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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.toolkit.internals1.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation"> Continued Fraction
Evaluation</a>
</h4></div></div></div>
<a name="math_toolkit.toolkit.internals1.cf.synopsis"></a><h5>
<a name="id1193644"></a>
<a class="link" href="cf.html#math_toolkit.toolkit.internals1.cf.synopsis">Synopsis</a>
</h5>
<p>
</p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
</pre>
<p>
</p>
<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">&lt;</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">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</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">&amp;</span> <span class="identifier">max_terms</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">tolerance</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</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">&amp;</span> <span class="identifier">max_terms</span><span class="special">)</span>
<span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</span> <span class="identifier">g</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</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">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">detail</span><span class="special">::</span><span class="identifier">fraction_traits</span><span class="special">&lt;</span><span class="identifier">Gen</span><span class="special">&gt;::</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">&amp;</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">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>
<span class="special">}}}</span> <span class="comment">// namespaces
</span></pre>
<a name="math_toolkit.toolkit.internals1.cf.description"></a><h5>
<a name="id1195904"></a>
<a class="link" href="cf.html#math_toolkit.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.png"></span>
</p>
<p>
and those with a "_b" suffix evaluate the fraction:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../equations/fraction1.png"></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&lt;&gt; 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>
<a name="math_toolkit.toolkit.internals1.cf.implementation"></a><h5>
<a name="id1196116"></a>
<a class="link" href="cf.html#math_toolkit.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>
<a name="math_toolkit.toolkit.internals1.cf.examples"></a><h5>
<a name="id1196132"></a>
<a class="link" href="cf.html#math_toolkit.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.png"></span>
</p>
<p>
We begin by defining a generator function:
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="keyword">double</span><span class="special">&gt;(),</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</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.png"></span>
</p>
<p>
So it's generator object would look like:
</p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="identifier">T</span><span class="special">,</span><span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</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 &#169; 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow,
Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan R&#229;de, Gautam Sewani and
Thijs van den Berg<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>
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