libs/math/doc/html/math_toolkit/internals/cf.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Continued Fraction Evaluation</title>
   5 <link rel="stylesheet" href="../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../internals.html" title="Internal tools">
   9 <link rel="prev" href="series_evaluation.html" title="Series Evaluation">
  10 <link rel="next" href="recurrence.html" title="Tools For 3-Term Recurrence Relations">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="series_evaluation.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="recurrence.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.internals.cf"></a><a class="link" href="cf.html" title="Continued Fraction Evaluation">Continued Fraction Evaluation</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.internals.cf.h0"></a>
  31         <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>
  32       </h5>
  33 <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>
  34 </pre>
  35 <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>
  36
  37 <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>
  38 <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>
  39    <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>
  40
  41 <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>
  42 <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>
  43    <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>
  44
  45 <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>
  46 <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>
  47    <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>
  48
  49 <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>
  50 <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>
  51    <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>
  52
  53 <span class="comment">//</span>
  54 <span class="comment">// These interfaces are present for legacy reasons, and are now deprecated:</span>
  55 <span class="comment">//</span>
  56 <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>
  57 <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>
  58    <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>
  59
  60 <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>
  61 <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>
  62    <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>
  63
  64 <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>
  65 <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>
  66    <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>
  67
  68 <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>
  69 <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>
  70    <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>
  71
  72 <span class="special">}}}</span> <span class="comment">// namespaces</span>
  73 </pre>
  74 <h5>
  75 <a name="math_toolkit.internals.cf.h1"></a>
  76         <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>
  77       </h5>
  78 <p>
  79         <a href="http://en.wikipedia.org/wiki/Continued_fraction" target="_top">Continued fractions
  80         are a common method of approximation. </a> These functions all evaluate
  81         the continued fraction described by the <span class="emphasis"><em>generator</em></span> type
  82         argument. The functions with an "_a" suffix evaluate the fraction:
  83       </p>
  84 <div class="blockquote"><blockquote class="blockquote"><p>
  85           <span class="inlinemediaobject"><img src="../../../equations/fraction2.svg"></span>
  86
  87         </p></blockquote></div>
  88 <p>
  89         and those with a "_b" suffix evaluate the fraction:
  90       </p>
  91 <div class="blockquote"><blockquote class="blockquote"><p>
  92           <span class="inlinemediaobject"><img src="../../../equations/fraction1.svg"></span>
  93
  94         </p></blockquote></div>
  95 <p>
  96         This latter form is somewhat more natural in that it corresponds with the
  97         usual definition of a continued fraction, but note that the first <span class="emphasis"><em>a</em></span>
  98         value returned by the generator is discarded. Further, often the first <span class="emphasis"><em>a</em></span>
  99         and <span class="emphasis"><em>b</em></span> values in a continued fraction have different
 100         defining equations to the remaining terms, which may make the "_a"
 101         suffixed form more appropriate.
 102       </p>
 103 <p>
 104         The generator type should be a function object which supports the following
 105         operations:
 106       </p>
 107 <div class="informaltable"><table class="table">
 108 <colgroup>
 109 <col>
 110 <col>
 111 </colgroup>
 112 <thead><tr>
 113 <th>
 114                 <p>
 115                   Expression
 116                 </p>
 117               </th>
 118 <th>
 119                 <p>
 120                   Description
 121                 </p>
 122               </th>
 123 </tr></thead>
 124 <tbody>
 125 <tr>
 126 <td>
 127                 <p>
 128                   Gen::result_type
 129                 </p>
 130               </td>
 131 <td>
 132                 <p>
 133                   The type that is the result of invoking operator(). This can be
 134                   either an arithmetic or complex type, or a std::pair&lt;&gt; of
 135                   arithmetic or complex types.
 136                 </p>
 137               </td>
 138 </tr>
 139 <tr>
 140 <td>
 141                 <p>
 142                   g()
 143                 </p>
 144               </td>
 145 <td>
 146                 <p>
 147                   Returns an object of type Gen::result_type.
 148                 </p>
 149                 <p>
 150                   Each time this operator is called then the next pair of <span class="emphasis"><em>a</em></span>
 151                   and <span class="emphasis"><em>b</em></span> values is returned. Or, if result_type
 152                   is an arithmetic type, then the next <span class="emphasis"><em>b</em></span> value
 153                   is returned and all the <span class="emphasis"><em>a</em></span> values are assumed
 154                   to 1.
 155                 </p>
 156               </td>
 157 </tr>
 158 </tbody>
 159 </table></div>
 160 <p>
 161         In all the continued fraction evaluation functions the <span class="emphasis"><em>tolerance</em></span>
 162         parameter is the precision desired in the result, evaluation of the fraction
 163         will continue until the last term evaluated leaves the relative error in
 164         the result less than <span class="emphasis"><em>tolerance</em></span>. The deprecated interfaces
 165         take a number of digits precision here, internally they just convert this
 166         to a tolerance and forward call.
 167       </p>
 168 <p>
 169         If the optional <span class="emphasis"><em>max_terms</em></span> parameter is specified then
 170         no more than <span class="emphasis"><em>max_terms</em></span> calls to the generator will be
 171         made, and on output, <span class="emphasis"><em>max_terms</em></span> will be set to actual
 172         number of calls made. This facility is particularly useful when profiling
 173         a continued fraction for convergence.
 174       </p>
 175 <h5>
 176 <a name="math_toolkit.internals.cf.h2"></a>
 177         <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>
 178       </h5>
 179 <p>
 180         Internally these algorithms all use the modified Lentz algorithm: refer to
 181         Numeric Recipes in C++, W. H. Press et all, chapter 5, (especially 5.2 Evaluation
 182         of continued fractions, p 175 - 179) for more information, also Lentz, W.J.
 183         1976, Applied Optics, vol. 15, pp. 668-671.
 184       </p>
 185 <h5>
 186 <a name="math_toolkit.internals.cf.h3"></a>
 187         <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>
 188       </h5>
 189 <p>
 190         All of these examples are in <a href="../../../../example/continued_fractions.cpp" target="_top">continued_fractions.cpp</a>.
 191       </p>
 192 <p>
 193         The <a href="http://en.wikipedia.org/wiki/Golden_ratio" target="_top">golden ratio phi
 194         = 1.618033989...</a> can be computed from the simplest continued fraction
 195         of all:
 196       </p>
 197 <div class="blockquote"><blockquote class="blockquote"><p>
 198           <span class="inlinemediaobject"><img src="../../../equations/fraction3.svg"></span>
 199
 200         </p></blockquote></div>
 201 <p>
 202         We begin by defining a generator function:
 203       </p>
 204 <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>
 205 <span class="keyword">struct</span> <span class="identifier">golden_ratio_fraction</span>
 206 <span class="special">{</span>
 207    <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
 208
 209    <span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()()</span>
 210    <span class="special">{</span>
 211       <span class="keyword">return</span> <span class="number">1</span><span class="special">;</span>
 212    <span class="special">}</span>
 213 <span class="special">};</span>
 214 </pre>
 215 <p>
 216         The golden ratio can then be computed to double precision using:
 217       </p>
 218 <pre class="programlisting"><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">func</span><span class="special">;</span>
 219 <span class="keyword">double</span> <span class="identifier">gr</span> <span class="special">=</span> <span class="identifier">continued_fraction_a</span><span class="special">(</span>
 220    <span class="identifier">func</span><span class="special">,</span>
 221    <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>
 222 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"The golden ratio is: "</span> <span class="special">&lt;&lt;</span> <span class="identifier">gr</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 223 </pre>
 224 <p>
 225         It's more usual though to have to define both the <span class="emphasis"><em>a</em></span>'s
 226         and the <span class="emphasis"><em>b</em></span>'s when evaluating special functions by continued
 227         fractions, for example the tan function is defined by:
 228       </p>
 229 <div class="blockquote"><blockquote class="blockquote"><p>
 230           <span class="inlinemediaobject"><img src="../../../equations/fraction4.svg"></span>
 231
 232         </p></blockquote></div>
 233 <p>
 234         So its generator object would look like:
 235       </p>
 236 <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>
 237 <span class="keyword">struct</span> <span class="identifier">tan_fraction</span>
 238 <span class="special">{</span>
 239 <span class="keyword">private</span><span class="special">:</span>
 240    <span class="identifier">T</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">;</span>
 241 <span class="keyword">public</span><span class="special">:</span>
 242    <span class="identifier">tan_fraction</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">v</span><span class="special">)</span>
 243       <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>
 244    <span class="special">{}</span>
 245
 246    <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>
 247
 248    <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>
 249    <span class="special">{</span>
 250       <span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
 251       <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>
 252    <span class="special">}</span>
 253 <span class="special">};</span>
 254 </pre>
 255 <p>
 256         Notice that if the continuant is subtracted from the <span class="emphasis"><em>b</em></span>
 257         terms, as is the case here, then all the <span class="emphasis"><em>a</em></span> terms returned
 258         by the generator will be negative. The tangent function can now be evaluated
 259         using:
 260       </p>
 261 <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>
 262 <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>
 263 <span class="special">{</span>
 264    <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>
 265    <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>
 266 <span class="special">}</span>
 267 </pre>
 268 <p>
 269         Notice that this time we're using the "_b" suffixed version to
 270         evaluate the fraction: we're removing the leading <span class="emphasis"><em>a</em></span>
 271         term during fraction evaluation as it's different from all the others.
 272       </p>
 273 <p>
 274         Now we'll look at a couple of complex number examples, starting with the
 275         exponential integral which can be calculated via:
 276       </p>
 277 <div class="blockquote"><blockquote class="blockquote"><p>
 278           <span class="inlinemediaobject"><img src="../../../equations/expint_n_3.svg"></span>
 279
 280         </p></blockquote></div>
 281 <p>
 282         So our functor looks like this:
 283       </p>
 284 <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>
 285 <span class="keyword">struct</span> <span class="identifier">expint_fraction</span>
 286 <span class="special">{</span>
 287    <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>
 288    <span class="identifier">expint_fraction</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n_</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z_</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">b</span><span class="special">(</span><span class="identifier">z_</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">i</span><span class="special">(-</span><span class="number">1</span><span class="special">),</span> <span class="identifier">n</span><span class="special">(</span><span class="identifier">n_</span><span class="special">)</span> <span class="special">{}</span>
 289    <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>
 290    <span class="special">{</span>
 291       <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</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">make_pair</span><span class="special">(-</span><span class="keyword">static_cast</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;((</span><span class="identifier">i</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">n</span> <span class="special">+</span> <span class="identifier">i</span><span class="special">)),</span> <span class="identifier">b</span><span class="special">);</span>
 292       <span class="identifier">b</span> <span class="special">+=</span> <span class="number">2</span><span class="special">;</span>
 293       <span class="special">++</span><span class="identifier">i</span><span class="special">;</span>
 294       <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
 295    <span class="special">}</span>
 296 <span class="keyword">private</span><span class="special">:</span>
 297    <span class="identifier">T</span> <span class="identifier">b</span><span class="special">;</span>
 298    <span class="keyword">int</span> <span class="identifier">i</span><span class="special">;</span>
 299    <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">;</span>
 300 <span class="special">};</span>
 301 </pre>
 302 <p>
 303         We can finish the example by wrapping everything up in a function:
 304       </p>
 305 <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>
 306 <span class="keyword">inline</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">expint_as_fraction</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">const</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">)</span>
 307 <span class="special">{</span>
 308    <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>
 309    <span class="identifier">expint_fraction</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="identifier">z</span><span class="special">);</span>
 310    <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">result</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">continued_fraction_b</span><span class="special">(</span>
 311       <span class="identifier">f</span><span class="special">,</span>
 312       <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</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="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()),</span>
 313       <span class="identifier">max_iter</span><span class="special">);</span>
 314    <span class="identifier">result</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">z</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">result</span><span class="special">;</span>
 315    <span class="keyword">return</span> <span class="identifier">result</span><span class="special">;</span>
 316 <span class="special">}</span>
 317 </pre>
 318 <p>
 319         Notice how the termination condition is still expressed as a complex number,
 320         albeit one with zero imaginary part.
 321       </p>
 322 <p>
 323         Our final example will use <code class="literal">continued_fraction_a</code>, in fact
 324         there is only one special function in our code which uses that variant, and
 325         it's the upper incomplete gamma function (Q), which can be calculated via:
 326       </p>
 327 <div class="blockquote"><blockquote class="blockquote"><p>
 328           <span class="inlinemediaobject"><img src="../../../equations/igamma9.svg"></span>
 329
 330         </p></blockquote></div>
 331 <p>
 332         In this case the first couple of terms are different from the rest, so our
 333         fraction will start with the first "regular" a term:
 334       </p>
 335 <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>
 336 <span class="keyword">struct</span> <span class="identifier">upper_incomplete_gamma_fract</span>
 337 <span class="special">{</span>
 338 <span class="keyword">private</span><span class="special">:</span>
 339    <span class="keyword">typedef</span> <span class="keyword">typename</span> <span class="identifier">T</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">scalar_type</span><span class="special">;</span>
 340    <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">a</span><span class="special">;</span>
 341    <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
 342 <span class="keyword">public</span><span class="special">:</span>
 343    <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>
 344
 345    <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>
 346       <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="identifier">scalar_type</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>
 347    <span class="special">{</span>
 348    <span class="special">}</span>
 349
 350    <span class="identifier">result_type</span> <span class="keyword">operator</span><span class="special">()()</span>
 351    <span class="special">{</span>
 352       <span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
 353       <span class="identifier">z</span> <span class="special">+=</span> <span class="identifier">scalar_type</span><span class="special">(</span><span class="number">2</span><span class="special">);</span>
 354       <span class="keyword">return</span> <span class="identifier">result_type</span><span class="special">(</span><span class="identifier">scalar_type</span><span class="special">(</span><span class="identifier">k</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">scalar_type</span><span class="special">(</span><span class="identifier">k</span><span class="special">)),</span> <span class="identifier">z</span><span class="special">);</span>
 355    <span class="special">}</span>
 356 <span class="special">};</span>
 357 </pre>
 358 <p>
 359         So now we can implement Q, this time using <code class="literal">continued_fraction_a</code>:
 360       </p>
 361 <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>
 362 <span class="keyword">inline</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">gamma_Q_as_fraction</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">a</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;&amp;</span> <span class="identifier">z</span><span class="special">)</span>
 363 <span class="special">{</span>
 364    <span class="identifier">upper_incomplete_gamma_fract</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">z</span><span class="special">);</span>
 365    <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">eps</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>
 366    <span class="keyword">return</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">z</span><span class="special">,</span> <span class="identifier">a</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">exp</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span> <span class="special">*(</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">a</span> <span class="special">+</span> <span class="identifier">T</span><span class="special">(</span><span class="number">1</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">tools</span><span class="special">::</span><span class="identifier">continued_fraction_a</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">eps</span><span class="special">)));</span>
 367 <span class="special">}</span>
 368 </pre>
 369 </div>
 370 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 371 <td align="left"></td>
 372 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 373       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 374       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 375       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 376       Daryle Walker and Xiaogang Zhang<p>
 377         Distributed under the Boost Software License, Version 1.0. (See accompanying
 378         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>)
 379       </p>
 380 </div></td>
 381 </tr></table>
 382 <hr>
 383 <div class="spirit-nav">
 384 <a accesskey="p" href="series_evaluation.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="recurrence.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 385 </div>
 386 </body>
 387 </html>