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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.internals1.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation">Continued Fraction Evaluation</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.internals1.cf.h0"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.cf.synopsis"></a></span><a class="link" href="cf.html#math_toolkit.internals1.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.internals1.cf.h1"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.cf.description"></a></span><a class="link" href="cf.html#math_toolkit.internals1.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.internals1.cf.h2"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.cf.implementation"></a></span><a class="link" href="cf.html#math_toolkit.internals1.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.internals1.cf.h3"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.cf.examples"></a></span><a class="link" href="cf.html#math_toolkit.internals1.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, 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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