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<title>Continued Fraction Evaluation</title>
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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.internals.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation">Continued Fraction Evaluation</a>
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</h3></div></div></div>
|
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<h5>
|
||||
<a name="math_toolkit.internals.cf.h0"></a>
|
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<span class="phrase"><a name="math_toolkit.internals.cf.synopsis"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">fraction</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">)</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">)</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">)</span>
|
||||
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// These interfaces are present for legacy reasons, and are now deprecated:</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special"><</span><span class="identifier">Gen</span><span class="special">>::</span><span class="identifier">result_type</span>
|
||||
<span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">Gen</span><span class="special">&</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||||
|
||||
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.cf.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.cf.description"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
<a href="http://en.wikipedia.org/wiki/Continued_fraction" target="_top">Continued fractions
|
||||
are a common method of approximation. </a> These functions all evaluate
|
||||
the continued fraction described by the <span class="emphasis"><em>generator</em></span> type
|
||||
argument. The functions with an "_a" suffix evaluate the fraction:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/fraction2.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
and those with a "_b" suffix evaluate the fraction:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/fraction1.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
This latter form is somewhat more natural in that it corresponds with the
|
||||
usual definition of a continued fraction, but note that the first <span class="emphasis"><em>a</em></span>
|
||||
value returned by the generator is discarded. Further, often the first <span class="emphasis"><em>a</em></span>
|
||||
and <span class="emphasis"><em>b</em></span> values in a continued fraction have different
|
||||
defining equations to the remaining terms, which may make the "_a"
|
||||
suffixed form more appropriate.
|
||||
</p>
|
||||
<p>
|
||||
The generator type should be a function object which supports the following
|
||||
operations:
|
||||
</p>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Expression
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Description
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
Gen::result_type
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
The type that is the result of invoking operator(). This can be
|
||||
either an arithmetic type, or a std::pair<> of arithmetic
|
||||
types.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
g()
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Returns an object of type Gen::result_type.
|
||||
</p>
|
||||
<p>
|
||||
Each time this operator is called then the next pair of <span class="emphasis"><em>a</em></span>
|
||||
and <span class="emphasis"><em>b</em></span> values is returned. Or, if result_type
|
||||
is an arithmetic type, then the next <span class="emphasis"><em>b</em></span> value
|
||||
is returned and all the <span class="emphasis"><em>a</em></span> values are assumed
|
||||
to 1.
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<p>
|
||||
In all the continued fraction evaluation functions the <span class="emphasis"><em>tolerance</em></span>
|
||||
parameter is the precision desired in the result, evaluation of the fraction
|
||||
will continue until the last term evaluated leaves the relative error in
|
||||
the result less than <span class="emphasis"><em>tolerance</em></span>. The deprecated interfaces
|
||||
take a number of digits precision here, internally they just convert this
|
||||
to a tolerance and forward call.
|
||||
</p>
|
||||
<p>
|
||||
If the optional <span class="emphasis"><em>max_terms</em></span> parameter is specified then
|
||||
no more than <span class="emphasis"><em>max_terms</em></span> calls to the generator will be
|
||||
made, and on output, <span class="emphasis"><em>max_terms</em></span> will be set to actual
|
||||
number of calls made. This facility is particularly useful when profiling
|
||||
a continued fraction for convergence.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.cf.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.cf.implementation"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
Internally these algorithms all use the modified Lentz algorithm: refer to
|
||||
Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 Evaluation
|
||||
of continued fractions, p 175 - 179) for more information, also Lentz, W.J.
|
||||
1976, Applied Optics, vol. 15, pp. 668-671.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.cf.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.cf.examples"></a></span><a class="link" href="cf.html#math_toolkit.internals.cf.examples">Examples</a>
|
||||
</h5>
|
||||
<p>
|
||||
The <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">golden ratio phi
|
||||
= 1.618033989...</a> can be computed from the simplest continued fraction
|
||||
of all:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/fraction3.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
We begin by defining a generator function:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">golden_ratio_fraction</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">return</span> <span class="number">1</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
The golden ratio can then be computed to double precision using:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">continued_fraction_a</span><span class="special">(</span>
|
||||
<span class="identifier">golden_ratio_fraction</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
||||
</pre>
|
||||
<p>
|
||||
It's more usual though to have to define both the <span class="emphasis"><em>a</em></span>'s
|
||||
and the <span class="emphasis"><em>b</em></span>'s when evaluating special functions by continued
|
||||
fractions, for example the tan function is defined by:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/fraction4.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
So its generator object would look like:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">tan_fraction</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
<span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="identifier">tan_fraction</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">v</span><span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">a</span><span class="special">(-</span><span class="identifier">v</span><span class="special">*</span><span class="identifier">v</span><span class="special">),</span> <span class="identifier">b</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span>
|
||||
<span class="special">{}</span>
|
||||
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">result_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
|
||||
<span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
Notice that if the continuant is subtracted from the <span class="emphasis"><em>b</em></span>
|
||||
terms, as is the case here, then all the <span class="emphasis"><em>a</em></span> terms returned
|
||||
by the generator will be negative. The tangent function can now be evaluated
|
||||
using:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">T</span> <span class="identifier">tan</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">tan_fraction</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">fract</span><span class="special">(</span><span class="identifier">a</span><span class="special">);</span>
|
||||
<span class="keyword">return</span> <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">continued_fraction_b</span><span class="special">(</span><span class="identifier">fract</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
Notice that this time we're using the "_b" suffixed version to
|
||||
evaluate the fraction: we're removing the leading <span class="emphasis"><em>a</em></span>
|
||||
term during fraction evaluation as it's different from all the others.
|
||||
</p>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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>
|
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<div class="spirit-nav">
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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.internals.error_test"></a><a class="link" href="error_test.html" title="Relative Error and Testing">Relative Error and
|
||||
Testing</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.error_test.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.error_test.synopsis"></a></span><a class="link" href="error_test.html#math_toolkit.internals.error_test.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<div class="important"><table border="0" summary="Important">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../../../../../doc/src/images/important.png"></td>
|
||||
<th align="left">Important</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
The header <code class="computeroutput"><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test</span><span class="special">.</span><span class="identifier">hpp</span></code> is located under <code class="computeroutput"><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">include_private</span></code>
|
||||
and is not installed to the usual locations by default, you will need to
|
||||
add <code class="computeroutput"><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">include_private</span></code> to your compiler's include
|
||||
path in order to use this header.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">T</span> <span class="identifier">relative_error</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">F1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">F2</span><span class="special">></span>
|
||||
<span class="identifier">test_result</span><span class="special"><</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">></span> <span class="identifier">test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">A</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">F1</span> <span class="identifier">test_func</span><span class="special">,</span> <span class="identifier">F2</span> <span class="identifier">expect_func</span><span class="special">);</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.error_test.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.error_test.description"></a></span><a class="link" href="error_test.html#math_toolkit.internals.error_test.description">Description</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">T</span> <span class="identifier">relative_error</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">v</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Returns the relative error between <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>v</em></span>
|
||||
using the usual formula:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/error1.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
In addition the value returned is zero if:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
Both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>v</em></span> are infinite.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Both <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>v</em></span> are denormalised
|
||||
numbers or zero.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
Otherwise if only one of <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>v</em></span>
|
||||
is zero then the value returned is 1.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">A</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">F1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">F2</span><span class="special">></span>
|
||||
<span class="identifier">test_result</span><span class="special"><</span><span class="identifier">see</span><span class="special">-</span><span class="identifier">below</span><span class="special">></span> <span class="identifier">test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">A</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">F1</span> <span class="identifier">test_func</span><span class="special">,</span> <span class="identifier">F2</span> <span class="identifier">expect_func</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
This function is used for testing a function against tabulated test data.
|
||||
</p>
|
||||
<p>
|
||||
The return type contains statistical data on the relative errors (max, mean,
|
||||
variance, and the number of test cases etc), as well as the row of test data
|
||||
that caused the largest relative error. Public members of type test_result
|
||||
are:
|
||||
</p>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl class="variablelist">
|
||||
<dt><span class="term"><code class="computeroutput"><span class="keyword">unsigned</span> <span class="identifier">worst</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the row at which the worst error occurred.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">min</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the smallest relative error found.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">max</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the largest relative error found.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the mean error found.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the number of test cases.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">variance</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the variance of the errors found.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">variance1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the unbiased variance of the errors found.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">T</span> <span class="identifier">rms</span><span class="special">()</span><span class="keyword">const</span></code></span></dt>
|
||||
<dd><p>
|
||||
Returns the Root Mean Square, or quadratic mean of the errors.
|
||||
</p></dd>
|
||||
<dt><span class="term"><code class="computeroutput"><span class="identifier">test_result</span><span class="special">&</span>
|
||||
<span class="keyword">operator</span><span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">test_result</span><span class="special">&</span> <span class="identifier">t</span><span class="special">)</span></code></span></dt>
|
||||
<dd><p>
|
||||
Combines two test_result's into a single result.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<p>
|
||||
The template parameter of test_result, is the same type as the values in
|
||||
the two dimensional array passed to function <span class="emphasis"><em>test</em></span>, roughly
|
||||
that's <code class="computeroutput"><span class="identifier">A</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">::</span><span class="identifier">value_type</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
Parameter <span class="emphasis"><em>a</em></span> is a matrix of test data: and must be a
|
||||
standard library Sequence type, that contains another Sequence type: typically
|
||||
it will be a two dimensional instance of <code class="literal">boost::array</code>.
|
||||
Each row of <span class="emphasis"><em>a</em></span> should contain all the parameters that
|
||||
are passed to the function under test as well as the expected result.
|
||||
</p>
|
||||
<p>
|
||||
Parameter <span class="emphasis"><em>test_func</em></span> is the function under test, it is
|
||||
invoked with each row of test data in <span class="emphasis"><em>a</em></span>. Typically type
|
||||
F1 is created with Boost.Lambda: see the example below.
|
||||
</p>
|
||||
<p>
|
||||
Parameter <span class="emphasis"><em>expect_func</em></span> is a functor that extracts the
|
||||
expected result from a row of test data in <span class="emphasis"><em>a</em></span>. Typically
|
||||
type F2 is created with Boost.Lambda: see the example below.
|
||||
</p>
|
||||
<p>
|
||||
If the function under test returns a non-finite value when a finite result
|
||||
is expected, or if a gross error is found, then a message is sent to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cerr</span></code>,
|
||||
and a call to BOOST_ERROR() made (which means that including this header
|
||||
requires you use Boost.Test). This is mainly a debugging/development aid
|
||||
(and a good place for a breakpoint).
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.error_test.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.error_test.example"></a></span><a class="link" href="error_test.html#math_toolkit.internals.error_test.example">Example</a>
|
||||
</h5>
|
||||
<p>
|
||||
Suppose we want to test the tgamma and lgamma functions, we can create a
|
||||
two dimensional matrix of test data, each row is one test case, and contains
|
||||
three elements: the input value, and the expected results for the tgamma
|
||||
and lgamma functions respectively.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">,</span> <span class="number">3</span><span class="special">>,</span> <span class="identifier">NumberOfTests</span><span class="special">></span>
|
||||
<span class="identifier">factorials</span> <span class="special">=</span> <span class="special">{</span>
|
||||
<span class="comment">/* big array of test data goes here */</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
Now we can invoke the test function to test tgamma:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lambda</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// get a pointer to the function under test:</span>
|
||||
<span class="identifier">TestType</span> <span class="special">(*</span><span class="identifier">funcp</span><span class="special">)(</span><span class="identifier">TestType</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tgamma</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// declare something to hold the result:</span>
|
||||
<span class="identifier">test_result</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">></span> <span class="identifier">result</span><span class="special">;</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// and test tgamma against data:</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="identifier">result</span> <span class="special">=</span> <span class="identifier">test</span><span class="special">(</span>
|
||||
<span class="identifier">factorials</span><span class="special">,</span>
|
||||
<span class="identifier">bind</span><span class="special">(</span><span class="identifier">funcp</span><span class="special">,</span> <span class="identifier">ret</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">>(</span><span class="identifier">_1</span><span class="special">[</span><span class="number">0</span><span class="special">])),</span> <span class="comment">// calls tgamma with factorials[row][0]</span>
|
||||
<span class="identifier">ret</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">>(</span><span class="identifier">_1</span><span class="special">[</span><span class="number">1</span><span class="special">])</span> <span class="comment">// extracts the expected result from factorials[row][1]</span>
|
||||
<span class="special">);</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// Print out some results:</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The Mean was "</span> <span class="special"><<</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">mean</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The worst error was "</span> <span class="special"><<</span> <span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">max</span><span class="special">)()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"The worst error was at row "</span> <span class="special"><<</span> <span class="identifier">result</span><span class="special">.</span><span class="identifier">worst_case</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// same again with lgamma this time:</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="identifier">funcp</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lgamma</span><span class="special">;</span>
|
||||
<span class="identifier">result</span> <span class="special">=</span> <span class="identifier">test</span><span class="special">(</span>
|
||||
<span class="identifier">factorials</span><span class="special">,</span>
|
||||
<span class="identifier">bind</span><span class="special">(</span><span class="identifier">funcp</span><span class="special">,</span> <span class="identifier">ret</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">>(</span><span class="identifier">_1</span><span class="special">[</span><span class="number">0</span><span class="special">])),</span> <span class="comment">// calls tgamma with factorials[row][0]</span>
|
||||
<span class="identifier">ret</span><span class="special"><</span><span class="identifier">TestType</span><span class="special">>(</span><span class="identifier">_1</span><span class="special">[</span><span class="number">2</span><span class="special">])</span> <span class="comment">// extracts the expected result from factorials[row][2]</span>
|
||||
<span class="special">);</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// etc ...</span>
|
||||
<span class="comment">//</span>
|
||||
</pre>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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>
|
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<div class="spirit-nav">
|
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<a accesskey="p" href="minimax.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="test_data.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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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.internals.minimax"></a><a class="link" href="minimax.html" title="Minimax Approximations and the Remez Algorithm">Minimax Approximations
|
||||
and the Remez Algorithm</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
The directory libs/math/minimax contains a command line driven program for
|
||||
the generation of minimax approximations using the Remez algorithm. Both
|
||||
polynomial and rational approximations are supported, although the latter
|
||||
are tricky to converge: it is not uncommon for convergence of rational forms
|
||||
to fail. No such limitations are present for polynomial approximations which
|
||||
should always converge smoothly.
|
||||
</p>
|
||||
<p>
|
||||
It's worth stressing that developing rational approximations to functions
|
||||
is often not an easy task, and one to which many books have been devoted.
|
||||
To use this tool, you will need to have a reasonable grasp of what the Remez
|
||||
algorithm is, and the general form of the approximation you want to achieve.
|
||||
</p>
|
||||
<p>
|
||||
Unless you already familar with the Remez method, you should first read the
|
||||
<a class="link" href="../remez.html" title="The Remez Method">brief background article explaining the
|
||||
principles behind the Remez algorithm</a>.
|
||||
</p>
|
||||
<p>
|
||||
The program consists of two parts:
|
||||
</p>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl class="variablelist">
|
||||
<dt><span class="term">main.cpp</span></dt>
|
||||
<dd><p>
|
||||
Contains the command line parser, and all the calls to the Remez code.
|
||||
</p></dd>
|
||||
<dt><span class="term">f.cpp</span></dt>
|
||||
<dd><p>
|
||||
Contains the function to approximate.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</div>
|
||||
<p>
|
||||
Therefore to use this tool, you must modify f.cpp to return the function
|
||||
to approximate. The tools supports multiple function approximations within
|
||||
the same compiled program: each as a separate variant:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">&</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">variant</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Returns the value of the function <span class="emphasis"><em>variant</em></span> at point
|
||||
<span class="emphasis"><em>x</em></span>. So if you wish you can just add the function to approximate
|
||||
as a new variant after the existing examples.
|
||||
</p>
|
||||
<p>
|
||||
In addition to those two files, the program needs to be linked to a <a class="link" href="../high_precision/use_ntl.html" title="Using NTL Library">patched NTL library to compile</a>.
|
||||
</p>
|
||||
<p>
|
||||
Note that the function <span class="emphasis"><em>f</em></span> must return the rational part
|
||||
of the approximation: for example if you are approximating a function <span class="emphasis"><em>f(x)</em></span>
|
||||
then it is quite common to use:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">g</span><span class="special">(</span><span class="identifier">x</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">x</span><span class="special">))</span>
|
||||
</pre>
|
||||
<p>
|
||||
where <span class="emphasis"><em>g(x)</em></span> is the dominant part of <span class="emphasis"><em>f(x)</em></span>,
|
||||
<span class="emphasis"><em>Y</em></span> is some constant, and <span class="emphasis"><em>R(x)</em></span> is
|
||||
the rational approximation part, usually optimised for a low absolute error
|
||||
compared to |Y|.
|
||||
</p>
|
||||
<p>
|
||||
In this case you would define <span class="emphasis"><em>f</em></span> to return <span class="emphasis"><em>f(x)/g(x)</em></span>
|
||||
and then set the y-offset of the approximation to <span class="emphasis"><em>Y</em></span>
|
||||
(see command line options below).
|
||||
</p>
|
||||
<p>
|
||||
Many other forms are possible, but in all cases the objective is to split
|
||||
<span class="emphasis"><em>f(x)</em></span> into a dominant part that you can evaluate easily
|
||||
using standard math functions, and a smooth and slowly changing rational
|
||||
approximation part. Refer to your favourite textbook for more examples.
|
||||
</p>
|
||||
<p>
|
||||
Command line options for the program are as follows:
|
||||
</p>
|
||||
<div class="variablelist">
|
||||
<p class="title"><b></b></p>
|
||||
<dl class="variablelist">
|
||||
<dt><span class="term">variant N</span></dt>
|
||||
<dd><p>
|
||||
Sets the current function variant to N. This allows multiple functions
|
||||
that are to be approximated to be compiled into the same executable.
|
||||
Defaults to 0.
|
||||
</p></dd>
|
||||
<dt><span class="term">range a b</span></dt>
|
||||
<dd><p>
|
||||
Sets the domain for the approximation to the range [a,b], defaults
|
||||
to [0,1].
|
||||
</p></dd>
|
||||
<dt><span class="term">relative</span></dt>
|
||||
<dd><p>
|
||||
Sets the Remez code to optimise for relative error. This is the default
|
||||
at program startup. Note that relative error can only be used if f(x)
|
||||
has no roots over the range being optimised.
|
||||
</p></dd>
|
||||
<dt><span class="term">absolute</span></dt>
|
||||
<dd><p>
|
||||
Sets the Remez code to optimise for absolute error.
|
||||
</p></dd>
|
||||
<dt><span class="term">pin [true|false]</span></dt>
|
||||
<dd><p>
|
||||
"Pins" the code so that the rational approximation passes
|
||||
through the origin. Obviously only set this to <span class="emphasis"><em>true</em></span>
|
||||
if R(0) must be zero. This is typically used when trying to preserve
|
||||
a root at [0,0] while also optimising for relative error.
|
||||
</p></dd>
|
||||
<dt><span class="term">order N D</span></dt>
|
||||
<dd><p>
|
||||
Sets the order of the approximation to <span class="emphasis"><em>N</em></span> in the
|
||||
numerator and <span class="emphasis"><em>D</em></span> in the denominator. If <span class="emphasis"><em>D</em></span>
|
||||
is zero then the result will be a polynomial approximation. There will
|
||||
be N+D+2 coefficients in total, the first coefficient of the numerator
|
||||
is zero if <span class="emphasis"><em>pin</em></span> was set to true, and the first
|
||||
coefficient of the denominator is always one.
|
||||
</p></dd>
|
||||
<dt><span class="term">working-precision N</span></dt>
|
||||
<dd><p>
|
||||
Sets the working precision of NTL::RR to <span class="emphasis"><em>N</em></span> binary
|
||||
digits. Defaults to 250.
|
||||
</p></dd>
|
||||
<dt><span class="term">target-precision N</span></dt>
|
||||
<dd><p>
|
||||
Sets the precision of printed output to <span class="emphasis"><em>N</em></span> binary
|
||||
digits: set to the same number of digits as the type that will be used
|
||||
to evaluate the approximation. Defaults to 53 (for double precision).
|
||||
</p></dd>
|
||||
<dt><span class="term">skew val</span></dt>
|
||||
<dd><p>
|
||||
"Skews" the initial interpolated control points towards one
|
||||
end or the other of the range. Positive values skew the initial control
|
||||
points towards the left hand side of the range, and negative values
|
||||
towards the right hand side. If an approximation won't converge (a
|
||||
common situation) try adjusting the skew parameter until the first
|
||||
step yields the smallest possible error. <span class="emphasis"><em>val</em></span> should
|
||||
be in the range [-100,+100], the default is zero.
|
||||
</p></dd>
|
||||
<dt><span class="term">brake val</span></dt>
|
||||
<dd><p>
|
||||
Sets a brake on each step so that the change in the control points
|
||||
is braked by <span class="emphasis"><em>val%</em></span>. Defaults to 50, try a higher
|
||||
value if an approximation won't converge, or a lower value to get speedier
|
||||
convergence.
|
||||
</p></dd>
|
||||
<dt><span class="term">x-offset val</span></dt>
|
||||
<dd><p>
|
||||
Sets the x-offset to <span class="emphasis"><em>val</em></span>: the approximation will
|
||||
be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
|
||||
where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
|
||||
is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
|
||||
to zero. To avoid rounding errors, take care to specify a value that
|
||||
can be exactly represented as a floating point number.
|
||||
</p></dd>
|
||||
<dt><span class="term">x-scale val</span></dt>
|
||||
<dd><p>
|
||||
Sets the x-scale to <span class="emphasis"><em>val</em></span>: the approximation will
|
||||
be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
|
||||
where <span class="emphasis"><em>S</em></span> is the x-scale, <span class="emphasis"><em>X</em></span>
|
||||
is the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
|
||||
to one. To avoid rounding errors, take care to specify a value that
|
||||
can be exactly represented as a floating point number.
|
||||
</p></dd>
|
||||
<dt><span class="term">y-offset val</span></dt>
|
||||
<dd><p>
|
||||
Sets the y-offset to <span class="emphasis"><em>val</em></span>: the approximation will
|
||||
be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
|
||||
where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
|
||||
is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
|
||||
to zero. To avoid rounding errors, take care to specify a value that
|
||||
can be exactly represented as a floating point number.
|
||||
</p></dd>
|
||||
<dt><span class="term">y-offset auto</span></dt>
|
||||
<dd><p>
|
||||
Sets the y-offset to the average value of f(x) evaluated at the two
|
||||
endpoints of the range plus the midpoint of the range. The calculated
|
||||
value is deliberately truncated to <span class="emphasis"><em>float</em></span> precision
|
||||
(and should be stored as a <span class="emphasis"><em>float</em></span> in your code).
|
||||
The approximation will be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">Y</span></code> where <span class="emphasis"><em>X</em></span> is
|
||||
the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults to
|
||||
zero.
|
||||
</p></dd>
|
||||
<dt><span class="term">graph N</span></dt>
|
||||
<dd><p>
|
||||
Prints N evaluations of f(x) at evenly spaced points over the range
|
||||
being optimised. If unspecified then <span class="emphasis"><em>N</em></span> defaults
|
||||
to 3. Use to check that f(x) is indeed smooth over the range of interest.
|
||||
</p></dd>
|
||||
<dt><span class="term">step N</span></dt>
|
||||
<dd><p>
|
||||
Performs <span class="emphasis"><em>N</em></span> steps, or one step if <span class="emphasis"><em>N</em></span>
|
||||
is unspecified. After each step prints: the peek error at the extrema
|
||||
of the error function of the approximation, the theoretical error term
|
||||
solved for on the last step, and the maximum relative change in the
|
||||
location of the Chebyshev control points. The approximation is converged
|
||||
on the minimax solution when the two error terms are (approximately)
|
||||
equal, and the change in the control points has decreased to a suitably
|
||||
small value.
|
||||
</p></dd>
|
||||
<dt><span class="term">test [float|double|long]</span></dt>
|
||||
<dd><p>
|
||||
Tests the current approximation at float, double, or long double precision.
|
||||
Useful to check for rounding errors in evaluating the approximation
|
||||
at fixed precision. Tests are conducted at the extrema of the error
|
||||
function of the approximation, and at the zeros of the error function.
|
||||
</p></dd>
|
||||
<dt><span class="term">test [float|double|long] N</span></dt>
|
||||
<dd><p>
|
||||
Tests the current approximation at float, double, or long double precision.
|
||||
Useful to check for rounding errors in evaluating the approximation
|
||||
at fixed precision. Tests are conducted at N evenly spaced points over
|
||||
the range of the approximation. If none of [float|double|long] are
|
||||
specified then tests using NTL::RR, this can be used to obtain the
|
||||
error function of the approximation.
|
||||
</p></dd>
|
||||
<dt><span class="term">rescale a b</span></dt>
|
||||
<dd><p>
|
||||
Takes the current Chebeshev control points, and rescales them over
|
||||
a new interval [a,b]. Sometimes this can be used to obtain starting
|
||||
control points for an approximation that can not otherwise be converged.
|
||||
</p></dd>
|
||||
<dt><span class="term">rotate</span></dt>
|
||||
<dd><p>
|
||||
Moves one term from the numerator to the denominator, but keeps the
|
||||
Chebyshev control points the same. Sometimes this can be used to obtain
|
||||
starting control points for an approximation that can not otherwise
|
||||
be converged.
|
||||
</p></dd>
|
||||
<dt><span class="term">info</span></dt>
|
||||
<dd><p>
|
||||
Prints out the current approximation: the location of the zeros of
|
||||
the error function, the location of the Chebyshev control points, the
|
||||
x and y offsets, and of course the coefficients of the polynomials.
|
||||
</p></dd>
|
||||
</dl>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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="tuples.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="error_test.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,278 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Polynomials</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals.polynomials"></a><a class="link" href="polynomials.html" title="Polynomials">Polynomials</a>
|
||||
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<h5>
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||||
<a name="math_toolkit.internals.polynomials.h0"></a>
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||||
<span class="phrase"><a name="math_toolkit.internals.polynomials.synopsis"></a></span><a class="link" href="polynomials.html#math_toolkit.internals.polynomials.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">class</span> <span class="identifier">polynomial</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="comment">// typedefs:</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value_type</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">size_type</span> <span class="identifier">size_type</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// construct:</span>
|
||||
<span class="identifier">polynomial</span><span class="special">(){}</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">data</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">order</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">I</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">(</span><span class="identifier">I</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">I</span> <span class="identifier">last</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">explicit</span> <span class="identifier">polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">point</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// access:</span>
|
||||
<span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">size_type</span> <span class="identifier">degree</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">value_type</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">);</span>
|
||||
<span class="keyword">const</span> <span class="identifier">value_type</span><span class="special">&</span> <span class="keyword">operator</span><span class="special">[](</span><span class="identifier">size_type</span> <span class="identifier">i</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">data</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">();</span>
|
||||
|
||||
<span class="comment">// operators:</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">+=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">-=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">*=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">/=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special">&</span> <span class="keyword">operator</span> <span class="special">%=(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">U</span><span class="special">>&</span> <span class="identifier">value</span><span class="special">);</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">/</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">%</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">/</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">%</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">+</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">*</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span> <span class="special">-</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">==</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">!=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">&</span><span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">charT</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">traits</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="keyword">operator</span> <span class="special"><<</span>
|
||||
<span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">os</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">typename</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>,</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">></span>
|
||||
<span class="identifier">quotient_remainder</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
|
||||
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.polynomials.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.polynomials.description"></a></span><a class="link" href="polynomials.html#math_toolkit.internals.polynomials.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This is a somewhat trivial class for polynomial manipulation.
|
||||
</p>
|
||||
<p>
|
||||
Implementation is currently of the "naive" variety, with 𝑶(N<sup>2</sup>) multiplication,
|
||||
for example. This class should not be used in high-performance computing
|
||||
environments: it is intended for the simple manipulation of small polynomials,
|
||||
typically generated for special function approximation.
|
||||
</p>
|
||||
<p>
|
||||
It does has division for polynomials over a field (floating point, complex,
|
||||
etc) and over a unique factorization domain (integers). Division of polynomials
|
||||
over a field is compatible with Euclidean GCD.
|
||||
</p>
|
||||
<p>
|
||||
Advanced manipulations: the FFT, factorisation etc are not currently provided.
|
||||
Submissions for these are of course welcome :-)
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.polynomials.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.polynomials.polynomial_examples"></a></span><a class="link" href="polynomials.html#math_toolkit.internals.polynomials.polynomial_examples">Polynomial
|
||||
Arithmetic Examples</a>
|
||||
</h5>
|
||||
<p>
|
||||
First include the essential polynomial header (and others) to make the example:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">polynomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<p>
|
||||
and some using statements are convenient:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">abs</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">lexical_cast</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
Store the coefficients in a convenient way to access them, then create some
|
||||
polynomials using construction from an iterator range, and finally output
|
||||
in a 'pretty' formula format.
|
||||
</p>
|
||||
<div class="tip"><table border="0" summary="Tip">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../doc/src/images/tip.png"></td>
|
||||
<th align="left">Tip</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
Although we might conventionally write a polynomial from left to right
|
||||
in descending order of degree, Boost.Math stores in <span class="bold"><strong>ascending
|
||||
order of degree</strong></span>.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<pre class="programlisting"><span class="identifier">Read</span><span class="special">/</span><span class="identifier">write</span> <span class="keyword">for</span> <span class="identifier">humans</span><span class="special">:</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span>
|
||||
<span class="identifier">Boost</span> <span class="identifier">polynomial</span> <span class="identifier">storage</span><span class="special">:</span> <span class="special">[</span> <span class="number">10</span><span class="special">,</span> <span class="special">-</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">4</span><span class="special">,</span> <span class="number">3</span> <span class="special">]</span>
|
||||
</pre>
|
||||
<pre class="programlisting"> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">4</span><span class="special">></span> <span class="keyword">const</span> <span class="identifier">d3a</span> <span class="special">=</span> <span class="special">{{</span><span class="number">10</span><span class="special">,</span> <span class="special">-</span><span class="number">6</span><span class="special">,</span> <span class="special">-</span><span class="number">4</span><span class="special">,</span> <span class="number">3</span><span class="special">}};</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">2</span><span class="special">></span> <span class="keyword">const</span> <span class="identifier">d1a</span> <span class="special">=</span> <span class="special">{{-</span><span class="number">2</span><span class="special">,</span> <span class="number">1</span><span class="special">}};</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="keyword">const</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">d3a</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">d3a</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="keyword">const</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">d1a</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">d1a</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
|
||||
<span class="comment">// pretty_format() converts from Boost storage to human notation.</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span>
|
||||
<span class="special"><<</span> <span class="string">"\nb = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">b</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n\n"</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
for output:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">a</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">6</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">10</span>
|
||||
<span class="identifier">b</span> <span class="special">=</span> <span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="comment">// Now we can do arithmetic with the usual infix operators: + - * / and %.</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">s</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a + b = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">s</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a - b = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">d</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a * b = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">p</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">q</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">/</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a / b = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">q</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
<span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span><span class="special">;</span>
|
||||
<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"a % b = "</span> <span class="special"><<</span> <span class="identifier">pretty_format</span><span class="special">(</span><span class="identifier">r</span><span class="special">)</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
for output
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">5</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">8</span>
|
||||
<span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">-</span> <span class="number">4</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">-</span> <span class="number">7</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">12</span>
|
||||
<span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">4</span> <span class="special">-</span> <span class="number">10</span><span class="identifier">x</span><span class="special">^</span><span class="number">3</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">22</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">20</span>
|
||||
<span class="identifier">a</span> <span class="special">/</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">3</span><span class="identifier">x</span><span class="special">^</span><span class="number">2</span> <span class="special">+</span> <span class="number">2</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">2</span>
|
||||
<span class="identifier">a</span> <span class="special">%</span> <span class="identifier">b</span> <span class="special">=</span> <span class="number">6</span>
|
||||
</pre>
|
||||
<h6>
|
||||
<a name="math_toolkit.internals.polynomials.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.polynomials.division_quotient_and_remainder"></a></span><a class="link" href="polynomials.html#math_toolkit.internals.polynomials.division_quotient_and_remainder">Division,
|
||||
Quotient and Remainder</a>
|
||||
</h6>
|
||||
<p>
|
||||
Division is a special case where you can calculate two for the price of one.
|
||||
</p>
|
||||
<p>
|
||||
Actually, quotient and remainder are always calculated together due to the
|
||||
nature of the algorithm: the infix operators return one result and throw
|
||||
the other away.
|
||||
</p>
|
||||
<p>
|
||||
If you are doing a lot of division and want both the quotient and remainder,
|
||||
then you don't want to do twice the work necessary.
|
||||
</p>
|
||||
<p>
|
||||
In that case you can call the underlying function, <code class="literal">quotient_remainder</code>,
|
||||
to get both results together as a pair.
|
||||
</p>
|
||||
<pre class="programlisting"> <span class="identifier">pair</span><span class="special"><</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">>,</span> <span class="identifier">polynomial</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="special">></span> <span class="identifier">result</span><span class="special">;</span>
|
||||
<span class="identifier">result</span> <span class="special">=</span> <span class="identifier">quotient_remainder</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
|
||||
<span class="comment">// Reassure ourselves that the result is the same.</span>
|
||||
<span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">first</span> <span class="special">==</span> <span class="identifier">q</span><span class="special">);</span>
|
||||
<span class="identifier">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">result</span><span class="special">.</span><span class="identifier">second</span> <span class="special">==</span> <span class="identifier">r</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
The source code is at <a href="../../../../example/polynomial_arithmetic.cpp" target="_top">polynomial_arithmetic.cpp</a>
|
||||
</p>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
|
||||
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">
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
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<a accesskey="p" href="cf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.roots.rational"></a><a class="link" href="rational.html" title="Polynomial and Rational Function Evaluation">Polynomial and Rational
|
||||
Function Evaluation</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.roots.rational.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.roots.rational.synopsis"></a></span><a class="link" href="rational.html#math_toolkit.roots.rational.synopsis">synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">rational</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="comment">// Polynomials:</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// Even polynomials:</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// Odd polynomials </span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// Rational Functions:</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">b</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">denom</span><span class="special">,</span> <span class="identifier">V</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.roots.rational.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.roots.rational.description"></a></span><a class="link" href="rational.html#math_toolkit.roots.rational.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
Each of the functions come in three variants: a pair of overloaded functions
|
||||
where the order of the polynomial or rational function is evaluated at compile
|
||||
time, and an overload that accepts a runtime variable for the size of the
|
||||
coefficient array. Generally speaking, compile time evaluation of the array
|
||||
size results in better type safety, is less prone to programmer errors, and
|
||||
may result in better optimised code. The polynomial evaluation functions
|
||||
in particular, are specialised for various array sizes, allowing for loop
|
||||
unrolling, and one hopes, optimal inline expansion.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">val</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Evaluates the <a href="http://en.wikipedia.org/wiki/Polynomial" target="_top">polynomial</a>
|
||||
described by the coefficients stored in <span class="emphasis"><em>poly</em></span>.
|
||||
</p>
|
||||
<p>
|
||||
If the size of the array is specified at runtime, then the polynomial most
|
||||
have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> coefficients.
|
||||
Otherwise it has order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span>
|
||||
coefficients.
|
||||
</p>
|
||||
<p>
|
||||
Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms
|
||||
are in poly[i].
|
||||
</p>
|
||||
<p>
|
||||
The types of the coefficients and of variable <span class="emphasis"><em>z</em></span> may
|
||||
differ as long as <span class="emphasis"><em>*poly</em></span> is convertible to type <span class="emphasis"><em>U</em></span>.
|
||||
This allows, for example, for the coefficient table to be a table of integers
|
||||
if this is appropriate.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">poly</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">poly</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_even_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
As above, but evaluates an even polynomial: one where all the powers of
|
||||
<span class="emphasis"><em>z</em></span> are even numbers. Equivalent to calling <code class="computeroutput"><span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="identifier">poly</span><span class="special">,</span> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">count</span><span class="special">)</span></code>.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">a</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="identifier">U</span> <span class="identifier">evaluate_odd_polynomial</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">poly</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
As above but evaluates a polynomial where all the powers are odd numbers.
|
||||
Equivalent to <code class="computeroutput"><span class="identifier">evaluate_polynomial</span><span class="special">(</span><span class="identifier">poly</span><span class="special">+</span><span class="number">1</span><span class="special">,</span> <span class="identifier">z</span><span class="special">*</span><span class="identifier">z</span><span class="special">,</span>
|
||||
<span class="identifier">count</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">*</span>
|
||||
<span class="identifier">z</span> <span class="special">+</span> <span class="identifier">poly</span><span class="special">[</span><span class="number">0</span><span class="special">]</span></code>.
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">(&</span><span class="identifier">num</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">(&</span><span class="identifier">denom</span><span class="special">)[</span><span class="identifier">N</span><span class="special">],</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">size_t</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">U</span><span class="special">,</span><span class="identifier">N</span><span class="special">>&</span> <span class="identifier">denom</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="identifier">V</span> <span class="identifier">evaluate_rational</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span><span class="special">*</span> <span class="identifier">num</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">*</span> <span class="identifier">denom</span><span class="special">,</span> <span class="identifier">V</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">count</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Evaluates the rational function (the ratio of two polynomials) described
|
||||
by the coefficients stored in <span class="emphasis"><em>num</em></span> and <span class="emphasis"><em>demom</em></span>.
|
||||
</p>
|
||||
<p>
|
||||
If the size of the array is specified at runtime then both polynomials most
|
||||
have order <span class="emphasis"><em>count-1</em></span> with <span class="emphasis"><em>count</em></span> coefficients.
|
||||
Otherwise both polynomials have order <span class="emphasis"><em>N-1</em></span> with <span class="emphasis"><em>N</em></span>
|
||||
coefficients.
|
||||
</p>
|
||||
<p>
|
||||
Array <span class="emphasis"><em>num</em></span> describes the numerator, and <span class="emphasis"><em>demon</em></span>
|
||||
the denominator.
|
||||
</p>
|
||||
<p>
|
||||
Coefficients should be stored such that the coefficients for the x<sup>i </sup> terms
|
||||
are in num[i] and denom[i].
|
||||
</p>
|
||||
<p>
|
||||
The types of the coefficients and of variable <span class="emphasis"><em>v</em></span> may
|
||||
differ as long as <span class="emphasis"><em>*num</em></span> and <span class="emphasis"><em>*denom</em></span>
|
||||
are convertible to type <span class="emphasis"><em>V</em></span>. This allows, for example,
|
||||
for one or both of the coefficient tables to be a table of integers if this
|
||||
is appropriate.
|
||||
</p>
|
||||
<p>
|
||||
These functions are designed to safely evaluate the result, even when the
|
||||
value <span class="emphasis"><em>z</em></span> is very large. As such they do not take advantage
|
||||
of compile time array sizes to make any optimisations. These functions are
|
||||
best reserved for situations where <span class="emphasis"><em>z</em></span> may be large: if
|
||||
you can be sure that numerical overflow will not occur then polynomial evaluation
|
||||
with compile-time array sizes may offer slightly better performance.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.roots.rational.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.roots.rational.implementation"></a></span><a class="link" href="rational.html#math_toolkit.roots.rational.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
Polynomials are evaluated by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
|
||||
method</a>. If the array size is known at compile time then the functions
|
||||
dispatch to size-specific implementations that unroll the evaluation loop.
|
||||
</p>
|
||||
<p>
|
||||
Rational evaluation is by <a href="http://en.wikipedia.org/wiki/Horner_algorithm" target="_top">Horners
|
||||
method</a>: with the two polynomials being evaluated in parallel to make
|
||||
the most of the processors floating-point pipeline. If <span class="emphasis"><em>v</em></span>
|
||||
is greater than one, then the polynomials are evaluated in reverse order
|
||||
as polynomials in <span class="emphasis"><em>1/v</em></span>: this avoids unnecessary numerical
|
||||
overflow when the coefficients are large.
|
||||
</p>
|
||||
<p>
|
||||
Both the polynomial and rational function evaluation algorithms can be tuned
|
||||
using various configuration macros to provide optimal performance for a particular
|
||||
combination of compiler and platform. This includes support for second-order
|
||||
Horner's methods. The various options are <a class="link" href="../tuning.html" title="Performance Tuning Macros">documented
|
||||
here</a>. However, the performance benefits to be gained from these are
|
||||
marginal on most current hardware, consequently it's best to run the <a class="link" href="../perf_test_app.html" title="The Performance Test Applications">performance test application</a>
|
||||
before changing the default settings.
|
||||
</p>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
|
||||
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="cf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
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|
||||
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|
||||
</html>
|
||||
@@ -0,0 +1,197 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Series Evaluation</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
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<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="up" href="../internals.html" title="Internal tools">
|
||||
<link rel="prev" href="../internals.html" title="Internal tools">
|
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<link rel="next" href="cf.html" title="Continued Fraction Evaluation">
|
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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 align="center"><a href="../../../../../../index.html">Home</a></td>
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<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
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<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
|
||||
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<a accesskey="p" href="../internals.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="cf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.internals.series_evaluation"></a><a class="link" href="series_evaluation.html" title="Series Evaluation">Series Evaluation</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.series_evaluation.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.series_evaluation.synopsis"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">series</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">init_value</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">></span>
|
||||
<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// The following interfaces are now deprecated:</span>
|
||||
<span class="comment">// </span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
|
||||
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">);</span>
|
||||
|
||||
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.series_evaluation.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.series_evaluation.description"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
These algorithms are intended for the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" target="_top">summation
|
||||
of infinite series</a>.
|
||||
</p>
|
||||
<p>
|
||||
Each of the algorithms takes a nullary-function object as the first argument:
|
||||
the function object will be repeatedly invoked to pull successive terms from
|
||||
the series being summed.
|
||||
</p>
|
||||
<p>
|
||||
The second argument is the precision required, summation will stop when the
|
||||
next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The
|
||||
deprecated versions of sum_series take an integer number of bits here - internally
|
||||
they just convert this to a tolerance and forward the call.
|
||||
</p>
|
||||
<p>
|
||||
The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
|
||||
the number of terms of the series to evaluate. In addition, on exit the function
|
||||
will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of
|
||||
the series that were evaluated: this is particularly useful for profiling
|
||||
the convergence properties of a new series.
|
||||
</p>
|
||||
<p>
|
||||
The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial
|
||||
value of the sum to which the terms of the series should be added. This is
|
||||
useful in two situations:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
Where the first value of the series has a different formula to successive
|
||||
terms. In this case the first value in the series can be passed as the
|
||||
last argument and the logic of the function object can then be simplified
|
||||
to return subsequent terms.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Where the series is being added (or subtracted) from some other value:
|
||||
termination of the series will likely occur much more rapidly if that
|
||||
other value is passed as the last argument. For example, there are several
|
||||
functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where
|
||||
S(z) is an infinite series. In this case, pass -1 as the last argument
|
||||
and then negate the result of the summation to get the result of <span class="emphasis"><em>1
|
||||
- S(z)</em></span>.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms
|
||||
maintain a carry term that corrects for roundoff error during summation.
|
||||
They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan
|
||||
Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
|
||||
Every Computer Scientist Should Know About Floating-Point Arithmetic</a>.
|
||||
However, it should be pointed out that there are very few series that require
|
||||
summation in this way.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.series_evaluation.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.series_evaluation.example"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.example">Example</a>
|
||||
</h5>
|
||||
<p>
|
||||
Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
|
||||
infinite series,
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
|
||||
</p>
|
||||
<p>
|
||||
We begin by writing a small function object to return successive terms of
|
||||
the series:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">log1p_series</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// we must define a result_type typedef:</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">){}</span>
|
||||
|
||||
<span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// This is the function operator invoked by the summation</span>
|
||||
<span class="comment">// algorithm, the first call to this operator should return</span>
|
||||
<span class="comment">// the first term of the series, the second call the second </span>
|
||||
<span class="comment">// term and so on.</span>
|
||||
<span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
|
||||
<span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
<span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
|
||||
<span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span>
|
||||
<span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
Implementing log(1+x) is now fairly trivial:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">T</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">// We really should add some error checking on x here!</span>
|
||||
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special"><</span> <span class="number">1</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// Construct the series functor:</span>
|
||||
<span class="identifier">log1p_series</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
|
||||
<span class="comment">// Set a limit on how many iterations we permit:</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">max_iter</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
|
||||
<span class="comment">// Add it up, with enough precision for full machine precision:</span>
|
||||
<span class="keyword">return</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">(),</span> <span class="identifier">max_iter</span><span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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="../internals.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="cf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -0,0 +1,563 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Graphing, Profiling, and Generating Test Data for Special Functions</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
|
||||
<link rel="up" href="../internals.html" title="Internal tools">
|
||||
<link rel="prev" href="error_test.html" title="Relative Error and Testing">
|
||||
<link rel="next" href="../../using_udt.html" title="Chapter 14. Use with User-Defined Floating-Point Types - Boost.Multiprecision and others">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<table cellpadding="2" width="100%"><tr>
|
||||
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
|
||||
<td align="center"><a href="../../../../../../index.html">Home</a></td>
|
||||
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
|
||||
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
|
||||
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
|
||||
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="error_test.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="../../using_udt.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
<a name="math_toolkit.internals.test_data"></a><a class="link" href="test_data.html" title="Graphing, Profiling, and Generating Test Data for Special Functions">Graphing, Profiling,
|
||||
and Generating Test Data for Special Functions</a>
|
||||
</h3></div></div></div>
|
||||
<p>
|
||||
The class <code class="computeroutput"><span class="identifier">test_data</span></code> and associated
|
||||
helper functions are designed so that in just a few lines of code you should
|
||||
be able to:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
Profile a continued fraction, or infinite series for convergence and
|
||||
accuracy.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Generate csv data from a special function that can be imported into your
|
||||
favorite graphing program (or spreadsheet) for further analysis.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Generate high precision test data.
|
||||
</li>
|
||||
</ul></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.test_data.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.synopsis"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test_data</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<div class="important"><table border="0" summary="Important">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../../../../../doc/src/images/important.png"></td>
|
||||
<th align="left">Important</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
This is a non-core Boost.Math header that is predominantly used for internal
|
||||
maintenance of the library: as a result the library is located under <code class="computeroutput"><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">include_private</span></code> and you will need to
|
||||
add that directory to your include path in order to use this feature.
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
|
||||
|
||||
<span class="keyword">enum</span> <span class="identifier">parameter_type</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">random_in_range</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span>
|
||||
<span class="identifier">periodic_in_range</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span>
|
||||
<span class="identifier">power_series</span> <span class="special">=</span> <span class="number">2</span><span class="special">,</span>
|
||||
<span class="identifier">dummy_param</span> <span class="special">=</span> <span class="number">0x80</span><span class="special">,</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">parameter_info</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_random_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">start_range</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">end_range</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n_points</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_periodic_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">start_range</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">end_range</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n_points</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_power_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">basis</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">start_exponent</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">end_exponent</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">bool</span> <span class="identifier">get_user_parameter_info</span><span class="special">(</span><span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">info</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span> <span class="identifier">param_name</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">class</span> <span class="identifier">test_data</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">row_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">row_type</span> <span class="identifier">value_type</span><span class="special">;</span>
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">set</span><span class="special"><</span><span class="identifier">row_type</span><span class="special">></span> <span class="identifier">container_type</span><span class="special">;</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">reference</span> <span class="identifier">reference</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">const_reference</span> <span class="identifier">const_reference</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">iterator</span> <span class="identifier">iterator</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">const_iterator</span> <span class="identifier">const_iterator</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">difference_type</span> <span class="identifier">difference_type</span><span class="special">;</span>
|
||||
<span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">container_type</span><span class="special">::</span><span class="identifier">size_type</span> <span class="identifier">size_type</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// creation:</span>
|
||||
<span class="identifier">test_data</span><span class="special">(){}</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
|
||||
<span class="identifier">test_data</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg1</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// insertion:</span>
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
|
||||
<span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">insert</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg1</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
|
||||
<span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">insert</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg1</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg2</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">></span>
|
||||
<span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">insert</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg1</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg2</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">arg3</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">void</span> <span class="identifier">clear</span><span class="special">();</span>
|
||||
|
||||
<span class="comment">// access:</span>
|
||||
<span class="identifier">iterator</span> <span class="identifier">begin</span><span class="special">();</span>
|
||||
<span class="identifier">iterator</span> <span class="identifier">end</span><span class="special">();</span>
|
||||
<span class="identifier">const_iterator</span> <span class="identifier">begin</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">const_iterator</span> <span class="identifier">end</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">==(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">d</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span><span class="special">!=(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">d</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">void</span> <span class="identifier">swap</span><span class="special">(</span><span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">other</span><span class="special">);</span>
|
||||
<span class="identifier">size_type</span> <span class="identifier">size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="identifier">size_type</span> <span class="identifier">max_size</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="identifier">empty</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special"><</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">dat</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special"><=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">dat</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">></span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">dat</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="keyword">bool</span> <span class="keyword">operator</span> <span class="special">>=</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">test_data</span><span class="special">&</span> <span class="identifier">dat</span><span class="special">)</span><span class="keyword">const</span><span class="special">;</span>
|
||||
<span class="special">};</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">charT</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">traits</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">write_csv</span><span class="special">(</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">os</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">test_data</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">charT</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">traits</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">write_csv</span><span class="special">(</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">basic_ostream</span><span class="special"><</span><span class="identifier">charT</span><span class="special">,</span> <span class="identifier">traits</span><span class="special">>&</span> <span class="identifier">os</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">test_data</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">charT</span><span class="special">*</span> <span class="identifier">separator</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">write_code</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">ostream</span><span class="special">&</span> <span class="identifier">os</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="identifier">test_data</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">data</span><span class="special">,</span>
|
||||
<span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span> <span class="identifier">name</span><span class="special">);</span>
|
||||
|
||||
<span class="special">}}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.test_data.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.description"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This tool is best illustrated with the following series of examples.
|
||||
</p>
|
||||
<p>
|
||||
The functionality of test_data is split into the following parts:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
A functor that implements the function for which data is being generated:
|
||||
this is the bit you have to write.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
One of more parameters that are to be passed to the functor, these are
|
||||
described in fairly abstract terms: give me N points distributed like
|
||||
<span class="emphasis"><em>this</em></span> etc.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
The class test_data, that takes the functor and descriptions of the parameters
|
||||
and computes how ever many output points have been requested, these are
|
||||
stored in a sorted container.
|
||||
</li>
|
||||
<li class="listitem">
|
||||
Routines to iterate over the test_data container and output the data
|
||||
in either csv format, or as C++ source code (as a table using Boost.Array).
|
||||
</li>
|
||||
</ul></div>
|
||||
<h6>
|
||||
<a name="math_toolkit.internals.test_data.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.example_1_output_data_for_graph_"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.example_1_output_data_for_graph_">Example
|
||||
1: Output Data for Graph Plotting</a>
|
||||
</h6>
|
||||
<p>
|
||||
For example, lets say we want to graph the lgamma function between -3 and
|
||||
100, one could do this like so:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test_data</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
|
||||
<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// create an object to hold the data:</span>
|
||||
<span class="identifier">test_data</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">data</span><span class="special">;</span>
|
||||
|
||||
<span class="comment">// insert 500 points at uniform intervals between just after -3 and 100:</span>
|
||||
<span class="keyword">double</span> <span class="special">(*</span><span class="identifier">pf</span><span class="special">)(</span><span class="keyword">double</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lgamma</span><span class="special">;</span>
|
||||
<span class="identifier">data</span><span class="special">.</span><span class="identifier">insert</span><span class="special">(</span><span class="identifier">pf</span><span class="special">,</span> <span class="identifier">make_periodic_param</span><span class="special">(-</span><span class="number">3.0</span> <span class="special">+</span> <span class="number">0.00001</span><span class="special">,</span> <span class="number">100.0</span><span class="special">,</span> <span class="number">500</span><span class="special">));</span>
|
||||
|
||||
<span class="comment">// print out in csv format:</span>
|
||||
<span class="identifier">write_csv</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="identifier">data</span><span class="special">,</span> <span class="string">", "</span><span class="special">);</span>
|
||||
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
Which, when plotted, results in:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span>
|
||||
</p>
|
||||
<h6>
|
||||
<a name="math_toolkit.internals.test_data.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.example_2_creating_test_data"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.example_2_creating_test_data">Example
|
||||
2: Creating Test Data</a>
|
||||
</h6>
|
||||
<p>
|
||||
As a second example, let's suppose we want to create highly accurate test
|
||||
data for a special function. Since many special functions have two or more
|
||||
independent parameters, it's very hard to effectively cover all of the possible
|
||||
parameter space without generating gigabytes of data at great computational
|
||||
expense. A second best approach is to provide the tools by which a user (or
|
||||
the library maintainer) can quickly generate more data on demand to probe
|
||||
the function over a particular domain of interest.
|
||||
</p>
|
||||
<p>
|
||||
In this example we'll generate test data for the beta function using <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a> at 1000 bit precision.
|
||||
Rather than call our generic version of the beta function, we'll implement
|
||||
a deliberately naive version of the beta function using lgamma, and rely
|
||||
on the high precision of the data type used to get results accurate to at
|
||||
least 128-bit precision. In this way our test data is independent of whatever
|
||||
clever tricks we may wish to use inside the our beta function.
|
||||
</p>
|
||||
<p>
|
||||
To start with then, here's the function object that creates the test data:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">ntl</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test_data</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">fstream</span><span class="special">></span>
|
||||
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test_data</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">struct</span> <span class="identifier">beta_data_generator</span>
|
||||
<span class="special">{</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">b</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// If we throw a domain error then test_data will</span>
|
||||
<span class="comment">// ignore this input point. We'll use this to filter</span>
|
||||
<span class="comment">// out all cases where a < b since the beta function</span>
|
||||
<span class="comment">// is symmetrical in a and b:</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="keyword">if</span><span class="special">(</span><span class="identifier">a</span> <span class="special"><</span> <span class="identifier">b</span><span class="special">)</span>
|
||||
<span class="keyword">throw</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">domain_error</span><span class="special">(</span><span class="string">""</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// very naively calculate spots with lgamma:</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">g1</span><span class="special">,</span> <span class="identifier">g2</span><span class="special">,</span> <span class="identifier">g3</span><span class="special">;</span>
|
||||
<span class="keyword">int</span> <span class="identifier">s1</span><span class="special">,</span> <span class="identifier">s2</span><span class="special">,</span> <span class="identifier">s3</span><span class="special">;</span>
|
||||
<span class="identifier">g1</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="special">&</span><span class="identifier">s1</span><span class="special">);</span>
|
||||
<span class="identifier">g2</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">b</span><span class="special">,</span> <span class="special">&</span><span class="identifier">s2</span><span class="special">);</span>
|
||||
<span class="identifier">g3</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">,</span> <span class="special">&</span><span class="identifier">s3</span><span class="special">);</span>
|
||||
<span class="identifier">g1</span> <span class="special">+=</span> <span class="identifier">g2</span> <span class="special">-</span> <span class="identifier">g3</span><span class="special">;</span>
|
||||
<span class="identifier">g1</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(</span><span class="identifier">g1</span><span class="special">);</span>
|
||||
<span class="identifier">g1</span> <span class="special">*=</span> <span class="identifier">s1</span> <span class="special">*</span> <span class="identifier">s2</span> <span class="special">*</span> <span class="identifier">s3</span><span class="special">;</span>
|
||||
<span class="keyword">return</span> <span class="identifier">g1</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
To create the data, we'll need to input the domains for a and b for which
|
||||
the function will be tested: the function <code class="computeroutput"><span class="identifier">get_user_parameter_info</span></code>
|
||||
is designed for just that purpose. The start of main will look something
|
||||
like:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="comment">// Set the precision on RR:</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">::</span><span class="identifier">SetPrecision</span><span class="special">(</span><span class="number">1000</span><span class="special">);</span> <span class="comment">// bits.</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">::</span><span class="identifier">SetOutputPrecision</span><span class="special">(</span><span class="number">40</span><span class="special">);</span> <span class="comment">// decimal digits.</span>
|
||||
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">></span> <span class="identifier">arg1</span><span class="special">,</span> <span class="identifier">arg2</span><span class="special">;</span>
|
||||
<span class="identifier">test_data</span><span class="special"><</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">></span> <span class="identifier">data</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Welcome.\n"</span>
|
||||
<span class="string">"This program will generate spot tests for the beta function:\n"</span>
|
||||
<span class="string">" beta(a, b)\n\n"</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">bool</span> <span class="identifier">cont</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">string</span> <span class="identifier">line</span><span class="special">;</span>
|
||||
|
||||
<span class="keyword">do</span><span class="special">{</span>
|
||||
<span class="comment">// prompt the user for the domain of a and b to test:</span>
|
||||
<span class="identifier">get_user_parameter_info</span><span class="special">(</span><span class="identifier">arg1</span><span class="special">,</span> <span class="string">"a"</span><span class="special">);</span>
|
||||
<span class="identifier">get_user_parameter_info</span><span class="special">(</span><span class="identifier">arg2</span><span class="special">,</span> <span class="string">"b"</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// create the data:</span>
|
||||
<span class="identifier">data</span><span class="special">.</span><span class="identifier">insert</span><span class="special">(</span><span class="identifier">beta_data_generator</span><span class="special">(),</span> <span class="identifier">arg1</span><span class="special">,</span> <span class="identifier">arg2</span><span class="special">);</span>
|
||||
|
||||
<span class="comment">// see if the user want's any more domains tested:</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Any more data [y/n]?"</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">getline</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cin</span><span class="special">,</span> <span class="identifier">line</span><span class="special">);</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">algorithm</span><span class="special">::</span><span class="identifier">trim</span><span class="special">(</span><span class="identifier">line</span><span class="special">);</span>
|
||||
<span class="identifier">cont</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">line</span> <span class="special">==</span> <span class="string">"y"</span><span class="special">);</span>
|
||||
<span class="special">}</span><span class="keyword">while</span><span class="special">(</span><span class="identifier">cont</span><span class="special">);</span>
|
||||
</pre>
|
||||
<div class="caution"><table border="0" summary="Caution">
|
||||
<tr>
|
||||
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../doc/src/images/caution.png"></td>
|
||||
<th align="left">Caution</th>
|
||||
</tr>
|
||||
<tr><td align="left" valign="top"><p>
|
||||
At this point one potential stumbling block should be mentioned: test_data<>::insert
|
||||
will create a matrix of test data when there are two or more parameters,
|
||||
so if we have two parameters and we're asked for a thousand points on each,
|
||||
that's a <span class="emphasis"><em>million test points in total</em></span>. Don't say you
|
||||
weren't warned!
|
||||
</p></td></tr>
|
||||
</table></div>
|
||||
<p>
|
||||
There's just one final step now, and that's to write the test data to file:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Enter name of test data file [default=beta_data.ipp]"</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">getline</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cin</span><span class="special">,</span> <span class="identifier">line</span><span class="special">);</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">algorithm</span><span class="special">::</span><span class="identifier">trim</span><span class="special">(</span><span class="identifier">line</span><span class="special">);</span>
|
||||
<span class="keyword">if</span><span class="special">(</span><span class="identifier">line</span> <span class="special">==</span> <span class="string">""</span><span class="special">)</span>
|
||||
<span class="identifier">line</span> <span class="special">=</span> <span class="string">"beta_data.ipp"</span><span class="special">;</span>
|
||||
<span class="identifier">std</span><span class="special">::</span><span class="identifier">ofstream</span> <span class="identifier">ofs</span><span class="special">(</span><span class="identifier">line</span><span class="special">.</span><span class="identifier">c_str</span><span class="special">());</span>
|
||||
<span class="identifier">write_code</span><span class="special">(</span><span class="identifier">ofs</span><span class="special">,</span> <span class="identifier">data</span><span class="special">,</span> <span class="string">"beta_data"</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
The format of the test data looks something like:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#define</span> <span class="identifier">SC_</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">BOOST_JOIN</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="identifier">L</span><span class="special">))</span>
|
||||
<span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="number">3</span><span class="special">>,</span> <span class="number">1830</span><span class="special">></span>
|
||||
<span class="identifier">beta_med_data</span> <span class="special">=</span> <span class="special">{</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">0.4883005917072296142578125</span><span class="special">),</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">0.4883005917072296142578125</span><span class="special">),</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">3.245912809500479157065104747353807392371</span><span class="special">),</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">3.5808107852935791015625</span><span class="special">),</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">0.4883005917072296142578125</span><span class="special">),</span>
|
||||
<span class="identifier">SC_</span><span class="special">(</span><span class="number">1.007653173802923954909901438393379243537</span><span class="special">),</span>
|
||||
<span class="comment">/* ... lots of rows skipped */</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
The first two values in each row are the input parameters that were passed
|
||||
to our functor and the last value is the return value from the functor. Had
|
||||
our functor returned a <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>
|
||||
rather than a value, then we would have had one entry for each element in
|
||||
the tuple in addition to the input parameters.
|
||||
</p>
|
||||
<p>
|
||||
The first #define serves two purposes:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
It reduces the file sizes considerably: all those <code class="computeroutput"><span class="keyword">static_cast</span></code>'s
|
||||
add up to a lot of bytes otherwise (they are needed to suppress compiler
|
||||
warnings when <code class="computeroutput"><span class="identifier">T</span></code> is narrower
|
||||
than a <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>).
|
||||
</li>
|
||||
<li class="listitem">
|
||||
It provides a useful customisation point: for example if we were testing
|
||||
a user-defined type that has more precision than a <code class="computeroutput"><span class="keyword">long</span>
|
||||
<span class="keyword">double</span></code> we could change it to:
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
<code class="literal">#define SC_(x) lexical_cast<T>(BOOST_STRINGIZE(x))</code>
|
||||
</p>
|
||||
<p>
|
||||
in order to ensure that no truncation of the values occurs prior to conversion
|
||||
to <code class="computeroutput"><span class="identifier">T</span></code>. Note that this isn't
|
||||
used by default as it's rather hard on the compiler when the table is large.
|
||||
</p>
|
||||
<h6>
|
||||
<a name="math_toolkit.internals.test_data.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.example_3_profiling_a_continued_"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.example_3_profiling_a_continued_">Example
|
||||
3: Profiling a Continued Fraction for Convergence and Accuracy</a>
|
||||
</h6>
|
||||
<p>
|
||||
Alternatively, lets say we want to profile a continued fraction for convergence
|
||||
and error. As an example, we'll use the continued fraction for the upper
|
||||
incomplete gamma function, the following function object returns the next
|
||||
a<sub>N </sub> and b<sub>N </sub> of the continued fraction each time it's invoked:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">upper_incomplete_gamma_fract</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">private</span><span class="special">:</span>
|
||||
<span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">a</span><span class="special">;</span>
|
||||
<span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
|
||||
<span class="keyword">public</span><span class="special">:</span>
|
||||
<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">result_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">upper_incomplete_gamma_fract</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">a1</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z1</span><span class="special">)</span>
|
||||
<span class="special">:</span> <span class="identifier">z</span><span class="special">(</span><span class="identifier">z1</span><span class="special">-</span><span class="identifier">a1</span><span class="special">+</span><span class="number">1</span><span class="special">),</span> <span class="identifier">a</span><span class="special">(</span><span class="identifier">a1</span><span class="special">),</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="special">}</span>
|
||||
|
||||
<span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
|
||||
<span class="identifier">z</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
|
||||
<span class="keyword">return</span> <span class="identifier">result_type</span><span class="special">(</span><span class="identifier">k</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">k</span><span class="special">),</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
We want to measure both the relative error, and the rate of convergence of
|
||||
this fraction, so we'll write a functor that returns both as a <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a>:
|
||||
class test_data will unpack the tuple for us, and create one column of data
|
||||
for each element in the tuple (in addition to the input parameters):
|
||||
</p>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test_data</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">test</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">gamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">ntl</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">tuple</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">struct</span> <span class="identifier">profile_gamma_fraction</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">typedef</span> <a class="link" href="tuples.html" title="Tuples">boost::math::tuple</a><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">result_type</span><span class="special">;</span>
|
||||
|
||||
<span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="identifier">val</span><span class="special">)</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
<span class="comment">// estimate the true value, using arbitary precision</span>
|
||||
<span class="comment">// arithmetic and NTL::RR:</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">rval</span><span class="special">(</span><span class="identifier">val</span><span class="special">);</span>
|
||||
<span class="identifier">upper_incomplete_gamma_fract</span><span class="special"><</span><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">></span> <span class="identifier">f1</span><span class="special">(</span><span class="identifier">rval</span><span class="special">,</span> <span class="identifier">rval</span><span class="special">);</span>
|
||||
<span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">true_val</span> <span class="special">=</span> <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">f1</span><span class="special">,</span> <span class="number">1000</span><span class="special">);</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// Now get the aproximation at double precision, along with the number of</span>
|
||||
<span class="comment">// iterations required:</span>
|
||||
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">iters</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">>::</span><span class="identifier">max</span><span class="special">();</span>
|
||||
<span class="identifier">upper_incomplete_gamma_fract</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">f2</span><span class="special">(</span><span class="identifier">val</span><span class="special">,</span> <span class="identifier">val</span><span class="special">);</span>
|
||||
<span class="identifier">T</span> <span class="identifier">found_val</span> <span class="special">=</span> <span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">f2</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">,</span> <span class="identifier">iters</span><span class="special">);</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// Work out the relative error, as measured in units of epsilon:</span>
|
||||
<span class="identifier">T</span> <span class="identifier">err</span> <span class="special">=</span> <span class="identifier">real_cast</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">relative_error</span><span class="special">(</span><span class="identifier">true_val</span><span class="special">,</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">(</span><span class="identifier">found_val</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
||||
<span class="comment">//</span>
|
||||
<span class="comment">// now just return the results as a tuple:</span>
|
||||
<span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">make_tuple</span><span class="special">(</span><span class="identifier">err</span><span class="special">,</span> <span class="identifier">iters</span><span class="special">);</span>
|
||||
<span class="special">}</span>
|
||||
<span class="special">};</span>
|
||||
</pre>
|
||||
<p>
|
||||
Feeding that functor into test_data allows rapid output of csv data, for
|
||||
whatever type <code class="computeroutput"><span class="identifier">T</span></code> we may be
|
||||
interested in:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
|
||||
<span class="special">{</span>
|
||||
<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">;</span>
|
||||
<span class="comment">// create an object to hold the data:</span>
|
||||
<span class="identifier">test_data</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">data</span><span class="special">;</span>
|
||||
<span class="comment">// insert 500 points at uniform intervals between just after 0 and 100:</span>
|
||||
<span class="identifier">data</span><span class="special">.</span><span class="identifier">insert</span><span class="special">(</span><span class="identifier">profile_gamma_fraction</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span> <span class="identifier">make_periodic_param</span><span class="special">(</span><span class="number">0.01</span><span class="special">,</span> <span class="number">100.0</span><span class="special">,</span> <span class="number">100</span><span class="special">));</span>
|
||||
<span class="comment">// print out in csv format:</span>
|
||||
<span class="identifier">write_csv</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">,</span> <span class="identifier">data</span><span class="special">,</span> <span class="string">", "</span><span class="special">);</span>
|
||||
<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
|
||||
<span class="special">}</span>
|
||||
</pre>
|
||||
<p>
|
||||
This time there's no need to plot a graph, the first few rows are:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">a</span> <span class="keyword">and</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Error</span><span class="special">/</span><span class="identifier">epsilon</span><span class="special">,</span> <span class="identifier">Iterations</span> <span class="identifier">required</span>
|
||||
|
||||
<span class="number">0.01</span><span class="special">,</span> <span class="number">9723.14</span><span class="special">,</span> <span class="number">4726</span>
|
||||
<span class="number">1.0099</span><span class="special">,</span> <span class="number">9.54818</span><span class="special">,</span> <span class="number">87</span>
|
||||
<span class="number">2.0098</span><span class="special">,</span> <span class="number">3.84777</span><span class="special">,</span> <span class="number">40</span>
|
||||
<span class="number">3.0097</span><span class="special">,</span> <span class="number">0.728358</span><span class="special">,</span> <span class="number">25</span>
|
||||
<span class="number">4.0096</span><span class="special">,</span> <span class="number">2.39712</span><span class="special">,</span> <span class="number">21</span>
|
||||
<span class="number">5.0095</span><span class="special">,</span> <span class="number">0.233263</span><span class="special">,</span> <span class="number">16</span>
|
||||
</pre>
|
||||
<p>
|
||||
So it's pretty clear that this fraction shouldn't be used for small values
|
||||
of a and z.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.test_data.h5"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.test_data.reference"></a></span><a class="link" href="test_data.html#math_toolkit.internals.test_data.reference">reference</a>
|
||||
</h5>
|
||||
<p>
|
||||
Most of this tool has been described already in the examples above, we'll
|
||||
just add the following notes on the non-member functions:
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_random_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">start_range</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">end_range</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n_points</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Tells class test_data to test <span class="emphasis"><em>n_points</em></span> random values
|
||||
in the range [start_range,end_range].
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_periodic_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">start_range</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">end_range</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">n_points</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Tells class test_data to test <span class="emphasis"><em>n_points</em></span> evenly spaced
|
||||
values in the range [start_range,end_range].
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">make_power_param</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">basis</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">start_exponent</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">end_exponent</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Tells class test_data to test points of the form <span class="emphasis"><em>basis + R * 2<sup>expon</sup></em></span>
|
||||
for each <span class="emphasis"><em>expon</em></span> in the range [start_exponent, end_exponent],
|
||||
and <span class="emphasis"><em>R</em></span> a random number in [0.5, 1].
|
||||
</p>
|
||||
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
||||
<span class="keyword">bool</span> <span class="identifier">get_user_parameter_info</span><span class="special">(</span><span class="identifier">parameter_info</span><span class="special"><</span><span class="identifier">T</span><span class="special">>&</span> <span class="identifier">info</span><span class="special">,</span> <span class="keyword">const</span> <span class="keyword">char</span><span class="special">*</span> <span class="identifier">param_name</span><span class="special">);</span>
|
||||
</pre>
|
||||
<p>
|
||||
Prompts the user for the parameter range and form to use.
|
||||
</p>
|
||||
<p>
|
||||
Finally, if we don't want the parameter to be included in the output, we
|
||||
can tell test_data by setting it a "dummy parameter":
|
||||
</p>
|
||||
<pre class="programlisting"><span class="identifier">parameter_info</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">make_random_param</span><span class="special">(</span><span class="number">2.0</span><span class="special">,</span> <span class="number">5.0</span><span class="special">,</span> <span class="number">10</span><span class="special">);</span>
|
||||
<span class="identifier">p</span><span class="special">.</span><span class="identifier">type</span> <span class="special">|=</span> <span class="identifier">dummy_param</span><span class="special">;</span>
|
||||
</pre>
|
||||
<p>
|
||||
This is useful when the functor used transforms the parameter in some way
|
||||
before passing it to the function under test, usually the functor will then
|
||||
return both the transformed input and the result in a tuple, so there's no
|
||||
need for the original pseudo-parameter to be included in program output.
|
||||
</p>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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>
|
||||
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|
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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.internals.tuples"></a><a class="link" href="tuples.html" title="Tuples">Tuples</a>
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</h3></div></div></div>
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<h5>
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||||
<a name="math_toolkit.internals.tuples.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.tuples.synopsis"></a></span><a class="link" href="tuples.html#math_toolkit.internals.tuples.synopsis">Synopsis</a>
|
||||
</h5>
|
||||
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">tuple</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.internals.tuples.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.internals.tuples.description"></a></span><a class="link" href="tuples.html#math_toolkit.internals.tuples.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
This header defines the type <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tuple</span></code>,
|
||||
the associated free functions <code class="computeroutput"><span class="identifier">ignore</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">tie</span></code>, <code class="computeroutput"><span class="identifier">make_tuple</span></code>,
|
||||
<code class="computeroutput"><span class="identifier">get</span></code>, and associated types
|
||||
<code class="computeroutput"><span class="identifier">tuple_size</span></code> and <code class="computeroutput"><span class="identifier">tuple_element</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
These types and functions are aliases for:
|
||||
</p>
|
||||
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tuple</span></code> etc when available, otherwise:
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">tr1</span><span class="special">::</span><span class="identifier">tuple</span></code> etc when available, otherwise:
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">fusion</span><span class="special">::</span><span class="identifier">tuple</span></code> etc if the compiler supports
|
||||
it, otherwise:
|
||||
</li>
|
||||
<li class="listitem">
|
||||
<code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">tuple</span></code>.
|
||||
</li>
|
||||
</ul></div>
|
||||
<p>
|
||||
So this <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tuple</span></code> is strongly recommended for maximum
|
||||
portability.
|
||||
</p>
|
||||
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
||||
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
||||
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
||||
Benjamin Sobotta, 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">
|
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<a accesskey="p" href="cf.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.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="minimax.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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