
 533                   3
 534                 </p>
 535               </td>
 536 <td>
 537                 <p>
 538                   734
 539                 </p>
 540               </td>
 541 <td>
 542                 <p>
 543                   <span class="blue">1.00</span>
 544                 </p>
 545               </td>
 546 <td>
 547                 <p>
 548                   3
 549                 </p>
 550               </td>
 551 <td>
 552               </td>
 553 <td>
 554                 <p>
 555                   3
 556                 </p>
 557               </td>
 558 <td>
 559                 <p>
 560                   734
 561                 </p>
 562               </td>
 563 <td>
 564                 <p>
 565                   <span class="blue">1.00</span>
 566                 </p>
 567               </td>
 568 <td>
 569                 <p>
 570                   3
 571                 </p>
 572               </td>
 573 <td>
 574               </td>
 575 <td>
 576                 <p>
 577                   4
 578                 </p>
 579               </td>
 580 <td>
 581                 <p>
 582                   1109375
 583                 </p>
 584               </td>
 585 <td>
 586                 <p>
 587                   <span class="blue">1.00</span>
 588                 </p>
 589               </td>
 590 <td>
 591                 <p>
 592                   0
 593                 </p>
 594               </td>
 595 <td>
 596               </td>
 597 </tr>
 598 <tr>
 599 <td>
 600                 <p>
 601                   Schr&#246;der
 602                 </p>
 603               </td>
 604 <td>
 605                 <p>
 606                   3
 607                 </p>
 608               </td>
 609 <td>
 610                 <p>
 611                   671
 612                 </p>
 613               </td>
 614 <td>
 615                 <p>
 616                   1.54
 617                 </p>
 618               </td>
 619 <td>
 620                 <p>
 621                   -1
 622                 </p>
 623               </td>
 624 <td>
 625               </td>
 626 <td>
 627                 <p>
 628                   6
 629                 </p>
 630               </td>
 631 <td>
 632                 <p>
 633                   1296
 634                 </p>
 635               </td>
 636 <td>
 637                 <p>
 638                   1.77
 639                 </p>
 640               </td>
 641 <td>
 642                 <p>
 643                   1
 644                 </p>
 645               </td>
 646 <td>
 647               </td>
 648 <td>
 649                 <p>
 650                   6
 651                 </p>
 652               </td>
 653 <td>
 654                 <p>
 655                   1406
 656                 </p>
 657               </td>
 658 <td>
 659                 <p>
 660                   1.92
 661                 </p>
 662               </td>
 663 <td>
 664                 <p>
 665                   1
 666                 </p>
 667               </td>
 668 <td>
 669               </td>
 670 <td>
 671                 <p>
 672                   5
 673                 </p>
 674               </td>
 675 <td>
 676                 <p>
 677                   1462500
 678                 </p>
 679               </td>
 680 <td>
 681                 <p>
 682                   1.32
 683                 </p>
 684               </td>
 685 <td>
 686                 <p>
 687                   -2
 688                 </p>
 689               </td>
 690 <td>
 691               </td>
 692 </tr>
 693 </tbody>
 694 </table></div>
 695 </div>
 696 <br class="table-break"><h4>
 697 <a name="math_toolkit.root_comparison.elliptic_comparison.h1"></a>
 698         <span class="phrase"><a name="math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_f0"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_f0">Program
 699         root_elliptic_finding.cpp,
 700         Microsoft Visual C++ version 12.0, Dinkumware standard library version 610,
 701         Win32 Compiled in optimise mode., _X64_AVX</a>
 702       </h4>
 703 <div class="table">
 704 <a name="math_toolkit.root_comparison.elliptic_comparison.elliptic0"></a><p class="title"><b>Table&#160;10.13.&#160;root with radius 28 and arc length 300) for float, double, long double
 705         and cpp_bin_float_50 types, using _X64_AVX</b></p>
 706 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
 707         and cpp_bin_float_50 types, using _X64_AVX">
 708 <colgroup>
 709 <col>
 710 <col>
 711 <col>
 712 <col>
 713 <col>
 714 <col>
 715 <col>
 716 <col>
 717 <col>
 718 <col>
 719 <col>
 720 <col>
 721 <col>
 722 <col>
 723 <col>
 724 <col>
 725 <col>
 726 <col>
 727 <col>
 728 <col>
 729 <col>
 730 </colgroup>
 731 <thead><tr>
 732 <th>
 733               </th>
 734 <th>
 735                 <p>
 736                   float
 737                 </p>
 738               </th>
 739 <th>
 740               </th>
 741 <th>
 742               </th>
 743 <th>
 744               </th>
 745 <th>
 746               </th>
 747 <th>
 748                 <p>
 749                   double
 750                 </p>
 751               </th>
 752 <th>
 753               </th>
 754 <th>
 755               </th>
 756 <th>
 757               </th>
 758 <th>
 759               </th>
 760 <th>
 761                 <p>
 762                   long d
 763                 </p>
 764               </th>
 765 <th>
 766               </th>
 767 <th>
 768               </th>
 769 <th>
 770               </th>
 771 <th>
 772               </th>
 773 <th>
 774                 <p>
 775                   cpp50
 776                 </p>
 777               </th>
 778 <th>
 779               </th>
 780 <th>
 781               </th>
 782 <td class="auto-generated">&#160;</td>
 783 <td class="auto-generated">&#160;</td>
 784 </tr></thead>
 785 <tbody>
 786 <tr>
 787 <td>
 788                 <p>
 789                   Algo
 790                 </p>
 791               </td>
 792 <td>
 793                 <p>
 794                   Its
 795                 </p>
 796               </td>
 797 <td>
 798                 <p>
 799                   Times
 800                 </p>
 801               </td>
 802 <td>
 803                 <p>
 804                   Norm
 805                 </p>
 806               </td>
 807 <td>
 808                 <p>
 809                   Dis
 810                 </p>
 811               </td>
 812 <td>
 813               </td>
 814 <td>
 815                 <p>
 816                   Its
 817                 </p>
 818               </td>
 819 <td>
 820                 <p>
 821                   Times
 822                 </p>
 823               </td>
 824 <td>
 825                 <p>
 826                   Norm
 827                 </p>
 828               </td>
 829 <td>
 830                 <p>
 831                   Dis
 832                 </p>
 833               </td>
 834 <td>
 835               </td>
 836 <td>
 837                 <p>
 838                   Its
 839                 </p>
 840               </td>
 841 <td>
 842                 <p>
 843                   Times
 844                 </p>
 845               </td>
 846 <td>
 847                 <p>
 848                   Norm
 849                 </p>
 850               </td>
 851 <td>
 852                 <p>
 853                   Dis
 854                 </p>
 855               </td>
 856 <td>
 857               </td>
 858 <td>
 859                 <p>
 860                   Its
 861                 </p>
 862               </td>
 863 <td>
 864                 <p>
 865                   Times
 866                 </p>
 867               </td>
 868 <td>
 869                 <p>
 870                   Norm
 871                 </p>
 872               </td>
 873 <td>
 874                 <p>
 875                   Dis
 876                 </p>
 877               </td>
 878 <td>
 879               </td>
 880 </tr>
 881 <tr>
 882 <td>
 883                 <p>
 884                   TOMS748
 885                 </p>
 886               </td>
 887 <td>
 888                 <p>
 889                   5
 890                 </p>
 891               </td>
 892 <td>
 893                 <p>
 894                   500
 895                 </p>
 896               </td>
 897 <td>
 898                 <p>
 899                   1.33
 900                 </p>
 901               </td>
 902 <td>
 903                 <p>
 904                   -1
 905                 </p>
 906               </td>
 907 <td>
 908               </td>
 909 <td>
 910                 <p>
 911                   9
 912                 </p>
 913               </td>
 914 <td>
 915                 <p>
 916                   1046
 917                 </p>
 918               </td>
 919 <td>
 920                 <p>
 921                   1.72
 922                 </p>
 923               </td>
 924 <td>
 925                 <p>
 926                   1
 927                 </p>
 928               </td>
 929 <td>
 930               </td>
 931 <td>
 932                 <p>
 933                   9
 934                 </p>
 935               </td>
 936 <td>
 937                 <p>
 938                   1062
 939                 </p>
 940               </td>
 941 <td>
 942                 <p>
 943                   1.70
 944                 </p>
 945               </td>
 946 <td>
 947                 <p>
 948                   1
 949                 </p>
 950               </td>
 951 <td>
 952               </td>
 953 <td>
 954                 <p>
 955                   11
 956                 </p>
 957               </td>
 958 <td>
 959                 <p>
 960                   698437
 961                 </p>
 962               </td>
 963 <td>
 964                 <p>
 965                   1.54
 966                 </p>
 967               </td>
 968 <td>
 969                 <p>
 970                   -3
 971                 </p>
 972               </td>
 973 <td>
 974               </td>
 975 </tr>
 976 <tr>
 977 <td>
 978                 <p>
 979                   Newton
 980                 </p>
 981               </td>
 982 <td>
 983                 <p>
 984                   3
 985                 </p>
 986               </td>
 987 <td>
 988                 <p>
 989                   484
 990                 </p>
 991               </td>
 992 <td>
 993                 <p>
 994                   1.29
 995                 </p>
 996               </td>
 997 <td>
 998                 <p>
 999                   -1
1000                 </p>
1001               </td>
1002 <td>
1003               </td>
1004 <td>
1005                 <p>
1006                   4
1007                 </p>
1008               </td>
1009 <td>
1010                 <p>
1011                   734
1012                 </p>
1013               </td>
1014 <td>
1015                 <p>
1016                   1.21
1017                 </p>
1018               </td>
1019 <td>
1020                 <p>
1021                   1
1022                 </p>
1023               </td>
1024 <td>
1025               </td>
1026 <td>
1027                 <p>
1028                   4
1029                 </p>
1030               </td>
1031 <td>
1032                 <p>
1033                   687
1034                 </p>
1035               </td>
1036 <td>
1037                 <p>
1038                   1.10
1039                 </p>
1040               </td>
1041 <td>
1042                 <p>
1043                   1
1044                 </p>
1045               </td>
1046 <td>
1047               </td>
1048 <td>
1049                 <p>
1050                   5
1051                 </p>
1052               </td>
1053 <td>
1054                 <p>
1055                   545312
1056                 </p>
1057               </td>
1058 <td>
1059                 <p>
1060                   1.20
1061                 </p>
1062               </td>
1063 <td>
1064                 <p>
1065                   0
1066                 </p>
1067               </td>
1068 <td>
1069               </td>
1070 </tr>
1071 <tr>
1072 <td>
1073                 <p>
1074                   Halley
1075                 </p>
1076               </td>
1077 <td>
1078                 <p>
1079                   2
1080                 </p>
1081               </td>
1082 <td>
1083                 <p>
1084                   375
1085                 </p>
1086               </td>
1087 <td>
1088                 <p>
1089                   <span class="blue">1.00</span>
1090                 </p>
1091               </td>
1092 <td>
1093                 <p>
1094                   0
1095                 </p>
1096               </td>
1097 <td>
1098               </td>
1099 <td>
1100                 <p>
1101                   3
1102                 </p>
1103               </td>
1104 <td>
1105                 <p>
1106                   609
1107                 </p>
1108               </td>
1109 <td>
1110                 <p>
1111                   <span class="blue">1.00</span>
1112                 </p>
1113               </td>
1114 <td>
1115                 <p>
1116                   3
1117                 </p>
1118               </td>
1119 <td>
1120               </td>
1121 <td>
1122                 <p>
1123                   3
1124                 </p>
1125               </td>
1126 <td>
1127                 <p>
1128                   625
1129                 </p>
1130               </td>
1131 <td>
1132                 <p>
1133                   <span class="blue">1.00</span>
1134                 </p>
1135               </td>
1136 <td>
1137                 <p>
1138                   3
1139                 </p>
1140               </td>
1141 <td>
1142               </td>
1143 <td>
1144                 <p>
1145                   4
1146                 </p>
1147               </td>
1148 <td>
1149                 <p>
1150                   453125
1151                 </p>
1152               </td>
1153 <td>
1154                 <p>
1155                   <span class="blue">1.00</span>
1156                 </p>
1157               </td>
1158 <td>
1159                 <p>
1160                   0
1161                 </p>
1162               </td>
1163 <td>
1164               </td>
1165 </tr>
1166 <tr>
1167 <td>
1168                 <p>
1169                   Schr&#246;der
1170                 </p>
1171               </td>
1172 <td>
1173                 <p>
1174                   3
1175                 </p>
1176               </td>
1177 <td>
1178                 <p>
1179                   546
1180                 </p>
1181               </td>
1182 <td>
1183                 <p>
1184                   1.46
1185                 </p>
1186               </td>
1187 <td>
1188                 <p>
1189                   -1
1190                 </p>
1191               </td>
1192 <td>
1193               </td>
1194 <td>
1195                 <p>
1196                   6
1197                 </p>
1198               </td>
1199 <td>
1200                 <p>
1201                   1109
1202                 </p>
1203               </td>
1204 <td>
1205                 <p>
1206                   1.82
1207                 </p>
1208               </td>
1209 <td>
1210                 <p>
1211                   1
1212                 </p>
1213               </td>
1214 <td>
1215               </td>
1216 <td>
1217                 <p>
1218                   6
1219                 </p>
1220               </td>
1221 <td>
1222                 <p>
1223                   1187
1224                 </p>
1225               </td>
1226 <td>
1227                 <p>
1228                   1.90
1229                 </p>
1230               </td>
1231 <td>
1232                 <p>
1233                   1
1234                 </p>
1235               </td>
1236 <td>
1237               </td>
1238 <td>
1239                 <p>
1240                   5
1241                 </p>
1242               </td>
1243 <td>
1244                 <p>
1245                   564062
1246                 </p>
1247               </td>
1248 <td>
1249                 <p>
1250                   1.24
1251                 </p>
1252               </td>
1253 <td>
1254                 <p>
1255                   -2
1256                 </p>
1257               </td>
1258 <td>
1259               </td>
1260 </tr>
1261 </tbody>
1262 </table></div>
1263 </div>
1264 <br class="table-break"><h4>
1265 <a name="math_toolkit.root_comparison.elliptic_comparison.h2"></a>
1266         <span class="phrase"><a name="math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_f1"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.root_comparison.elliptic_comparison.program_example_root_elliptic_f1">Program
1267         root_elliptic_finding.cpp,
1268         GNU C++ version 7.1.0, GNU libstdc++ version 20170502, Win32 Compiled in
1269         optimise mode., _X64_SSE2</a>
1270       </h4>
1271 <div class="table">
1272 <a name="math_toolkit.root_comparison.elliptic_comparison.elliptic1"></a><p class="title"><b>Table&#160;10.14.&#160;root with radius 28 and arc length 300) for float, double, long double
1273         and cpp_bin_float_50 types, using _X64_SSE2</b></p>
1274 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
1275         and cpp_bin_float_50 types, using _X64_SSE2">
1276 <colgroup>
1277 <col>
1278 <col>
1279 <col>
1280 <col>
1281 <col>
1282 <col>
1283 <col>
1284 <col>
1285 <col>
1286 <col>
1287 <col>
1288 <col>
1289 <col>
1290 <col>
1291 <col>
1292 <col>
1293 <col>
1294 <col>
1295 <col>
1296 <col>
1297 <col>
1298 </colgroup>
1299 <thead><tr>
1300 <th>
1301               </th>
1302 <th>
1303                 <p>
1304                   float
1305                 </p>
1306               </th>
1307 <th>
1308               </th>
1309 <th>
1310               </th>
1311 <th>
1312               </th>
1313 <th>
1314               </th>
1315 <th>
1316                 <p>
1317                   double
1318                 </p>
1319               </th>
1320 <th>
1321               </th>
1322 <th>
1323               </th>
1324 <th>
1325               </th>
1326 <th>
1327               </th>
1328 <th>
1329                 <p>
1330                   long d
1331                 </p>
1332               </th>
1333 <th>
1334               </th>
1335 <th>
1336               </th>
1337 <th>
1338               </th>
1339 <th>
1340               </th>
1341 <th>
1342                 <p>
1343                   cpp50
1344                 </p>
1345               </th>
1346 <th>
1347               </th>
1348 <th>
1349               </th>
1350 <td class="auto-generated">&#160;</td>
1351 <td class="auto-generated">&#160;</td>
1352 </tr></thead>
1353 <tbody>
1354 <tr>
1355 <td>
1356                 <p>
1357                   Algo
1358                 </p>
1359               </td>
1360 <td>
1361                 <p>
1362                   Its
1363                 </p>
1364               </td>
1365 <td>
1366                 <p>
1367                   Times
1368                 </p>
1369               </td>
1370 <td>
1371                 <p>
1372                   Norm
1373                 </p>
1374               </td>
1375 <td>
1376                 <p>
1377                   Dis
1378                 </p>
1379               </td>
1380 <td>
1381               </td>
1382 <td>
1383                 <p>
1384                   Its
1385                 </p>
1386               </td>
1387 <td>
1388                 <p>
1389                   Times
1390                 </p>
1391               </td>
1392 <td>
1393                 <p>
1394                   Norm
1395                 </p>
1396               </td>
1397 <td>
1398                 <p>
1399                   Dis
1400                 </p>
1401               </td>
1402 <td>
1403               </td>
1404 <td>
1405                 <p>
1406                   Its
1407                 </p>
1408               </td>
1409 <td>
1410                 <p>
1411                   Times
1412                 </p>
1413               </td>
1414 <td>
1415                 <p>
1416                   Norm
1417                 </p>
1418               </td>
1419 <td>
1420                 <p>
1421                   Dis
1422                 </p>
1423               </td>
1424 <td>
1425               </td>
1426 <td>
1427                 <p>
1428                   Its
1429                 </p>
1430               </td>
1431 <td>
1432                 <p>
1433                   Times
1434                 </p>
1435               </td>
1436 <td>
1437                 <p>
1438                   Norm
1439                 </p>
1440               </td>
1441 <td>
1442                 <p>
1443                   Dis
1444                 </p>
1445               </td>
1446 <td>
1447               </td>
1448 </tr>
1449 <tr>
1450 <td>
1451                 <p>
1452                   TOMS748
1453                 </p>
1454               </td>
1455 <td>
1456                 <p>
1457                   5
1458                 </p>
1459               </td>
1460 <td>
1461                 <p>
1462                   328
1463                 </p>
1464               </td>
1465 <td>
1466                 <p>
1467                   1.24
1468                 </p>
1469               </td>
1470 <td>
1471                 <p>
1472                   -1
1473                 </p>
1474               </td>
1475 <td>
1476               </td>
1477 <td>
1478                 <p>
1479                   8
1480                 </p>
1481               </td>
1482 <td>
1483                 <p>
1484                   890
1485                 </p>
1486               </td>
1487 <td>
1488                 <p>
1489                   1.50
1490                 </p>
1491               </td>
1492 <td>
1493                 <p>
1494                   0
1495                 </p>
1496               </td>
1497 <td>
1498               </td>
1499 <td>
1500                 <p>
1501                   8
1502                 </p>
1503               </td>
1504 <td>
1505                 <p>
1506                   1234
1507                 </p>
1508               </td>
1509 <td>
1510                 <p>
1511                   1.61
1512                 </p>
1513               </td>
1514 <td>
1515                 <p>
1516                   4
1517                 </p>
1518               </td>
1519 <td>
1520               </td>
1521 <td>
1522                 <p>
1523                   11
1524                 </p>
1525               </td>
1526 <td>
1527                 <p>
1528                   487500
1529                 </p>
1530               </td>
1531 <td>
1532                 <p>
1533                   1.57
1534                 </p>
1535               </td>
1536 <td>
1537                 <p>
1538                   -3
1539                 </p>
1540               </td>
1541 <td>
1542               </td>
1543 </tr>
1544 <tr>
1545 <td>
1546                 <p>
1547                   Newton
1548                 </p>
1549               </td>
1550 <td>
1551                 <p>
1552                   3
1553                 </p>
1554               </td>
1555 <td>
1556                 <p>
1557                   359
1558                 </p>
1559               </td>
1560 <td>
1561                 <p>
1562                   1.35
1563                 </p>
1564               </td>
1565 <td>
1566                 <p>
1567                   -1
1568                 </p>
1569               </td>
1570 <td>
1571               </td>
1572 <td>
1573                 <p>
1574                   4
1575                 </p>
1576               </td>
1577 <td>
1578                 <p>
1579                   718
1580                 </p>
1581               </td>
1582 <td>
1583                 <p>
1584                   1.21
1585                 </p>
1586               </td>
1587 <td>
1588                 <p>
1589                   1
1590                 </p>
1591               </td>
1592 <td>
1593               </td>
1594 <td>
1595                 <p>
1596                   4
1597                 </p>
1598               </td>
1599 <td>
1600                 <p>
1601                   843
1602                 </p>
1603               </td>
1604 <td>
1605                 <p>
1606                   1.10
1607                 </p>
1608               </td>
1609 <td>
1610                 <p>
1611                   1
1612                 </p>
1613               </td>
1614 <td>
1615               </td>
1616 <td>
1617                 <p>
1618                   5
1619                 </p>
1620               </td>
1621 <td>
1622                 <p>
1623                   379687
1624                 </p>
1625               </td>
1626 <td>
1627                 <p>
1628                   1.22
1629                 </p>
1630               </td>
1631 <td>
1632                 <p>
1633                   0
1634                 </p>
1635               </td>
1636 <td>
1637               </td>
1638 </tr>
1639 <tr>
1640 <td>
1641                 <p>
1642                   Halley
1643                 </p>
1644               </td>
1645 <td>
1646                 <p>
1647                   2
1648                 </p>
1649               </td>
1650 <td>
1651                 <p>
1652                   265
1653                 </p>
1654               </td>
1655 <td>
1656                 <p>
1657                   <span class="blue">1.00</span>
1658                 </p>
1659               </td>
1660 <td>
1661                 <p>
1662                   0
1663                 </p>
1664               </td>
1665 <td>
1666               </td>
1667 <td>
1668                 <p>
1669                   3
1670                 </p>
1671               </td>
1672 <td>
1673                 <p>
1674                   593
1675                 </p>
1676               </td>
1677 <td>
1678                 <p>
1679                   <span class="blue">1.00</span>
1680                 </p>
1681               </td>
1682 <td>
1683                 <p>
1684                   1
1685                 </p>
1686               </td>
1687 <td>
1688               </td>
1689 <td>
1690                 <p>
1691                   3
1692                 </p>
1693               </td>
1694 <td>
1695                 <p>
1696                   765
1697                 </p>
1698               </td>
1699 <td>
1700                 <p>
1701                   <span class="blue">1.00</span>
1702                 </p>
1703               </td>
1704 <td>
1705                 <p>
1706                   7
1707                 </p>
1708               </td>
1709 <td>
1710               </td>
1711 <td>
1712                 <p>
1713                   4
1714                 </p>
1715               </td>
1716 <td>
1717                 <p>
1718                   310937
1719                 </p>
1720               </td>
1721 <td>
1722                 <p>
1723                   <span class="blue">1.00</span>
1724                 </p>
1725               </td>
1726 <td>
1727                 <p>
1728                   0
1729                 </p>
1730               </td>
1731 <td>
1732               </td>
1733 </tr>
1734 <tr>
1735 <td>
1736                 <p>
1737                   Schr&#246;der
1738                 </p>
1739               </td>
1740 <td>
1741                 <p>
1742                   3
1743                 </p>
1744               </td>
1745 <td>
1746                 <p>
1747                   343
1748                 </p>
1749               </td>
1750 <td>
1751                 <p>
1752                   1.29
1753                 </p>
1754               </td>
1755 <td>
1756                 <p>
1757                   -1
1758                 </p>
1759               </td>
1760 <td>
1761               </td>
1762 <td>
1763                 <p>
1764                   4
1765                 </p>
1766               </td>
1767 <td>
1768                 <p>
1769                   812
1770                 </p>
1771               </td>
1772 <td>
1773                 <p>
1774                   1.37
1775                 </p>
1776               </td>
1777 <td>
1778                 <p>
1779                   0
1780                 </p>
1781               </td>
1782 <td>
1783               </td>
1784 <td>
1785                 <p>
1786                   4
1787                 </p>
1788               </td>
1789 <td>
1790                 <p>
1791                   1046
1792                 </p>
1793               </td>
1794 <td>
1795                 <p>
1796                   1.37
1797                 </p>
1798               </td>
1799 <td>
1800                 <p>
1801                   3
1802                 </p>
1803               </td>
1804 <td>
1805               </td>
1806 <td>
1807                 <p>
1808                   5
1809                 </p>
1810               </td>
1811 <td>
1812                 <p>
1813                   390625
1814                 </p>
1815               </td>
1816 <td>
1817                 <p>
1818                   1.26
1819                 </p>
1820               </td>
1821 <td>
1822                 <p>
1823                   -2
1824                 </p>
1825               </td>
1826 <td>
1827               </td>
1828 </tr>
1829 </tbody>
1830 </table></div>
1831 </div>
1832 <br class="table-break"><p>
1833         Remarks:
1834       </p>
1835 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1836 <li class="listitem">
1837             The function being solved is now moderately expensive to call, and twice
1838             as expensive to call when obtaining the derivative than when not. Consequently
1839             there is very little improvement in moving from a derivative free method,
1840             to Newton iteration. However, once you've calculated the first derivative
1841             the second comes almost for free, consequently the third order methods
1842             (Halley) does much the best.
1843           </li>
1844 <li class="listitem">
1845             Of the two second order methods, Halley does best as would be expected:
1846             the Schroder method offers better guarantees of <span class="emphasis"><em>quadratic</em></span>
1847             convergence, while Halley relies on a smooth function with a single root
1848             to give <span class="emphasis"><em>cubic</em></span> convergence. It's not entirely clear
1849             why Schroder iteration often does worse than Newton.
1850           </li>
1851 </ul></div>
1852 </div>
1853 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1854 <td align="left"></td>
1855 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
1856       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
1857       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
1858       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
1859       Daryle Walker and Xiaogang Zhang<p>
1860         Distributed under the Boost Software License, Version 1.0. (See accompanying
1861         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
1862       </p>
1863 </div></td>
1864 </tr></table>
1865 <hr>
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