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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.hypergeometric.hypergeometric_pfq"></a><a class="link" href="hypergeometric_pfq.html" title="Hypergeometric pFq">Hypergeometric
  28       <sub>p</sub>F<sub>q</sub></a>
  29 </h3></div></div></div>
  30 <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">special_functions</span><span class="special">/</span><span class="identifier">hypergeometric_pFq</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  31
  32 <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>
  33
  34 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  35 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</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">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  38 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span> <span class="special">=</span> <span class="number">0</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">R</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  41 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">R</span><span class="special">&gt;&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">R</span><span class="special">&gt;&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</span><span class="special">);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">R</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  44 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">R</span><span class="special">&gt;&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">R</span><span class="special">&gt;&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  47 <span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</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">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">&gt;</span>
  50 <span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span> <span class="special">=</span> <span class="number">0.5</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">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  53 <span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&amp;</span> <span class="identifier">pol</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">Real</span><span class="special">&gt;</span>
  56 <span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;&amp;</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&amp;</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">);</span>
  57
  58 <span class="special">}}</span> <span class="comment">// namespaces</span>
  59 </pre>
  60 <h5>
  61 <a name="math_toolkit.hypergeometric.hypergeometric_pfq.h0"></a>
  62         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_pfq.description"></a></span><a class="link" href="hypergeometric_pfq.html#math_toolkit.hypergeometric.hypergeometric_pfq.description">Description</a>
  63       </h5>
  64 <p>
  65         The function <code class="computeroutput"><span class="identifier">hypergeometric_pFq</span></code>
  66         returns the result of:
  67       </p>
  68 <div class="blockquote"><blockquote class="blockquote"><p>
  69           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_pfq_1.svg" width="334" height="49"></object></span>
  70         </p></blockquote></div>
  71 <p>
  72         It is most naturally used via initializer lists as in:
  73       </p>
  74 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="identifier">hypergeometric_pFq</span><span class="special">({</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">},</span> <span class="special">{</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">,</span><span class="number">8</span><span class="special">},</span> <span class="identifier">z</span><span class="special">);</span>  <span class="comment">// Calculate 3F4</span>
  75 </pre>
  76 <p>
  77         The optional <span class="emphasis"><em>p_abs_error</em></span> argument calculates an estimate
  78         of the absolute error in the result from the L1 norm of the sum, plus some
  79         other factors (see implementation below).
  80       </p>
  81 <p>
  82         You should divide this value by the result to get an estimate of <span class="emphasis"><em>relative
  83         error</em></span>.
  84       </p>
  85 <p>
  86         It should be noted that the error estimates are rather crude - the error
  87         can be significantly underestimated in some circumstances, and over-estimated
  88         in others.
  89       </p>
  90 <p>
  91         The <code class="computeroutput"><span class="identifier">hypergeometric_pFq_precision</span></code>
  92         functions will calculate <code class="computeroutput"><span class="identifier">pFq</span></code>
  93         to a specified number of decimal places, and if <code class="computeroutput"><span class="identifier">timeout</span></code>
  94         is reached then they raise an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>.
  95         Note that while these functions are defined as templates, they require type
  96         <code class="computeroutput"><span class="identifier">Real</span></code> to be a <span class="bold"><strong>variable-precision</strong></span>
  97         floating-point type: in practice the only type currently supported (as of
  98         July 2019) is <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">mpfr_float</span></code>. Typical usage would be:
  99       </p>
 100 <pre class="programlisting"><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">mpfr_float</span> <span class="identifier">mp_type</span><span class="special">;</span>
 101 <span class="comment">//</span>
 102 <span class="comment">// Calculate 2F2 to 20 decimal places using a 10 second timeout:</span>
 103 <span class="comment">//</span>
 104 <span class="identifier">mp_type</span> <span class="identifier">result</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">hypergeometric_pFq_precision</span><span class="special">(</span>
 105   <span class="special">{</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">a1</span><span class="special">),</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">a2</span><span class="special">)</span> <span class="special">},</span> <span class="special">{</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">b1</span><span class="special">),</span> <span class="identifier">m_type</span><span class="special">(</span><span class="identifier">b2</span><span class="special">)</span> <span class="special">},</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">z</span><span class="special">),</span> <span class="number">20</span><span class="special">,</span> <span class="number">10.0</span>
 106   <span class="special">);</span>
 107 <span class="comment">//</span>
 108 <span class="comment">// Convert the result back to a double:</span>
 109 <span class="comment">//</span>
 110 <span class="keyword">double</span> <span class="identifier">d_result</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="identifier">result</span><span class="special">);</span>
 111 </pre>
 112 <h5>
 113 <a name="math_toolkit.hypergeometric.hypergeometric_pfq.h1"></a>
 114         <span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_pfq.implementation"></a></span><a class="link" href="hypergeometric_pfq.html#math_toolkit.hypergeometric.hypergeometric_pfq.implementation">Implementation</a>
 115       </h5>
 116 <p>
 117         This function is implemented by direct summation of the series; summation
 118         normally starts with the zeroth term, but will skip forward and sum outward
 119         (ie in both forward and backward directions) from some term <span class="emphasis"><em>N</em></span>
 120         when required. This is particularly important when some of the <span class="emphasis"><em>b</em></span>
 121         arguments are negative, as in this situation the sum undergoes "false-convergence",
 122         and then diverges again as each b<sub>j</sub> passes the origin. Consequently, were you
 123         to plot the magnitude of the terms in the sum, you would see them pass through
 124         a series of minima and maxima before finally converging to zero at infinity.
 125         For some values of <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span> we can
 126         compute where the maxima occur, and therefore sum only the terms that will
 127         have an impact on the result. For other <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span>
 128         values, predicting the exact locations of the maxima is not so easy, and
 129         we simply restart summation at the point where each b<sub>j</sub> passes the origin:
 130         this will eventually reach all the significant terms of the sum, but the
 131         key word may well be "eventually".
 132       </p>
 133 <p>
 134         The error term <span class="emphasis"><em>p_abs_error</em></span> is normally simply the sum
 135         of the absolute values of each term multiplied by machine epsilon for type
 136         <code class="computeroutput"><span class="identifier">Real</span></code>. However, if it was
 137         necessary for the summation to skip forward, then <span class="emphasis"><em>p_abs_error</em></span>
 138         is adjusted to account for the error inherent in calculating the N'th term
 139         via logarithms.
 140       </p>
 141 </div>
 142 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 143 <td align="left"></td>
 144 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 145       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 146       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 147       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 148       Daryle Walker and Xiaogang Zhang<p>
 149         Distributed under the Boost Software License, Version 1.0. (See accompanying
 150         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>)
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