1871 lines
37 KiB
HTML
1871 lines
37 KiB
HTML
<html>
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<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
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<title>Comparison of Elliptic Integral Root Finding Algoritghms</title>
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../root_comparison.html" title="Comparison of Root Finding Algorithms">
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<link rel="prev" href="root_n_comparison.html" title="Comparison of Nth-root Finding Algorithms">
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<link rel="next" href="../../poly.html" title="Chapter 11. Polynomials and Rational Functions">
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<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
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<table cellpadding="2" width="100%"><tr>
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<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
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<hr>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<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
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of Elliptic Integral Root Finding Algoritghms</a>
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</h3></div></div></div>
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<p>
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A second example compares four root finding algorithms for locating the second
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radius of an ellipse with first radius 28 and arc length 300, for four floating-point
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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>
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<span class="keyword">double</span></code> and a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
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type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>.
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</p>
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<p>
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Which is to say we're solving:
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</p>
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<pre class="programlisting">4xE(sqrt(1 - 28<sup>2</sup> / x<sup>2</sup>)) - 300 = 0</pre>
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<p>
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In each case the target accuracy was set using our "recomended"
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accuracy limits (or at least limits that make a good starting point - which
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is likely to give close to full accuracy without resorting to unnecessary
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iterations).
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Precision Requested
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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TOMS748
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</p>
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</td>
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<td>
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<p>
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numeric_limits<T>::digits - 2
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Newton
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.6)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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Halley
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.4)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
|
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Schröder
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</p>
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</td>
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<td>
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<p>
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floor(numeric_limits<T>::digits * 0.4)
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<p>
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Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source
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code <a href="../../../../example/root_elliptic_finding.cpp" target="_top">root_elliptic_finding.cpp</a>.
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</p>
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<p>
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The timing uncertainty (especially using MSVC) is at least 5% of normalized
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time 'Norm'.
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</p>
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<p>
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To pick out the 'best' and 'worst' algorithms are highlighted in blue and
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red. More than one result can be 'best' when normalized times are indistinguishable
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within the uncertainty.
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</p>
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<h4>
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<a name="math_toolkit.root_comparison.elliptic_comparison.h0"></a>
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<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
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root_elliptic_finding.cpp,
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Microsoft Visual C++ version 14.1, Dinkumware standard library version 650,
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Win32 Compiled in optimise mode., _X86_SSE2</a>
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</h4>
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<div class="table">
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<a name="math_toolkit.root_comparison.elliptic_comparison.elliptic"></a><p class="title"><b>Table 10.12. root with radius 28 and arc length 300) for float, double, long double
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and cpp_bin_float_50 types, using _X86_SSE2</b></p>
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<div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
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and cpp_bin_float_50 types, using _X86_SSE2">
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<colgroup>
|
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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float
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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double
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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long d
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</p>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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</th>
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<th>
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<p>
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cpp50
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</p>
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</th>
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<th>
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</th>
|
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<th>
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</th>
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<td class="auto-generated"> </td>
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<td class="auto-generated"> </td>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Algo
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</p>
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</td>
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<td>
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<p>
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Its
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</p>
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</td>
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<td>
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<p>
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Times
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</p>
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</td>
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<td>
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<p>
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Norm
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</p>
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</td>
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<td>
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<p>
|
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Dis
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</p>
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</td>
|
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<td>
|
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</td>
|
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<td>
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<p>
|
|
Its
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</p>
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</td>
|
|
<td>
|
|
<p>
|
|
Times
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|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
TOMS748
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
906
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2.07
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
9
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1312
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.79
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
9
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1281
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.75
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1690625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.52
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Newton
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
640
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.46
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
875
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.19
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
843
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.15
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1368750
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.23
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Halley
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
437
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
734
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
734
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1109375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Schröder
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
671
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.54
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1296
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.77
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1406
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.92
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1462500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.32
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><h4>
|
|
<a name="math_toolkit.root_comparison.elliptic_comparison.h1"></a>
|
|
<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
|
|
root_elliptic_finding.cpp,
|
|
Microsoft Visual C++ version 12.0, Dinkumware standard library version 610,
|
|
Win32 Compiled in optimise mode., _X64_AVX</a>
|
|
</h4>
|
|
<div class="table">
|
|
<a name="math_toolkit.root_comparison.elliptic_comparison.elliptic0"></a><p class="title"><b>Table 10.13. root with radius 28 and arc length 300) for float, double, long double
|
|
and cpp_bin_float_50 types, using _X64_AVX</b></p>
|
|
<div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
|
|
and cpp_bin_float_50 types, using _X64_AVX">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
float
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
long d
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
cpp50
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<td class="auto-generated"> </td>
|
|
<td class="auto-generated"> </td>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Algo
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
TOMS748
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.33
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
9
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1046
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.72
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
9
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1062
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.70
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
698437
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.54
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Newton
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
484
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.29
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
734
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.21
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
687
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.10
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
545312
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.20
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Halley
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
375
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
609
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
453125
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Schröder
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
546
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.46
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1109
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.82
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
6
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1187
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.90
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
564062
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.24
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><h4>
|
|
<a name="math_toolkit.root_comparison.elliptic_comparison.h2"></a>
|
|
<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
|
|
root_elliptic_finding.cpp,
|
|
GNU C++ version 7.1.0, GNU libstdc++ version 20170502, Win32 Compiled in
|
|
optimise mode., _X64_SSE2</a>
|
|
</h4>
|
|
<div class="table">
|
|
<a name="math_toolkit.root_comparison.elliptic_comparison.elliptic1"></a><p class="title"><b>Table 10.14. root with radius 28 and arc length 300) for float, double, long double
|
|
and cpp_bin_float_50 types, using _X64_SSE2</b></p>
|
|
<div class="table-contents"><table class="table" summary="root with radius 28 and arc length 300) for float, double, long double
|
|
and cpp_bin_float_50 types, using _X64_SSE2">
|
|
<colgroup>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
<col>
|
|
</colgroup>
|
|
<thead><tr>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
float
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
long d
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
cpp50
|
|
</p>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<th>
|
|
</th>
|
|
<td class="auto-generated"> </td>
|
|
<td class="auto-generated"> </td>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Algo
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Its
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Times
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Norm
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
Dis
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
TOMS748
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
328
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.24
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
890
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.50
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
8
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1234
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.61
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
11
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
487500
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.57
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Newton
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
359
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.35
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
718
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.21
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
843
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.10
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
379687
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.22
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Halley
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
265
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
593
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
765
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
7
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
310937
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">1.00</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Schröder
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
343
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.29
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-1
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
812
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.37
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
0
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
4
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1046
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.37
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
3
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
5
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
390625
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
1.26
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
-2
|
|
</p>
|
|
</td>
|
|
<td>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table></div>
|
|
</div>
|
|
<br class="table-break"><p>
|
|
Remarks:
|
|
</p>
|
|
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
|
<li class="listitem">
|
|
The function being solved is now moderately expensive to call, and twice
|
|
as expensive to call when obtaining the derivative than when not. Consequently
|
|
there is very little improvement in moving from a derivative free method,
|
|
to Newton iteration. However, once you've calculated the first derivative
|
|
the second comes almost for free, consequently the third order methods
|
|
(Halley) does much the best.
|
|
</li>
|
|
<li class="listitem">
|
|
Of the two second order methods, Halley does best as would be expected:
|
|
the Schroder method offers better guarantees of <span class="emphasis"><em>quadratic</em></span>
|
|
convergence, while Halley relies on a smooth function with a single root
|
|
to give <span class="emphasis"><em>cubic</em></span> convergence. It's not entirely clear
|
|
why Schroder iteration often does worse than Newton.
|
|
</li>
|
|
</ul></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
|
<td align="left"></td>
|
|
<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
|
|
Daryle Walker and Xiaogang Zhang<p>
|
|
Distributed under the Boost Software License, Version 1.0. (See accompanying
|
|
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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
|
<div class="spirit-nav">
|
|
<a accesskey="p" href="root_n_comparison.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_comparison.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../../poly.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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|
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</html>
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