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  27 <a name="boost_multiprecision.tut.floats.fp_eg.gi"></a><a class="link" href="gi.html" title="Calculating an Integral">Calculating
  28           an Integral</a>
  29 </h5></div></div></div>
  30 <p>
  31             Similar to the generic derivative example, we can calculate integrals
  32             in a similar manner:
  33           </p>
  34 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">value_type</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">function_type</span><span class="special">&gt;</span>
  35 <span class="keyword">inline</span> <span class="identifier">value_type</span> <span class="identifier">integral</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">a</span><span class="special">,</span>
  36                            <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">b</span><span class="special">,</span>
  37                            <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">tol</span><span class="special">,</span>
  38                            <span class="identifier">function_type</span> <span class="identifier">func</span><span class="special">)</span>
  39 <span class="special">{</span>
  40    <span class="keyword">unsigned</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">1U</span><span class="special">;</span>
  41
  42    <span class="identifier">value_type</span> <span class="identifier">h</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">);</span>
  43    <span class="identifier">value_type</span> <span class="identifier">I</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">func</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">func</span><span class="special">(</span><span class="identifier">b</span><span class="special">))</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">h</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span>
  44
  45    <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">k</span> <span class="special">=</span> <span class="number">0U</span><span class="special">;</span> <span class="identifier">k</span> <span class="special">&lt;</span> <span class="number">8U</span><span class="special">;</span> <span class="identifier">k</span><span class="special">++)</span>
  46    <span class="special">{</span>
  47       <span class="identifier">h</span> <span class="special">/=</span> <span class="number">2</span><span class="special">;</span>
  48
  49       <span class="identifier">value_type</span> <span class="identifier">sum</span><span class="special">(</span><span class="number">0</span><span class="special">);</span>
  50       <span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">j</span> <span class="special">=</span> <span class="number">1U</span><span class="special">;</span> <span class="identifier">j</span> <span class="special">&lt;=</span> <span class="identifier">n</span><span class="special">;</span> <span class="identifier">j</span><span class="special">++)</span>
  51       <span class="special">{</span>
  52          <span class="identifier">sum</span> <span class="special">+=</span> <span class="identifier">func</span><span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">value_type</span><span class="special">((</span><span class="identifier">j</span> <span class="special">*</span> <span class="number">2</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">h</span><span class="special">));</span>
  53       <span class="special">}</span>
  54
  55       <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">I0</span> <span class="special">=</span> <span class="identifier">I</span><span class="special">;</span>
  56       <span class="identifier">I</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">I</span> <span class="special">/</span> <span class="number">2</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">h</span> <span class="special">*</span> <span class="identifier">sum</span><span class="special">);</span>
  57
  58       <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">ratio</span>     <span class="special">=</span> <span class="identifier">I0</span> <span class="special">/</span> <span class="identifier">I</span><span class="special">;</span>
  59       <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">delta</span>     <span class="special">=</span> <span class="identifier">ratio</span> <span class="special">-</span> <span class="number">1</span><span class="special">;</span>
  60       <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">delta_abs</span> <span class="special">=</span> <span class="special">((</span><span class="identifier">delta</span> <span class="special">&lt;</span> <span class="number">0</span><span class="special">)</span> <span class="special">?</span> <span class="special">-</span><span class="identifier">delta</span> <span class="special">:</span> <span class="identifier">delta</span><span class="special">);</span>
  61
  62       <span class="keyword">if</span><span class="special">((</span><span class="identifier">k</span> <span class="special">&gt;</span> <span class="number">1U</span><span class="special">)</span> <span class="special">&amp;&amp;</span> <span class="special">(</span><span class="identifier">delta_abs</span> <span class="special">&lt;</span> <span class="identifier">tol</span><span class="special">))</span>
  63       <span class="special">{</span>
  64          <span class="keyword">break</span><span class="special">;</span>
  65       <span class="special">}</span>
  66
  67       <span class="identifier">n</span> <span class="special">*=</span> <span class="number">2U</span><span class="special">;</span>
  68    <span class="special">}</span>
  69
  70    <span class="keyword">return</span> <span class="identifier">I</span><span class="special">;</span>
  71 <span class="special">}</span>
  72 </pre>
  73 <p>
  74             The following sample program shows how the function can be called, we
  75             begin by defining a function object, which when integrated should yield
  76             the Bessel J function:
  77           </p>
  78 <pre class="programlisting"><span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">value_type</span><span class="special">&gt;</span>
  79 <span class="keyword">class</span> <span class="identifier">cyl_bessel_j_integral_rep</span>
  80 <span class="special">{</span>
  81 <span class="keyword">public</span><span class="special">:</span>
  82    <span class="identifier">cyl_bessel_j_integral_rep</span><span class="special">(</span><span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">,</span>
  83       <span class="keyword">const</span> <span class="identifier">value_type</span><span class="special">&amp;</span> <span class="identifier">X</span><span class="special">)</span> <span class="special">:</span> <span class="identifier">n</span><span class="special">(</span><span class="identifier">N</span><span class="special">),</span> <span class="identifier">x</span><span class="special">(</span><span class="identifier">X</span><span class="special">)</span> <span class="special">{</span> <span class="special">}</span>
  84
  85    <span class="identifier">value_type</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">const</span> <span class="identifier">value_type</span><span class="special">&amp;</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span>
  86    <span class="special">{</span>
  87       <span class="comment">// pi * Jn(x) = Int_0^pi [cos(x * sin(t) - n*t) dt]</span>
  88       <span class="keyword">return</span> <span class="identifier">cos</span><span class="special">(</span><span class="identifier">x</span> <span class="special">*</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">t</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">n</span> <span class="special">*</span> <span class="identifier">t</span><span class="special">));</span>
  89    <span class="special">}</span>
  90
  91 <span class="keyword">private</span><span class="special">:</span>
  92    <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">;</span>
  93    <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">x</span><span class="special">;</span>
  94 <span class="special">};</span>
  95 </pre>
  96 <pre class="programlisting">   <span class="comment">/* The function can now be called as follows: */</span>
  97 <span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="keyword">int</span><span class="special">,</span> <span class="keyword">char</span><span class="special">**)</span>
  98 <span class="special">{</span>
  99    <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">constants</span><span class="special">::</span><span class="identifier">pi</span><span class="special">;</span>
 100    <span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span> <span class="identifier">mp_type</span><span class="special">;</span>
 101
 102    <span class="keyword">const</span> <span class="keyword">float</span> <span class="identifier">j2_f</span> <span class="special">=</span>
 103       <span class="identifier">integral</span><span class="special">(</span><span class="number">0.0F</span><span class="special">,</span>
 104       <span class="identifier">pi</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;(),</span>
 105       <span class="number">0.01F</span><span class="special">,</span>
 106       <span class="identifier">cyl_bessel_j_integral_rep</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;(</span><span class="number">2U</span><span class="special">,</span> <span class="number">1.23F</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">pi</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;();</span>
 107
 108    <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">j2_d</span> <span class="special">=</span>
 109       <span class="identifier">integral</span><span class="special">(</span><span class="number">0.0</span><span class="special">,</span>
 110       <span class="identifier">pi</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(),</span>
 111       <span class="number">0.0001</span><span class="special">,</span>
 112       <span class="identifier">cyl_bessel_j_integral_rep</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">2U</span><span class="special">,</span> <span class="number">1.23</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">pi</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;();</span>
 113
 114    <span class="keyword">const</span> <span class="identifier">mp_type</span> <span class="identifier">j2_mp</span> <span class="special">=</span>
 115       <span class="identifier">integral</span><span class="special">(</span><span class="identifier">mp_type</span><span class="special">(</span><span class="number">0</span><span class="special">),</span>
 116       <span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">mp_type</span><span class="special">&gt;(),</span>
 117       <span class="identifier">mp_type</span><span class="special">(</span><span class="number">1.0E-20</span><span class="special">),</span>
 118       <span class="identifier">cyl_bessel_j_integral_rep</span><span class="special">&lt;</span><span class="identifier">mp_type</span><span class="special">&gt;(</span><span class="number">2U</span><span class="special">,</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="number">123</span><span class="special">)</span> <span class="special">/</span> <span class="number">100</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">pi</span><span class="special">&lt;</span><span class="identifier">mp_type</span><span class="special">&gt;();</span>
 119
 120    <span class="comment">// 0.166369</span>
 121    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span>
 122       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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="keyword">float</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
 123       <span class="special">&lt;&lt;</span> <span class="identifier">j2_f</span>
 124       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 125
 126    <span class="comment">// 0.166369383786814</span>
 127    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span>
 128       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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="keyword">double</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
 129       <span class="special">&lt;&lt;</span> <span class="identifier">j2_d</span>
 130       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 131
 132    <span class="comment">// 0.16636938378681407351267852431513159437103348245333</span>
 133    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span>
 134       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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">mp_type</span><span class="special">&gt;::</span><span class="identifier">digits10</span><span class="special">)</span>
 135       <span class="special">&lt;&lt;</span> <span class="identifier">j2_mp</span>
 136       <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 137
 138    <span class="comment">//</span>
 139    <span class="comment">// Print true value for comparison:</span>
 140    <span class="comment">// 0.166369383786814073512678524315131594371033482453329</span>
 141    <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">cyl_bessel_j</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="number">123</span><span class="special">)</span> <span class="special">/</span> <span class="number">100</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
 142 <span class="special">}</span>
 143 </pre>
 144 </div>
 145 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 146 <td align="left"></td>
 147 <td align="right"><div class="copyright-footer">Copyright &#169; 2002-2013 John Maddock and Christopher Kormanyos<p>
 148         Distributed under the Boost Software License, Version 1.0. (See accompanying
 149         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>)
 150       </p>
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