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   4 <title>Comparison of Elliptic Integral Root Finding Algoritghms</title>
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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.roots.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 </h4></div></div></div>
  30 <p>
  31           A second example compares four root finding algorithms for locating the
  32           second radius of an ellipse with first radius 28 and arc length 300, for
  33           four floating-point types, <code class="computeroutput"><span class="keyword">float</span></code>,
  34           <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
  35           <span class="keyword">double</span></code> and a <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
  36           type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>.
  37         </p>
  38 <p>
  39           Which is to say we're solving:
  40         </p>
  41 <pre class="programlisting">4xE(sqrt(1 - 28<sup>2</sup> / x<sup>2</sup>)) - 300 = 0</pre>
  42 <p>
  43           In each case the target accuracy was set using our "recomended"
  44           accuracy limits (or at least limits that make a good starting point - which
  45           is likely to give close to full accuracy without resorting to unnecessary
  46           iterations).
  47         </p>
  48 <div class="informaltable"><table class="table">
  49 <colgroup>
  50 <col>
  51 <col>
  52 </colgroup>
  53 <thead><tr>
  54 <th>
  55                   <p>
  56                     Function
  57                   </p>
  58                 </th>
  59 <th>
  60                   <p>
  61                     Precision Requested
  62                   </p>
  63                 </th>
  64 </tr></thead>
  65 <tbody>
  66 <tr>
  67 <td>
  68                   <p>
  69                     TOMS748
  70                   </p>
  71                 </td>
  72 <td>
  73                   <p>
  74                     numeric_limits&lt;T&gt;::digits - 2
  75                   </p>
  76                 </td>
  77 </tr>
  78 <tr>
  79 <td>
  80                   <p>
  81                     Newton
  82                   </p>
  83                 </td>
  84 <td>
  85                   <p>
  86                     floor(numeric_limits&lt;T&gt;::digits * 0.6)
  87                   </p>
  88                 </td>
  89 </tr>
  90 <tr>
  91 <td>
  92                   <p>
  93                     Halley
  94                   </p>
  95                 </td>
  96 <td>
  97                   <p>
  98                     floor(numeric_limits&lt;T&gt;::digits * 0.4)
  99                   </p>
 100                 </td>
 101 </tr>
 102 <tr>
 103 <td>
 104                   <p>
 105                     Schr&#246;der
 106                   </p>
 107                 </td>
 108 <td>
 109                   <p>
 110                     floor(numeric_limits&lt;T&gt;::digits * 0.4)
 111                   </p>
 112                 </td>
 113 </tr>
 114 </tbody>
 115 </table></div>
 116 <p>
 117           Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using
 118           source code <a href="../../../../../example/root_elliptic_finding.cpp" target="_top">root_elliptic_finding.cpp</a>.
 119         </p>
 120 <p>
 121           The timing uncertainty (especially using MSVC) is at least 5% of normalized
 122           time 'Norm'.
 123         </p>
 124 <p>
 125           To pick out the 'best' and 'worst' algorithms are highlighted in blue and
 126           red. More than one result can be 'best' when normalized times are indistinguishable
 127           within the uncertainty.
 128         </p>
 129 <h5>
 130 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.h0"></a>
 131           <span class="phrase"><a name="math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_fi"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_fi">Program
 132           root_elliptic_finding.cpp,
 133           Microsoft Visual C++ version 12.0, Dinkumware standard library version
 134           610, Win32 Compiled in optimise mode., _X86_SSE2</a>
 135         </h5>
 136 <div class="table">
 137 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.elliptic"></a><p class="title"><b>Table&#160;12.12.&#160;root with radius 28 and arc length 300) for float, double, long
 138           double and cpp_bin_float_50 types, using _X86_SSE2</b></p>
 139 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long
 140           double and cpp_bin_float_50 types, using _X86_SSE2">
 141 <colgroup>
 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 <col>
 163 </colgroup>
 164 <thead><tr>
 165 <th>
 166                 </th>
 167 <th>
 168                   <p>
 169                     float
 170                   </p>
 171                 </th>
 172 <th>
 173                 </th>
 174 <th>
 175                 </th>
 176 <th>
 177                 </th>
 178 <th>
 179                 </th>
 180 <th>
 181                   <p>
 182                     double
 183                   </p>
 184                 </th>
 185 <th>
 186                 </th>
 187 <th>
 188                 </th>
 189 <th>
 190                 </th>
 191 <th>
 192                 </th>
 193 <th>
 194                   <p>
 195                     long d
 196                   </p>
 197                 </th>
 198 <th>
 199                 </th>
 200 <th>
 201                 </th>
 202 <th>
 203                 </th>
 204 <th>
 205                 </th>
 206 <th>
 207                   <p>
 208                     cpp50
 209                   </p>
 210                 </th>
 211 <th>
 212                 </th>
 213 <th>
 214                 </th>
 215 <td class="auto-generated">&#160;</td>
 216 <td class="auto-generated">&#160;</td>
 217 </tr></thead>
 218 <tbody>
 219 <tr>
 220 <td>
 221                   <p>
 222                     Algo
 223                   </p>
 224                 </td>
 225 <td>
 226                   <p>
 227                     Its
 228                   </p>
 229                 </td>
 230 <td>
 231                   <p>
 232                     Times
 233                   </p>
 234                 </td>
 235 <td>
 236                   <p>
 237                     Norm
 238                   </p>
 239                 </td>
 240 <td>
 241                   <p>
 242                     Dis
 243                   </p>
 244                 </td>
 245 <td>
 246                 </td>
 247 <td>
 248                   <p>
 249                     Its
 250                   </p>
 251                 </td>
 252 <td>
 253                   <p>
 254                     Times
 255                   </p>
 256                 </td>
 257 <td>
 258                   <p>
 259                     Norm
 260                   </p>
 261                 </td>
 262 <td>
 263                   <p>
 264                     Dis
 265                   </p>
 266                 </td>
 267 <td>
 268                 </td>
 269 <td>
 270                   <p>
 271                     Its
 272                   </p>
 273                 </td>
 274 <td>
 275                   <p>
 276                     Times
 277                   </p>
 278                 </td>
 279 <td>
 280                   <p>
 281                     Norm
 282                   </p>
 283                 </td>
 284 <td>
 285                   <p>
 286                     Dis
 287                   </p>
 288                 </td>
 289 <td>
 290                 </td>
 291 <td>
 292                   <p>
 293                     Its
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     Times
 299                   </p>
 300                 </td>
 301 <td>
 302                   <p>
 303                     Norm
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     Dis
 309                   </p>
 310                 </td>
 311 <td>
 312                 </td>
 313 </tr>
 314 <tr>
 315 <td>
 316                   <p>
 317                     TOMS748
 318                   </p>
 319                 </td>
 320 <td>
 321                   <p>
 322                     5
 323                   </p>
 324                 </td>
 325 <td>
 326                   <p>
 327                     515
 328                   </p>
 329                 </td>
 330 <td>
 331                   <p>
 332                     1.43
 333                   </p>
 334                 </td>
 335 <td>
 336                   <p>
 337                     -1
 338                   </p>
 339                 </td>
 340 <td>
 341                 </td>
 342 <td>
 343                   <p>
 344                     9
 345                   </p>
 346                 </td>
 347 <td>
 348                   <p>
 349                     968
 350                   </p>
 351                 </td>
 352 <td>
 353                   <p>
 354                     1.82
 355                   </p>
 356                 </td>
 357 <td>
 358                   <p>
 359                     1
 360                   </p>
 361                 </td>
 362 <td>
 363                 </td>
 364 <td>
 365                   <p>
 366                     9
 367                   </p>
 368                 </td>
 369 <td>
 370                   <p>
 371                     968
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     1.82
 377                   </p>
 378                 </td>
 379 <td>
 380                   <p>
 381                     1
 382                   </p>
 383                 </td>
 384 <td>
 385                 </td>
 386 <td>
 387                   <p>
 388                     11
 389                   </p>
 390                 </td>
 391 <td>
 392                   <p>
 393                     871875
 394                   </p>
 395                 </td>
 396 <td>
 397                   <p>
 398                     1.53
 399                   </p>
 400                 </td>
 401 <td>
 402                   <p>
 403                     -3
 404                   </p>
 405                 </td>
 406 <td>
 407                 </td>
 408 </tr>
 409 <tr>
 410 <td>
 411                   <p>
 412                     Newton
 413                   </p>
 414                 </td>
 415 <td>
 416                   <p>
 417                     3
 418                   </p>
 419                 </td>
 420 <td>
 421                   <p>
 422                     453
 423                   </p>
 424                 </td>
 425 <td>
 426                   <p>
 427                     1.26
 428                   </p>
 429                 </td>
 430 <td>
 431                   <p>
 432                     -1
 433                   </p>
 434                 </td>
 435 <td>
 436                 </td>
 437 <td>
 438                   <p>
 439                     4
 440                   </p>
 441                 </td>
 442 <td>
 443                   <p>
 444                     640
 445                   </p>
 446                 </td>
 447 <td>
 448                   <p>
 449                     1.21
 450                   </p>
 451                 </td>
 452 <td>
 453                   <p>
 454                     1
 455                   </p>
 456                 </td>
 457 <td>
 458                 </td>
 459 <td>
 460                   <p>
 461                     4
 462                   </p>
 463                 </td>
 464 <td>
 465                   <p>
 466                     640
 467                   </p>
 468                 </td>
 469 <td>
 470                   <p>
 471                     1.21
 472                   </p>
 473                 </td>
 474 <td>
 475                   <p>
 476                     1
 477                   </p>
 478                 </td>
 479 <td>
 480                 </td>
 481 <td>
 482                   <p>
 483                     5
 484                   </p>
 485                 </td>
 486 <td>
 487                   <p>
 488                     685937
 489                   </p>
 490                 </td>
 491 <td>
 492                   <p>
 493                     1.20
 494                   </p>
 495                 </td>
 496 <td>
 497                   <p>
 498                     0
 499                   </p>
 500                 </td>
 501 <td>
 502                 </td>
 503 </tr>
 504 <tr>
 505 <td>
 506                   <p>
 507                     Halley
 508                   </p>
 509                 </td>
 510 <td>
 511                   <p>
 512                     2
 513                   </p>
 514                 </td>
 515 <td>
 516                   <p>
 517                     359
 518                   </p>
 519                 </td>
 520 <td>
 521                   <p>
 522                     <span class="blue">1.00</span>
 523                   </p>
 524                 </td>
 525 <td>
 526                   <p>
 527                     0
 528                   </p>
 529                 </td>
 530 <td>
 531                 </td>
 532 <td>
 533                   <p>
 534                     3
 535                   </p>
 536                 </td>
 537 <td>
 538                   <p>
 539                     531
 540                   </p>
 541                 </td>
 542 <td>
 543                   <p>
 544                     <span class="blue">1.00</span>
 545                   </p>
 546                 </td>
 547 <td>
 548                   <p>
 549                     3
 550                   </p>
 551                 </td>
 552 <td>
 553                 </td>
 554 <td>
 555                   <p>
 556                     3
 557                   </p>
 558                 </td>
 559 <td>
 560                   <p>
 561                     531
 562                   </p>
 563                 </td>
 564 <td>
 565                   <p>
 566                     <span class="blue">1.00</span>
 567                   </p>
 568                 </td>
 569 <td>
 570                   <p>
 571                     3
 572                   </p>
 573                 </td>
 574 <td>
 575                 </td>
 576 <td>
 577                   <p>
 578                     4
 579                   </p>
 580                 </td>
 581 <td>
 582                   <p>
 583                     570312
 584                   </p>
 585                 </td>
 586 <td>
 587                   <p>
 588                     <span class="blue">1.00</span>
 589                   </p>
 590                 </td>
 591 <td>
 592                   <p>
 593                     0
 594                   </p>
 595                 </td>
 596 <td>
 597                 </td>
 598 </tr>
 599 <tr>
 600 <td>
 601                   <p>
 602                     Schr&#246;der
 603                   </p>
 604                 </td>
 605 <td>
 606                   <p>
 607                     3
 608                   </p>
 609                 </td>
 610 <td>
 611                   <p>
 612                     484
 613                   </p>
 614                 </td>
 615 <td>
 616                   <p>
 617                     1.35
 618                   </p>
 619                 </td>
 620 <td>
 621                   <p>
 622                     -1
 623                   </p>
 624                 </td>
 625 <td>
 626                 </td>
 627 <td>
 628                   <p>
 629                     6
 630                   </p>
 631                 </td>
 632 <td>
 633                   <p>
 634                     1000
 635                   </p>
 636                 </td>
 637 <td>
 638                   <p>
 639                     1.88
 640                   </p>
 641                 </td>
 642 <td>
 643                   <p>
 644                     1
 645                   </p>
 646                 </td>
 647 <td>
 648                 </td>
 649 <td>
 650                   <p>
 651                     6
 652                   </p>
 653                 </td>
 654 <td>
 655                   <p>
 656                     984
 657                   </p>
 658                 </td>
 659 <td>
 660                   <p>
 661                     1.85
 662                   </p>
 663                 </td>
 664 <td>
 665                   <p>
 666                     1
 667                   </p>
 668                 </td>
 669 <td>
 670                 </td>
 671 <td>
 672                   <p>
 673                     5
 674                   </p>
 675                 </td>
 676 <td>
 677                   <p>
 678                     742187
 679                   </p>
 680                 </td>
 681 <td>
 682                   <p>
 683                     1.30
 684                   </p>
 685                 </td>
 686 <td>
 687                   <p>
 688                     -2
 689                   </p>
 690                 </td>
 691 <td>
 692                 </td>
 693 </tr>
 694 </tbody>
 695 </table></div>
 696 </div>
 697 <br class="table-break"><h5>
 698 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.h1"></a>
 699           <span class="phrase"><a name="math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_f0"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_f0">Program
 700           root_elliptic_finding.cpp,
 701           Microsoft Visual C++ version 12.0, Dinkumware standard library version
 702           610, Win32 Compiled in optimise mode., _X64_AVX</a>
 703         </h5>
 704 <div class="table">
 705 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.elliptic0"></a><p class="title"><b>Table&#160;12.13.&#160;root with radius 28 and arc length 300) for float, double, long
 706           double and cpp_bin_float_50 types, using _X64_AVX</b></p>
 707 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long
 708           double and cpp_bin_float_50 types, using _X64_AVX">
 709 <colgroup>
 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 <col>
 731 </colgroup>
 732 <thead><tr>
 733 <th>
 734                 </th>
 735 <th>
 736                   <p>
 737                     float
 738                   </p>
 739                 </th>
 740 <th>
 741                 </th>
 742 <th>
 743                 </th>
 744 <th>
 745                 </th>
 746 <th>
 747                 </th>
 748 <th>
 749                   <p>
 750                     double
 751                   </p>
 752                 </th>
 753 <th>
 754                 </th>
 755 <th>
 756                 </th>
 757 <th>
 758                 </th>
 759 <th>
 760                 </th>
 761 <th>
 762                   <p>
 763                     long d
 764                   </p>
 765                 </th>
 766 <th>
 767                 </th>
 768 <th>
 769                 </th>
 770 <th>
 771                 </th>
 772 <th>
 773                 </th>
 774 <th>
 775                   <p>
 776                     cpp50
 777                   </p>
 778                 </th>
 779 <th>
 780                 </th>
 781 <th>
 782                 </th>
 783 <td class="auto-generated">&#160;</td>
 784 <td class="auto-generated">&#160;</td>
 785 </tr></thead>
 786 <tbody>
 787 <tr>
 788 <td>
 789                   <p>
 790                     Algo
 791                   </p>
 792                 </td>
 793 <td>
 794                   <p>
 795                     Its
 796                   </p>
 797                 </td>
 798 <td>
 799                   <p>
 800                     Times
 801                   </p>
 802                 </td>
 803 <td>
 804                   <p>
 805                     Norm
 806                   </p>
 807                 </td>
 808 <td>
 809                   <p>
 810                     Dis
 811                   </p>
 812                 </td>
 813 <td>
 814                 </td>
 815 <td>
 816                   <p>
 817                     Its
 818                   </p>
 819                 </td>
 820 <td>
 821                   <p>
 822                     Times
 823                   </p>
 824                 </td>
 825 <td>
 826                   <p>
 827                     Norm
 828                   </p>
 829                 </td>
 830 <td>
 831                   <p>
 832                     Dis
 833                   </p>
 834                 </td>
 835 <td>
 836                 </td>
 837 <td>
 838                   <p>
 839                     Its
 840                   </p>
 841                 </td>
 842 <td>
 843                   <p>
 844                     Times
 845                   </p>
 846                 </td>
 847 <td>
 848                   <p>
 849                     Norm
 850                   </p>
 851                 </td>
 852 <td>
 853                   <p>
 854                     Dis
 855                   </p>
 856                 </td>
 857 <td>
 858                 </td>
 859 <td>
 860                   <p>
 861                     Its
 862                   </p>
 863                 </td>
 864 <td>
 865                   <p>
 866                     Times
 867                   </p>
 868                 </td>
 869 <td>
 870                   <p>
 871                     Norm
 872                   </p>
 873                 </td>
 874 <td>
 875                   <p>
 876                     Dis
 877                   </p>
 878                 </td>
 879 <td>
 880                 </td>
 881 </tr>
 882 <tr>
 883 <td>
 884                   <p>
 885                     TOMS748
 886                   </p>
 887                 </td>
 888 <td>
 889                   <p>
 890                     5
 891                   </p>
 892                 </td>
 893 <td>
 894                   <p>
 895                     500
 896                   </p>
 897                 </td>
 898 <td>
 899                   <p>
 900                     1.33
 901                   </p>
 902                 </td>
 903 <td>
 904                   <p>
 905                     -1
 906                   </p>
 907                 </td>
 908 <td>
 909                 </td>
 910 <td>
 911                   <p>
 912                     9
 913                   </p>
 914                 </td>
 915 <td>
 916                   <p>
 917                     1046
 918                   </p>
 919                 </td>
 920 <td>
 921                   <p>
 922                     1.72
 923                   </p>
 924                 </td>
 925 <td>
 926                   <p>
 927                     1
 928                   </p>
 929                 </td>
 930 <td>
 931                 </td>
 932 <td>
 933                   <p>
 934                     9
 935                   </p>
 936                 </td>
 937 <td>
 938                   <p>
 939                     1062
 940                   </p>
 941                 </td>
 942 <td>
 943                   <p>
 944                     1.70
 945                   </p>
 946                 </td>
 947 <td>
 948                   <p>
 949                     1
 950                   </p>
 951                 </td>
 952 <td>
 953                 </td>
 954 <td>
 955                   <p>
 956                     11
 957                   </p>
 958                 </td>
 959 <td>
 960                   <p>
 961                     698437
 962                   </p>
 963                 </td>
 964 <td>
 965                   <p>
 966                     1.54
 967                   </p>
 968                 </td>
 969 <td>
 970                   <p>
 971                     -3
 972                   </p>
 973                 </td>
 974 <td>
 975                 </td>
 976 </tr>
 977 <tr>
 978 <td>
 979                   <p>
 980                     Newton
 981                   </p>
 982                 </td>
 983 <td>
 984                   <p>
 985                     3
 986                   </p>
 987                 </td>
 988 <td>
 989                   <p>
 990                     484
 991                   </p>
 992                 </td>
 993 <td>
 994                   <p>
 995                     1.29
 996                   </p>
 997                 </td>
 998 <td>
 999                   <p>
1000                     -1
1001                   </p>
1002                 </td>
1003 <td>
1004                 </td>
1005 <td>
1006                   <p>
1007                     4
1008                   </p>
1009                 </td>
1010 <td>
1011                   <p>
1012                     734
1013                   </p>
1014                 </td>
1015 <td>
1016                   <p>
1017                     1.21
1018                   </p>
1019                 </td>
1020 <td>
1021                   <p>
1022                     1
1023                   </p>
1024                 </td>
1025 <td>
1026                 </td>
1027 <td>
1028                   <p>
1029                     4
1030                   </p>
1031                 </td>
1032 <td>
1033                   <p>
1034                     687
1035                   </p>
1036                 </td>
1037 <td>
1038                   <p>
1039                     1.10
1040                   </p>
1041                 </td>
1042 <td>
1043                   <p>
1044                     1
1045                   </p>
1046                 </td>
1047 <td>
1048                 </td>
1049 <td>
1050                   <p>
1051                     5
1052                   </p>
1053                 </td>
1054 <td>
1055                   <p>
1056                     545312
1057                   </p>
1058                 </td>
1059 <td>
1060                   <p>
1061                     1.20
1062                   </p>
1063                 </td>
1064 <td>
1065                   <p>
1066                     0
1067                   </p>
1068                 </td>
1069 <td>
1070                 </td>
1071 </tr>
1072 <tr>
1073 <td>
1074                   <p>
1075                     Halley
1076                   </p>
1077                 </td>
1078 <td>
1079                   <p>
1080                     2
1081                   </p>
1082                 </td>
1083 <td>
1084                   <p>
1085                     375
1086                   </p>
1087                 </td>
1088 <td>
1089                   <p>
1090                     <span class="blue">1.00</span>
1091                   </p>
1092                 </td>
1093 <td>
1094                   <p>
1095                     0
1096                   </p>
1097                 </td>
1098 <td>
1099                 </td>
1100 <td>
1101                   <p>
1102                     3
1103                   </p>
1104                 </td>
1105 <td>
1106                   <p>
1107                     609
1108                   </p>
1109                 </td>
1110 <td>
1111                   <p>
1112                     <span class="blue">1.00</span>
1113                   </p>
1114                 </td>
1115 <td>
1116                   <p>
1117                     3
1118                   </p>
1119                 </td>
1120 <td>
1121                 </td>
1122 <td>
1123                   <p>
1124                     3
1125                   </p>
1126                 </td>
1127 <td>
1128                   <p>
1129                     625
1130                   </p>
1131                 </td>
1132 <td>
1133                   <p>
1134                     <span class="blue">1.00</span>
1135                   </p>
1136                 </td>
1137 <td>
1138                   <p>
1139                     3
1140                   </p>
1141                 </td>
1142 <td>
1143                 </td>
1144 <td>
1145                   <p>
1146                     4
1147                   </p>
1148                 </td>
1149 <td>
1150                   <p>
1151                     453125
1152                   </p>
1153                 </td>
1154 <td>
1155                   <p>
1156                     <span class="blue">1.00</span>
1157                   </p>
1158                 </td>
1159 <td>
1160                   <p>
1161                     0
1162                   </p>
1163                 </td>
1164 <td>
1165                 </td>
1166 </tr>
1167 <tr>
1168 <td>
1169                   <p>
1170                     Schr&#246;der
1171                   </p>
1172                 </td>
1173 <td>
1174                   <p>
1175                     3
1176                   </p>
1177                 </td>
1178 <td>
1179                   <p>
1180                     546
1181                   </p>
1182                 </td>
1183 <td>
1184                   <p>
1185                     1.46
1186                   </p>
1187                 </td>
1188 <td>
1189                   <p>
1190                     -1
1191                   </p>
1192                 </td>
1193 <td>
1194                 </td>
1195 <td>
1196                   <p>
1197                     6
1198                   </p>
1199                 </td>
1200 <td>
1201                   <p>
1202                     1109
1203                   </p>
1204                 </td>
1205 <td>
1206                   <p>
1207                     1.82
1208                   </p>
1209                 </td>
1210 <td>
1211                   <p>
1212                     1
1213                   </p>
1214                 </td>
1215 <td>
1216                 </td>
1217 <td>
1218                   <p>
1219                     6
1220                   </p>
1221                 </td>
1222 <td>
1223                   <p>
1224                     1187
1225                   </p>
1226                 </td>
1227 <td>
1228                   <p>
1229                     1.90
1230                   </p>
1231                 </td>
1232 <td>
1233                   <p>
1234                     1
1235                   </p>
1236                 </td>
1237 <td>
1238                 </td>
1239 <td>
1240                   <p>
1241                     5
1242                   </p>
1243                 </td>
1244 <td>
1245                   <p>
1246                     564062
1247                   </p>
1248                 </td>
1249 <td>
1250                   <p>
1251                     1.24
1252                   </p>
1253                 </td>
1254 <td>
1255                   <p>
1256                     -2
1257                   </p>
1258                 </td>
1259 <td>
1260                 </td>
1261 </tr>
1262 </tbody>
1263 </table></div>
1264 </div>
1265 <br class="table-break"><h5>
1266 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.h2"></a>
1267           <span class="phrase"><a name="math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_f1"></a></span><a class="link" href="elliptic_comparison.html#math_toolkit.roots.root_comparison.elliptic_comparison.program_example_root_elliptic_f1">Program
1268           root_elliptic_finding.cpp,
1269           GNU C++ version 4.9.2, GNU libstdc++ version 20141030, Win32 Compiled in
1270           optimise mode., _X64_SSE2</a>
1271         </h5>
1272 <div class="table">
1273 <a name="math_toolkit.roots.root_comparison.elliptic_comparison.elliptic1"></a><p class="title"><b>Table&#160;12.14.&#160;root with radius 28 and arc length 300) for float, double, long
1274           double and cpp_bin_float_50 types, using _X64_SSE2</b></p>
1275 <div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long
1276           double and cpp_bin_float_50 types, using _X64_SSE2">
1277 <colgroup>
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 <col>
1299 </colgroup>
1300 <thead><tr>
1301 <th>
1302                 </th>
1303 <th>
1304                   <p>
1305                     float
1306                   </p>
1307                 </th>
1308 <th>
1309                 </th>
1310 <th>
1311                 </th>
1312 <th>
1313                 </th>
1314 <th>
1315                 </th>
1316 <th>
1317                   <p>
1318                     double
1319                   </p>
1320                 </th>
1321 <th>
1322                 </th>
1323 <th>
1324                 </th>
1325 <th>
1326                 </th>
1327 <th>
1328                 </th>
1329 <th>
1330                   <p>
1331                     long d
1332                   </p>
1333                 </th>
1334 <th>
1335                 </th>
1336 <th>
1337                 </th>
1338 <th>
1339                 </th>
1340 <th>
1341                 </th>
1342 <th>
1343                   <p>
1344                     cpp50
1345                   </p>
1346                 </th>
1347 <th>
1348                 </th>
1349 <th>
1350                 </th>
1351 <td class="auto-generated">&#160;</td>
1352 <td class="auto-generated">&#160;</td>
1353 </tr></thead>
1354 <tbody>
1355 <tr>
1356 <td>
1357                   <p>
1358                     Algo
1359                   </p>
1360                 </td>
1361 <td>
1362                   <p>
1363                     Its
1364                   </p>
1365                 </td>
1366 <td>
1367                   <p>
1368                     Times
1369                   </p>
1370                 </td>
1371 <td>
1372                   <p>
1373                     Norm
1374                   </p>
1375                 </td>
1376 <td>
1377                   <p>
1378                     Dis
1379                   </p>
1380                 </td>
1381 <td>
1382                 </td>
1383 <td>
1384                   <p>
1385                     Its
1386                   </p>
1387                 </td>
1388 <td>
1389                   <p>
1390                     Times
1391                   </p>
1392                 </td>
1393 <td>
1394                   <p>
1395                     Norm
1396                   </p>
1397                 </td>
1398 <td>
1399                   <p>
1400                     Dis
1401                   </p>
1402                 </td>
1403 <td>
1404                 </td>
1405 <td>
1406                   <p>
1407                     Its
1408                   </p>
1409                 </td>
1410 <td>
1411                   <p>
1412                     Times
1413                   </p>
1414                 </td>
1415 <td>
1416                   <p>
1417                     Norm
1418                   </p>
1419                 </td>
1420 <td>
1421                   <p>
1422                     Dis
1423                   </p>
1424                 </td>
1425 <td>
1426                 </td>
1427 <td>
1428                   <p>
1429                     Its
1430                   </p>
1431                 </td>
1432 <td>
1433                   <p>
1434                     Times
1435                   </p>
1436                 </td>
1437 <td>
1438                   <p>
1439                     Norm
1440                   </p>
1441                 </td>
1442 <td>
1443                   <p>
1444                     Dis
1445                   </p>
1446                 </td>
1447 <td>
1448                 </td>
1449 </tr>
1450 <tr>
1451 <td>
1452                   <p>
1453                     TOMS748
1454                   </p>
1455                 </td>
1456 <td>
1457                   <p>
1458                     5
1459                   </p>
1460                 </td>
1461 <td>
1462                   <p>
1463                     328
1464                   </p>
1465                 </td>
1466 <td>
1467                   <p>
1468                     1.31
1469                   </p>
1470                 </td>
1471 <td>
1472                   <p>
1473                     -1
1474                   </p>
1475                 </td>
1476 <td>
1477                 </td>
1478 <td>
1479                   <p>
1480                     8
1481                   </p>
1482                 </td>
1483 <td>
1484                   <p>
1485                     875
1486                   </p>
1487                 </td>
1488 <td>
1489                   <p>
1490                     1.51
1491                   </p>
1492                 </td>
1493 <td>
1494                   <p>
1495                     0
1496                   </p>
1497                 </td>
1498 <td>
1499                 </td>
1500 <td>
1501                   <p>
1502                     8
1503                   </p>
1504                 </td>
1505 <td>
1506                   <p>
1507                     1109
1508                   </p>
1509                 </td>
1510 <td>
1511                   <p>
1512                     1.69
1513                   </p>
1514                 </td>
1515 <td>
1516                   <p>
1517                     4
1518                   </p>
1519                 </td>
1520 <td>
1521                 </td>
1522 <td>
1523                   <p>
1524                     11
1525                   </p>
1526                 </td>
1527 <td>
1528                   <p>
1529                     479687
1530                   </p>
1531                 </td>
1532 <td>
1533                   <p>
1534                     1.49
1535                   </p>
1536                 </td>
1537 <td>
1538                   <p>
1539                     -3
1540                   </p>
1541                 </td>
1542 <td>
1543                 </td>
1544 </tr>
1545 <tr>
1546 <td>
1547                   <p>
1548                     Newton
1549                   </p>
1550                 </td>
1551 <td>
1552                   <p>
1553                     3
1554                   </p>
1555                 </td>
1556 <td>
1557                   <p>
1558                     328
1559                   </p>
1560                 </td>
1561 <td>
1562                   <p>
1563                     1.31
1564                   </p>
1565                 </td>
1566 <td>
1567                   <p>
1568                     -1
1569                   </p>
1570                 </td>
1571 <td>
1572                 </td>
1573 <td>
1574                   <p>
1575                     4
1576                   </p>
1577                 </td>
1578 <td>
1579                   <p>
1580                     671
1581                   </p>
1582                 </td>
1583 <td>
1584                   <p>
1585                     1.16
1586                   </p>
1587                 </td>
1588 <td>
1589                   <p>
1590                     1
1591                   </p>
1592                 </td>
1593 <td>
1594                 </td>
1595 <td>
1596                   <p>
1597                     4
1598                   </p>
1599                 </td>
1600 <td>
1601                   <p>
1602                     781
1603                   </p>
1604                 </td>
1605 <td>
1606                   <p>
1607                     1.19
1608                   </p>
1609                 </td>
1610 <td>
1611                   <p>
1612                     1
1613                   </p>
1614                 </td>
1615 <td>
1616                 </td>
1617 <td>
1618                   <p>
1619                     5
1620                   </p>
1621                 </td>
1622 <td>
1623                   <p>
1624                     387500
1625                   </p>
1626                 </td>
1627 <td>
1628                   <p>
1629                     1.20
1630                   </p>
1631                 </td>
1632 <td>
1633                   <p>
1634                     0
1635                   </p>
1636                 </td>
1637 <td>
1638                 </td>
1639 </tr>
1640 <tr>
1641 <td>
1642                   <p>
1643                     Halley
1644                   </p>
1645                 </td>
1646 <td>
1647                   <p>
1648                     2
1649                   </p>
1650                 </td>
1651 <td>
1652                   <p>
1653                     250
1654                   </p>
1655                 </td>
1656 <td>
1657                   <p>
1658                     <span class="blue">1.00</span>
1659                   </p>
1660                 </td>
1661 <td>
1662                   <p>
1663                     0
1664                   </p>
1665                 </td>
1666 <td>
1667                 </td>
1668 <td>
1669                   <p>
1670                     3
1671                   </p>
1672                 </td>
1673 <td>
1674                   <p>
1675                     578
1676                   </p>
1677                 </td>
1678 <td>
1679                   <p>
1680                     <span class="blue">1.00</span>
1681                   </p>
1682                 </td>
1683 <td>
1684                   <p>
1685                     1
1686                   </p>
1687                 </td>
1688 <td>
1689                 </td>
1690 <td>
1691                   <p>
1692                     3
1693                   </p>
1694                 </td>
1695 <td>
1696                   <p>
1697                     656
1698                   </p>
1699                 </td>
1700 <td>
1701                   <p>
1702                     <span class="blue">1.00</span>
1703                   </p>
1704                 </td>
1705 <td>
1706                   <p>
1707                     7
1708                   </p>
1709                 </td>
1710 <td>
1711                 </td>
1712 <td>
1713                   <p>
1714                     4
1715                   </p>
1716                 </td>
1717 <td>
1718                   <p>
1719                     321875
1720                   </p>
1721                 </td>
1722 <td>
1723                   <p>
1724                     <span class="blue">1.00</span>
1725                   </p>
1726                 </td>
1727 <td>
1728                   <p>
1729                     0
1730                   </p>
1731                 </td>
1732 <td>
1733                 </td>
1734 </tr>
1735 <tr>
1736 <td>
1737                   <p>
1738                     Schr&#246;der
1739                   </p>
1740                 </td>
1741 <td>
1742                   <p>
1743                     3
1744                   </p>
1745                 </td>
1746 <td>
1747                   <p>
1748                     375
1749                   </p>
1750                 </td>
1751 <td>
1752                   <p>
1753                     1.50
1754                   </p>
1755                 </td>
1756 <td>
1757                   <p>
1758                     -1
1759                   </p>
1760                 </td>
1761 <td>
1762                 </td>
1763 <td>
1764                   <p>
1765                     4
1766                   </p>
1767                 </td>
1768 <td>
1769                   <p>
1770                     734
1771                   </p>
1772                 </td>
1773 <td>
1774                   <p>
1775                     1.27
1776                   </p>
1777                 </td>
1778 <td>
1779                   <p>
1780                     0
1781                   </p>
1782                 </td>
1783 <td>
1784                 </td>
1785 <td>
1786                   <p>
1787                     4
1788                   </p>
1789                 </td>
1790 <td>
1791                   <p>
1792                     828
1793                   </p>
1794                 </td>
1795 <td>
1796                   <p>
1797                     1.26
1798                   </p>
1799                 </td>
1800 <td>
1801                   <p>
1802                     3
1803                   </p>
1804                 </td>
1805 <td>
1806                 </td>
1807 <td>
1808                   <p>
1809                     5
1810                   </p>
1811                 </td>
1812 <td>
1813                   <p>
1814                     414062
1815                   </p>
1816                 </td>
1817 <td>
1818                   <p>
1819                     1.29
1820                   </p>
1821                 </td>
1822 <td>
1823                   <p>
1824                     -2
1825                   </p>
1826                 </td>
1827 <td>
1828                 </td>
1829 </tr>
1830 </tbody>
1831 </table></div>
1832 </div>
1833 <br class="table-break"><p>
1834           Remarks:
1835         </p>
1836 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
1837 <li class="listitem">
1838               The function being solved is now moderately expensive to call, and
1839               twice as expensive to call when obtaining the derivative than when
1840               not. Consequently there is very little improvement in moving from a
1841               derivative free method, to Newton iteration. However, once you've calculated
1842               the first derivative the second comes almost for free, consequently
1843               the third order methods (Halley) does much the best.
1844             </li>
1845 <li class="listitem">
1846               Of the two second order methods, Halley does best as would be expected:
1847               the Schroder method offers better guarantees of <span class="emphasis"><em>quadratic</em></span>
1848               convergence, while Halley relies on a smooth function with a single
1849               root to give <span class="emphasis"><em>cubic</em></span> convergence. It's not entirely
1850               clear why Schroder iteration often does worse than Newton.
1851             </li>
1852 </ul></div>
1853 </div>
1854 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1855 <td align="left"></td>
1856 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
1857       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1858       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
1859       Benjamin Sobotta, Thijs van den Berg, 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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