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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.internals1.series_evaluation"></a><a class="link" href="series_evaluation.html" title="Series Evaluation">Series Evaluation</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.internals1.series_evaluation.h0"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.series_evaluation.synopsis"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals1.series_evaluation.synopsis">Synopsis</a> |
| </h5> |
| <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> |
| </pre> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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> |
| <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> |
| |
| <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> |
| <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="comment">//</span> |
| <span class="comment">// The following interfaces are now deprecated:</span> |
| <span class="comment">// </span> |
| <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">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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></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">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="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">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> |
| |
| <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">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> |
| |
| <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">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="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span> |
| <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> |
| |
| <span class="special">}}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.internals1.series_evaluation.h1"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.series_evaluation.description"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals1.series_evaluation.description">Description</a> |
| </h5> |
| <p> |
| These algorithms are intended for the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" target="_top">summation |
| of infinite series</a>. |
| </p> |
| <p> |
| Each of the algorithms takes a nullary-function object as the first argument: |
| the function object will be repeatedly invoked to pull successive terms from |
| the series being summed. |
| </p> |
| <p> |
| The second argument is the precision required, summation will stop when the |
| next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The |
| deprecated versions of sum_series take an integer number of bits here - internally |
| they just convert this to a tolerance and forward the call. |
| </p> |
| <p> |
| The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on |
| the number of terms of the series to evaluate. In addition, on exit the function |
| will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of |
| the series that were evaluated: this is particularly useful for profiling |
| the convergence properties of a new series. |
| </p> |
| <p> |
| The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial |
| value of the sum to which the terms of the series should be added. This is |
| useful in two situations: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Where the first value of the series has a different formula to successive |
| terms. In this case the first value in the series can be passed as the |
| last argument and the logic of the function object can then be simplified |
| to return subsequent terms. |
| </li> |
| <li class="listitem"> |
| Where the series is being added (or subtracted) from some other value: |
| termination of the series will likely occur much more rapidly if that |
| other value is passed as the last argument. For example, there are several |
| functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where |
| S(z) is an infinite series. In this case, pass -1 as the last argument |
| and then negate the result of the summation to get the result of <span class="emphasis"><em>1 |
| - S(z)</em></span>. |
| </li> |
| </ul></div> |
| <p> |
| The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms |
| maintain a carry term that corrects for roundoff error during summation. |
| They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan |
| Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What |
| Every Computer Scientist Should Know About Floating-Point Arithmetic</a>. |
| However, it should be pointed out that there are very few series that require |
| summation in this way. |
| </p> |
| <h5> |
| <a name="math_toolkit.internals1.series_evaluation.h2"></a> |
| <span class="phrase"><a name="math_toolkit.internals1.series_evaluation.example"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals1.series_evaluation.example">Example</a> |
| </h5> |
| <p> |
| Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its |
| infinite series, |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span> |
| </p> |
| <p> |
| We begin by writing a small function object to return successive terms of |
| the series: |
| </p> |
| <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="keyword">struct</span> <span class="identifier">log1p_series</span> |
| <span class="special">{</span> |
| <span class="comment">// we must define a result_type typedef:</span> |
| <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span> |
| |
| <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> |
| <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> |
| |
| <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span> |
| <span class="special">{</span> |
| <span class="comment">// This is the function operator invoked by the summation</span> |
| <span class="comment">// algorithm, the first call to this operator should return</span> |
| <span class="comment">// the first term of the series, the second call the second </span> |
| <span class="comment">// term and so on.</span> |
| <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span> |
| <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> |
| <span class="special">}</span> |
| |
| <span class="keyword">private</span><span class="special">:</span> |
| <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span> |
| <span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span> |
| <span class="special">};</span> |
| </pre> |
| <p> |
| Implementing log(1+x) is now fairly trivial: |
| </p> |
| <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">log1p</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="comment">// We really should add some error checking on x here!</span> |
| <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> |
| |
| <span class="comment">// Construct the series functor:</span> |
| <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> |
| <span class="comment">// Set a limit on how many iterations we permit:</span> |
| <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> |
| <span class="comment">// Add it up, with enough precision for full machine precision:</span> |
| <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> |
| <span class="special">}</span> |
| </pre> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
| </p> |
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