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  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 <code class="computeroutput"><span class="identifier">sum_series</span></code>
  83         take an integer number of bits here - internally they just convert this to
  84         a tolerance and forward the call.
  85       </p>
  86 <p>
  87         The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
  88         the number of terms of the series to evaluate. In addition, on exit the function
  89         will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of
  90         the series that were evaluated: this is particularly useful for profiling
  91         the convergence properties of a new series.
  92       </p>
  93 <p>
  94         The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial
  95         value of the sum to which the terms of the series should be added. This is
  96         useful in two situations:
  97       </p>
  98 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  99 <li class="listitem">
 100             Where the first value of the series has a different formula to successive
 101             terms. In this case the first value in the series can be passed as the
 102             last argument and the logic of the function object can then be simplified
 103             to return subsequent terms.
 104           </li>
 105 <li class="listitem">
 106             Where the series is being added (or subtracted) from some other value:
 107             termination of the series will likely occur much more rapidly if that
 108             other value is passed as the last argument. For example, there are several
 109             functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where
 110             S(z) is an infinite series. In this case, pass -1 as the last argument
 111             and then negate the result of the summation to get the result of <span class="emphasis"><em>1
 112             - S(z)</em></span>.
 113           </li>
 114 </ul></div>
 115 <p>
 116         The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms
 117         maintain a carry term that corrects for roundoff error during summation.
 118         They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan
 119         Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
 120         Every Computer Scientist Should Know About Floating-Point Arithmetic</a>.
 121         However, it should be pointed out that there are very few series that require
 122         summation in this way.
 123       </p>
 124 <h5>
 125 <a name="math_toolkit.internals.series_evaluation.h2"></a>
 126         <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>
 127       </h5>
 128 <p>
 129         These examples are all in <a href="../../../../example/series.cpp" target="_top">../../example/series.cpp</a>
 130       </p>
 131 <p>
 132         Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
 133         infinite series,
 134       </p>
 135 <div class="blockquote"><blockquote class="blockquote"><p>
 136           <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>
 137
 138         </p></blockquote></div>
 139 <p>
 140         We begin by writing a small function object to return successive terms of
 141         the series:
 142       </p>
 143 <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>
 144 <span class="keyword">struct</span> <span class="identifier">log1p_series</span>
 145 <span class="special">{</span>
 146    <span class="comment">// we must define a result_type typedef:</span>
 147    <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>
 148
 149    <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
 150       <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>
 151
 152    <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
 153    <span class="special">{</span>
 154       <span class="comment">// This is the function operator invoked by the summation</span>
 155       <span class="comment">// algorithm, the first call to this operator should return</span>
 156       <span class="comment">// the first term of the series, the second call the second</span>
 157       <span class="comment">// term and so on.</span>
 158       <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
 159       <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>
 160    <span class="special">}</span>
 161
 162 <span class="keyword">private</span><span class="special">:</span>
 163    <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
 164    <span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span>
 165    <span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span>
 166 <span class="special">};</span>
 167 </pre>
 168 <p>
 169         Implementing log(1+x) is now fairly trivial:
 170       </p>
 171 <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>
 172 <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>
 173 <span class="special">{</span>
 174    <span class="comment">// We really should add some error checking on x here!</span>
 175    <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">&lt;</span> <span class="number">1</span><span class="special">);</span>
 176
 177    <span class="comment">// Construct the series functor:</span>
 178    <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>
 179    <span class="comment">// Set a limit on how many iterations we permit:</span>
 180    <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>
 181    <span class="comment">// Add it up, with enough precision for full machine precision:</span>
 182    <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">&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>
 183 <span class="special">}</span>
 184 </pre>
 185 <p>
 186         We can almost use the code above for complex numbers as well - unfortunately
 187         we need a slightly different definition for epsilon, and within the functor,
 188         mixed complex and integer arithmetic is sadly not supported (as of C++17),
 189         so we need to cast out integers to floats:
 190       </p>
 191 <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>
 192 <span class="keyword">struct</span> <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span>
 193 <span class="special">{</span>
 194    <span class="comment">// we must define a result_type typedef:</span>
 195    <span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">result_type</span><span class="special">;</span>
 196
 197    <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
 198       <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>
 199
 200    <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span><span class="special">()()</span>
 201    <span class="special">{</span>
 202       <span class="comment">// This is the function operator invoked by the summation</span>
 203       <span class="comment">// algorithm, the first call to this operator should return</span>
 204       <span class="comment">// the first term of the series, the second call the second</span>
 205       <span class="comment">// term and so on.</span>
 206       <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
 207       <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>
 208    <span class="special">}</span>
 209
 210 <span class="keyword">private</span><span class="special">:</span>
 211    <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
 212    <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_mult</span><span class="special">;</span>
 213    <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_prod</span><span class="special">;</span>
 214 <span class="special">};</span>
 215
 216
 217 <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>
 218 <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
 219 <span class="special">{</span>
 220    <span class="comment">// We really should add some error checking on x here!</span>
 221    <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">&lt;</span> <span class="number">1</span><span class="special">);</span>
 222
 223    <span class="comment">// Construct the series functor:</span>
 224    <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
 225    <span class="comment">// Set a limit on how many iterations we permit:</span>
 226    <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>
 227    <span class="comment">// Add it up, with enough precision for full machine precision:</span>
 228    <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()),</span> <span class="identifier">max_iter</span><span class="special">);</span>
 229 <span class="special">}</span>
 230 </pre>
 231 <p>
 232         Of course with a few traits classes and a bit of meta-programming we could
 233         fold these two implementations into one, but that's beyond the scope of these
 234         examples.
 235       </p>
 236 </div>
 237 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 238 <td align="left"></td>
 239 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 240       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 241       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 242       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 243       Daryle Walker and Xiaogang Zhang<p>
 244         Distributed under the Boost Software License, Version 1.0. (See accompanying
 245         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>)
 246       </p>
 247 </div></td>
 248 </tr></table>
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