]>
Commit | Line | Data |
---|---|---|
7c673cae FG |
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"><</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">></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"><</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">></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">&</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&</span> <span class="identifier">init_value</span><span class="special">);</span> | |
39 | ||
40 | <span class="keyword">template</span> <span class="special"><</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">></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">&</span> <span class="identifier">func</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> | |
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"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></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">&</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"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></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">&</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">&</span> <span class="identifier">max_terms</span><span class="special">);</span> | |
51 | ||
52 | <span class="keyword">template</span> <span class="special"><</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> | |
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">&</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"><</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> | |
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">&</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">&</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"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></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">&</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"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></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">&</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">&</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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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"><</span><span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>::</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 © 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 Råde, 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> |