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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals.series_evaluation"></a><a class="link" href="series_evaluation.html" title="Series Evaluation">Series Evaluation</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.internals.series_evaluation.h0"></a>
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<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>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">series</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">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>
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<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>
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<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>
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<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>
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<span class="comment">//</span>
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<span class="comment">// The following interfaces are now deprecated:</span>
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<span class="comment">// </span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
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<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>
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<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>
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<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>
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<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>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">></span>
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<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>
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<span class="special">}}}</span> <span class="comment">// namespaces</span>
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</pre>
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<h5>
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<a name="math_toolkit.internals.series_evaluation.h1"></a>
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<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>
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</h5>
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<p>
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These algorithms are intended for the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" target="_top">summation
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of infinite series</a>.
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</p>
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<p>
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Each of the algorithms takes a nullary-function object as the first argument:
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the function object will be repeatedly invoked to pull successive terms from
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the series being summed.
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</p>
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<p>
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The second argument is the precision required, summation will stop when the
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next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The
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deprecated versions of <code class="computeroutput"><span class="identifier">sum_series</span></code>
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take an integer number of bits here - internally they just convert this to
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a tolerance and forward the call.
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</p>
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<p>
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The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
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the number of terms of the series to evaluate. In addition, on exit the function
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will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of
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the series that were evaluated: this is particularly useful for profiling
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the convergence properties of a new series.
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</p>
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<p>
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The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial
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value of the sum to which the terms of the series should be added. This is
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useful in two situations:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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Where the first value of the series has a different formula to successive
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terms. In this case the first value in the series can be passed as the
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last argument and the logic of the function object can then be simplified
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to return subsequent terms.
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</li>
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<li class="listitem">
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Where the series is being added (or subtracted) from some other value:
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termination of the series will likely occur much more rapidly if that
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other value is passed as the last argument. For example, there are several
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functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where
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S(z) is an infinite series. In this case, pass -1 as the last argument
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and then negate the result of the summation to get the result of <span class="emphasis"><em>1
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- S(z)</em></span>.
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</li>
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</ul></div>
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<p>
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The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms
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maintain a carry term that corrects for roundoff error during summation.
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They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan
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Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
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Every Computer Scientist Should Know About Floating-Point Arithmetic</a>.
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However, it should be pointed out that there are very few series that require
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summation in this way.
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</p>
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<h5>
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<a name="math_toolkit.internals.series_evaluation.h2"></a>
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<span class="phrase"><a name="math_toolkit.internals.series_evaluation.examples"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.examples">Examples</a>
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</h5>
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<p>
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These examples are all in <a href="../../../../example/series.cpp" target="_top">../../example/series.cpp</a>
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</p>
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<p>
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Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
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infinite series,
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
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</p></blockquote></div>
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<p>
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We begin by writing a small function object to return successive terms of
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the series:
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="keyword">struct</span> <span class="identifier">log1p_series</span>
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<span class="special">{</span>
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<span class="comment">// we must define a result_type typedef:</span>
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<span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
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<span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
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<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="special">{}</span>
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<span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
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<span class="special">{</span>
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<span class="comment">// This is the function operator invoked by the summation</span>
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<span class="comment">// algorithm, the first call to this operator should return</span>
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<span class="comment">// the first term of the series, the second call the second</span>
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<span class="comment">// term and so on.</span>
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<span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
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<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>
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<span class="special">}</span>
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<span class="keyword">private</span><span class="special">:</span>
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<span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
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<span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span>
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<span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span>
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<span class="special">};</span>
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</pre>
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<p>
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Implementing log(1+x) is now fairly trivial:
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="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>
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<span class="special">{</span>
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<span class="comment">// We really should add some error checking on x here!</span>
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<span class="identifier">BOOST_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>
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<span class="comment">// Construct the series functor:</span>
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<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>
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<span class="comment">// Set a limit on how many iterations we permit:</span>
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<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>
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<span class="comment">// Add it up, with enough precision for full machine precision:</span>
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<span class="keyword">return</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">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>
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<span class="special">}</span>
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</pre>
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<p>
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We can almost use the code above for complex numbers as well - unfortunately
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we need a slightly different definition for epsilon, and within the functor,
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mixed complex and integer arithmetic is sadly not supported (as of C++17),
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so we need to cast out integers to floats:
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</p>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="keyword">struct</span> <span class="identifier">log1p_series</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="special">></span>
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<span class="special">{</span>
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<span class="comment">// we must define a result_type typedef:</span>
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<span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">result_type</span><span class="special">;</span>
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<span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">x</span><span class="special">)</span>
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<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="special">{}</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="keyword">operator</span><span class="special">()()</span>
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<span class="special">{</span>
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<span class="comment">// This is the function operator invoked by the summation</span>
|
|
<span class="comment">// algorithm, the first call to this operator should return</span>
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<span class="comment">// the first term of the series, the second call the second</span>
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<span class="comment">// term and so on.</span>
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<span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
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<span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="identifier">T</span><span class="special">(++</span><span class="identifier">k</span><span class="special">);</span>
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<span class="special">}</span>
|
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<span class="keyword">private</span><span class="special">:</span>
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<span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
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<span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">m_mult</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">m_prod</span><span class="special">;</span>
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<span class="special">};</span>
|
|
|
|
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">BOOST_ASSERT</span><span class="special">(</span><span class="identifier">abs</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">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="identifier">T</span><span class="special">></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">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">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">complex</span><span class="special"><</span><span class="identifier">T</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>
|
|
<p>
|
|
Of course with a few traits classes and a bit of meta-programming we could
|
|
fold these two implementations into one, but that's beyond the scope of these
|
|
examples.
|
|
</p>
|
|
</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 © 2006-2019 Nikhar
|
|
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
|
|
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
|
|
Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|
|
</div></td>
|
|
</tr></table>
|
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