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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.hypergeometric.hypergeometric_pfq"></a><a class="link" href="hypergeometric_pfq.html" title="Hypergeometric pFq">Hypergeometric
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<sub>p</sub>F<sub>q</sub></a>
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</h3></div></div></div>
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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">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">></span>
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<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>
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<span class="keyword">template</span> <span class="special"><</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>
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<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">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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">&</span> <span class="identifier">pol</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">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
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<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">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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>
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<span class="keyword">template</span> <span class="special"><</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">></span>
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<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"><</span><span class="identifier">R</span><span class="special">>&</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"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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">&</span> <span class="identifier">pol</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">R</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
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<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"><</span><span class="identifier">R</span><span class="special">>&</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"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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>
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<span class="keyword">template</span> <span class="special"><</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">></span>
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<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">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</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">&</span> <span class="identifier">pol</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">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
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<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">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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>
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<span class="keyword">template</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">></span>
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<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"><</span><span class="identifier">Real</span><span class="special">>&</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"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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">&</span> <span class="identifier">pol</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">Real</span><span class="special">></span>
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<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"><</span><span class="identifier">Real</span><span class="special">>&</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"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</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>
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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.hypergeometric.hypergeometric_pfq.h0"></a>
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<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>
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</h5>
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<p>
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The function <code class="computeroutput"><span class="identifier">hypergeometric_pFq</span></code>
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returns the result of:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_pfq_1.svg" width="334" height="49"></object></span>
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</p></blockquote></div>
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<p>
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It is most naturally used via initializer lists as in:
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</p>
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<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>
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</pre>
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<p>
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The optional <span class="emphasis"><em>p_abs_error</em></span> argument calculates an estimate
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of the absolute error in the result from the L1 norm of the sum, plus some
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other factors (see implementation below).
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</p>
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<p>
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You should divide this value by the result to get an estimate of <span class="emphasis"><em>relative
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error</em></span>.
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</p>
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<p>
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It should be noted that the error estimates are rather crude - the error
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can be significantly underestimated in some circumstances, and over-estimated
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in others.
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</p>
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<p>
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The <code class="computeroutput"><span class="identifier">hypergeometric_pFq_precision</span></code>
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functions will calculate <code class="computeroutput"><span class="identifier">pFq</span></code>
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to a specified number of decimal places, and if <code class="computeroutput"><span class="identifier">timeout</span></code>
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is reached then they raise an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>.
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Note that while these functions are defined as templates, they require type
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<code class="computeroutput"><span class="identifier">Real</span></code> to be a <span class="bold"><strong>variable-precision</strong></span>
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floating-point type: in practice the only type currently supported (as of
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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:
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</p>
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<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>
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<span class="comment">//</span>
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<span class="comment">// Calculate 2F2 to 20 decimal places using a 10 second timeout:</span>
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<span class="comment">//</span>
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<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>
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<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>
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<span class="special">);</span>
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<span class="comment">//</span>
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<span class="comment">// Convert the result back to a double:</span>
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<span class="comment">//</span>
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<span class="keyword">double</span> <span class="identifier">d_result</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="identifier">result</span><span class="special">);</span>
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</pre>
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<h5>
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<a name="math_toolkit.hypergeometric.hypergeometric_pfq.h1"></a>
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<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>
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</h5>
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<p>
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This function is implemented by direct summation of the series; summation
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normally starts with the zeroth term, but will skip forward and sum outward
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(ie in both forward and backward directions) from some term <span class="emphasis"><em>N</em></span>
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when required. This is particularly important when some of the <span class="emphasis"><em>b</em></span>
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arguments are negative, as in this situation the sum undergoes "false-convergence",
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and then diverges again as each b<sub>j</sub> passes the origin. Consequently, were you
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to plot the magnitude of the terms in the sum, you would see them pass through
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a series of minima and maxima before finally converging to zero at infinity.
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For some values of <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span> we can
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compute where the maxima occur, and therefore sum only the terms that will
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have an impact on the result. For other <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span>
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values, predicting the exact locations of the maxima is not so easy, and
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we simply restart summation at the point where each b<sub>j</sub> passes the origin:
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this will eventually reach all the significant terms of the sum, but the
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key word may well be "eventually".
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</p>
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<p>
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The error term <span class="emphasis"><em>p_abs_error</em></span> is normally simply the sum
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of the absolute values of each term multiplied by machine epsilon for type
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<code class="computeroutput"><span class="identifier">Real</span></code>. However, if it was
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necessary for the summation to skip forward, then <span class="emphasis"><em>p_abs_error</em></span>
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is adjusted to account for the error inherent in calculating the N'th term
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via logarithms.
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</p>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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</p>
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</div></td>
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</tr></table>
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