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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>
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<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>
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</h5>
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<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>
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</pre>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="comment">//</span>
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<span class="comment">// These interfaces are present for legacy reasons, and are now deprecated:</span>
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<span class="comment">//</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Gen</span><span class="special">></span>
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<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>
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<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>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.internals.cf.h1"></a>
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<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>
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</h5>
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<p>
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<a href="http://en.wikipedia.org/wiki/Continued_fraction" target="_top">Continued fractions
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are a common method of approximation. </a> These functions all evaluate
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the continued fraction described by the <span class="emphasis"><em>generator</em></span> type
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argument. The functions with an "_a" suffix evaluate the fraction:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/fraction2.svg"></span>
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</p>
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<p>
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and those with a "_b" suffix evaluate the fraction:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/fraction1.svg"></span>
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</p>
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<p>
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This latter form is somewhat more natural in that it corresponds with the
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usual definition of a continued fraction, but note that the first <span class="emphasis"><em>a</em></span>
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value returned by the generator is discarded. Further, often the first <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span> values in a continued fraction have different
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defining equations to the remaining terms, which may make the "_a"
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suffixed form more appropriate.
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</p>
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<p>
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The generator type should be a function object which supports the following
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operations:
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Expression
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</p>
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</th>
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<th>
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<p>
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Description
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Gen::result_type
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</p>
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</td>
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<td>
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<p>
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The type that is the result of invoking operator(). This can be
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either an arithmetic type, or a std::pair<> of arithmetic
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types.
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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g()
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</p>
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</td>
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<td>
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<p>
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Returns an object of type Gen::result_type.
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</p>
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<p>
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Each time this operator is called then the next pair of <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>b</em></span> values is returned. Or, if result_type
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is an arithmetic type, then the next <span class="emphasis"><em>b</em></span> value
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is returned and all the <span class="emphasis"><em>a</em></span> values are assumed
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to 1.
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<p>
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In all the continued fraction evaluation functions the <span class="emphasis"><em>tolerance</em></span>
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parameter is the precision desired in the result, evaluation of the fraction
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will continue until the last term evaluated leaves the relative error in
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the result less than <span class="emphasis"><em>tolerance</em></span>. The deprecated interfaces
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take a number of digits precision here, internally they just convert this
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to a tolerance and forward call.
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</p>
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<p>
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If the optional <span class="emphasis"><em>max_terms</em></span> parameter is specified then
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no more than <span class="emphasis"><em>max_terms</em></span> calls to the generator will be
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made, and on output, <span class="emphasis"><em>max_terms</em></span> will be set to actual
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number of calls made. This facility is particularly useful when profiling
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a continued fraction for convergence.
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</p>
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<h5>
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<a name="math_toolkit.internals.cf.h2"></a>
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<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>
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</h5>
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<p>
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Internally these algorithms all use the modified Lentz algorithm: refer to
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Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 Evaluation
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of continued fractions, p 175 - 179) for more information, also Lentz, W.J.
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1976, Applied Optics, vol. 15, pp. 668-671.
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</p>
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<h5>
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<a name="math_toolkit.internals.cf.h3"></a>
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<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>
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</h5>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">golden ratio phi
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= 1.618033989...</a> can be computed from the simplest continued fraction
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of all:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/fraction3.svg"></span>
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</p>
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<p>
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We begin by defining a generator function:
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</p>
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<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>
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<span class="keyword">struct</span> <span class="identifier">golden_ratio_fraction</span>
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<span class="special">{</span>
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<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
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<span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()</span>
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<span class="special">{</span>
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<span class="keyword">return</span> <span class="number">1</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="special">};</span>
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</pre>
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<p>
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The golden ratio can then be computed to double precision using:
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</p>
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<pre class="programlisting"><span class="identifier">continued_fraction_a</span><span class="special">(</span>
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<span class="identifier">golden_ratio_fraction</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(),</span>
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<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>
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</pre>
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<p>
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It's more usual though to have to define both the <span class="emphasis"><em>a</em></span>'s
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and the <span class="emphasis"><em>b</em></span>'s when evaluating special functions by continued
|
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fractions, for example the tan function is defined by:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/fraction4.svg"></span>
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</p>
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<p>
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So its generator object would look like:
|
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</p>
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<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>
|
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<span class="keyword">struct</span> <span class="identifier">tan_fraction</span>
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<span class="special">{</span>
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<span class="keyword">private</span><span class="special">:</span>
|
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<span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span>
|
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<span class="keyword">public</span><span class="special">:</span>
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<span class="identifier">tan_fraction</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">v</span><span class="special">)</span>
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<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>
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<span class="special">{}</span>
|
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<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>
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<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>
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<span class="special">{</span>
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<span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
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<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>
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<span class="special">}</span>
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<span class="special">};</span>
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</pre>
|
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<p>
|
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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:
|
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</p>
|
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<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>
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|
<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>
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<span class="special">}</span>
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</pre>
|
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<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>
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|
<td align="left"></td>
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|
<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
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Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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