ceph/src/boost/libs/math/doc/html/math_toolkit/internals/series_evaluation.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Series Evaluation</title>
   5 <link rel="stylesheet" href="../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../internals.html" title="Internal tools">
   9 <link rel="prev" href="../internals.html" title="Internal tools">
  10 <link rel="next" href="cf.html" title="Continued Fraction Evaluation">
  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="../internals.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.internals.series_evaluation"></a><a class="link" href="series_evaluation.html" title="Series Evaluation">Series Evaluation</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.internals.series_evaluation.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.internals.series_evaluation.synopsis"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.synopsis">Synopsis</a>
  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">series</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">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  38 <span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</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">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">init_value</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
  41 <span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</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>
  42
  43 <span class="comment">//</span>
  44 <span class="comment">// The following interfaces are now deprecated:</span>
  45 <span class="comment">//   </span>
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
  47 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</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">Functor</span><span class="special">&gt;</span>
  50 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>
  51
  52 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  53 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>
  54
  55 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
  56 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>
  57
  58 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
  59 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
  60
  61 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
  62 <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>
  63
  64 <span class="special">}}}</span> <span class="comment">// namespaces</span>
  65 </pre>
  66 <h5>
  67 <a name="math_toolkit.internals.series_evaluation.h1"></a>
  68         <span class="phrase"><a name="math_toolkit.internals.series_evaluation.description"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.description">Description</a>
  69       </h5>
  70 <p>
  71         These algorithms are intended for the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" target="_top">summation
  72         of infinite series</a>.
  73       </p>
  74 <p>
  75         Each of the algorithms takes a nullary-function object as the first argument:
  76         the function object will be repeatedly invoked to pull successive terms from
  77         the series being summed.
  78       </p>
  79 <p>
  80         The second argument is the precision required, summation will stop when the
  81         next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The
  82         deprecated versions of sum_series take an integer number of bits here - internally
  83         they just convert this to a tolerance and forward the call.
  84       </p>
  85 <p>
  86         The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
  87         the number of terms of the series to evaluate. In addition, on exit the function
  88         will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of
  89         the series that were evaluated: this is particularly useful for profiling
  90         the convergence properties of a new series.
  91       </p>
  92 <p>
  93         The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial
  94         value of the sum to which the terms of the series should be added. This is
  95         useful in two situations:
  96       </p>
  97 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  98 <li class="listitem">
  99             Where the first value of the series has a different formula to successive
 100             terms. In this case the first value in the series can be passed as the
 101             last argument and the logic of the function object can then be simplified
 102             to return subsequent terms.
 103           </li>
 104 <li class="listitem">
 105             Where the series is being added (or subtracted) from some other value:
 106             termination of the series will likely occur much more rapidly if that
 107             other value is passed as the last argument. For example, there are several
 108             functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where
 109             S(z) is an infinite series. In this case, pass -1 as the last argument
 110             and then negate the result of the summation to get the result of <span class="emphasis"><em>1
 111             - S(z)</em></span>.
 112           </li>
 113 </ul></div>
 114 <p>
 115         The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms
 116         maintain a carry term that corrects for roundoff error during summation.
 117         They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan
 118         Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
 119         Every Computer Scientist Should Know About Floating-Point Arithmetic</a>.
 120         However, it should be pointed out that there are very few series that require
 121         summation in this way.
 122       </p>
 123 <h5>
 124 <a name="math_toolkit.internals.series_evaluation.h2"></a>
 125         <span class="phrase"><a name="math_toolkit.internals.series_evaluation.example"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.example">Example</a>
 126       </h5>
 127 <p>
 128         Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
 129         infinite series,
 130       </p>
 131 <p>
 132         <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
 133       </p>
 134 <p>
 135         We begin by writing a small function object to return successive terms of
 136         the series:
 137       </p>
 138 <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>
 139 <span class="keyword">struct</span> <span class="identifier">log1p_series</span>
 140 <span class="special">{</span>
 141    <span class="comment">// we must define a result_type typedef:</span>
 142    <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
 143
 144    <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
 145       <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">){}</span>
 146
 147    <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
 148    <span class="special">{</span>
 149       <span class="comment">// This is the function operator invoked by the summation</span>
 150       <span class="comment">// algorithm, the first call to this operator should return</span>
 151       <span class="comment">// the first term of the series, the second call the second </span>
 152       <span class="comment">// term and so on.</span>
 153       <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
 154       <span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
 155    <span class="special">}</span>
 156
 157 <span class="keyword">private</span><span class="special">:</span>
 158    <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
 159    <span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span>
 160    <span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span>
 161 <span class="special">};</span>
 162 </pre>
 163 <p>
 164         Implementing log(1+x) is now fairly trivial:
 165       </p>
 166 <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>
 167 <span class="identifier">T</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
 168 <span class="special">{</span>
 169    <span class="comment">// We really should add some error checking on x here!</span>
 170    <span class="identifier">assert</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>
 171
 172    <span class="comment">// Construct the series functor:</span>
 173    <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
 174    <span class="comment">// Set a limit on how many iterations we permit:</span>
 175    <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>
 176    <span class="comment">// Add it up, with enough precision for full machine precision:</span>
 177    <span class="keyword">return</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">(),</span> <span class="identifier">max_iter</span><span class="special">);</span>
 178 <span class="special">}</span>
 179 </pre>
 180 </div>
 181 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 182 <td align="left"></td>
 183 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 184       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 185       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 186       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 187         Distributed under the Boost Software License, Version 1.0. (See accompanying
 188         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>)
 189       </p>
 190 </div></td>
 191 </tr></table>
 192 <hr>
 193 <div class="spirit-nav">
 194 <a accesskey="p" href="../internals.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="cf.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 195 </div>
 196 </body>
 197 </html>