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<title>Comparison of Elliptic Integral Root Finding Algorithms</title>
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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.root_comparison.elliptic_comparison"></a><a class="link" href="elliptic_comparison.html" title="Comparison of Elliptic Integral Root Finding Algorithms">Comparison
of Elliptic Integral Root Finding Algorithms</a>
</h3></div></div></div>
<p>
A second example compares four root finding algorithms for locating the second
radius of an ellipse with first radius 28 and arc length 300, for four floating-point
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>
<span class="keyword">double</span></code> and a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>.
</p>
<p>
Which is to say we're solving:
</p>
<pre class="programlisting">4xE(sqrt(1 - 28<sup>2</sup> / x<sup>2</sup>)) - 300 = 0</pre>
<p>
In each case the target accuracy was set using our "recommended"
accuracy limits (or at least limits that make a good starting point - which
is likely to give close to full accuracy without resorting to unnecessary
iterations).
</p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Function
</p>
</th>
<th>
<p>
Precision Requested
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
TOMS748
</p>
</td>
<td>
<p>
numeric_limits&lt;T&gt;::digits - 2
</p>
</td>
</tr>
<tr>
<td>
<p>
Newton
</p>
</td>
<td>
<p>
floor(numeric_limits&lt;T&gt;::digits * 0.6)
</p>
</td>
</tr>
<tr>
<td>
<p>
Halley
</p>
</td>
<td>
<p>
floor(numeric_limits&lt;T&gt;::digits * 0.4)
</p>
</td>
</tr>
<tr>
<td>
<p>
Schröder
</p>
</td>
<td>
<p>
floor(numeric_limits&lt;T&gt;::digits * 0.4)
</p>
</td>
</tr>
</tbody>
</table></div>
<p>
Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source
code <a href="../../../../example/root_elliptic_finding.cpp" target="_top">root_elliptic_finding.cpp</a>.
</p>
<p>
The timing uncertainty (especially using MSVC) is at least 5% of normalized
time 'Norm'.
</p>
<p>
To pick out the 'best' and 'worst' algorithms are highlighted in blue and
red. More than one result can be 'best' when normalized times are indistinguishable
within the uncertainty.
</p>
<h4>
<a name="math_toolkit.root_comparison.elliptic_comparison.h0"></a>
<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
root_elliptic_finding.cpp,
Microsoft Visual C++ version 14.1, Dinkumware standard library version 650,
Win32 Compiled in optimise mode., _X86_SSE2</a>
</h4>
<div class="table">
<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
and cpp_bin_float_50 types, using _X86_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 _X86_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>
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>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, 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>
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