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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.factorials.sf_rising_factorial"></a><a class="link" href="sf_rising_factorial.html" title="Rising Factorial">Rising Factorial</a>
  28 </h3></div></div></div>
  29 <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">factorials</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  30 </pre>
  31 <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>
  32
  33 <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>
  34 <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">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">);</span>
  35
  36 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  37 <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">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  38
  39 <span class="special">}}</span> <span class="comment">// namespaces</span>
  40 </pre>
  41 <p>
  42         Returns the rising factorial of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span>:
  43       </p>
  44 <div class="blockquote"><blockquote class="blockquote"><p>
  45           <span class="serif_italic"><span class="emphasis"><em>rising_factorial(x, i) = &#915;(x + i)
  46           / &#915;(x)</em></span></span>
  47         </p></blockquote></div>
  48 <p>
  49         or
  50       </p>
  51 <div class="blockquote"><blockquote class="blockquote"><p>
  52           <span class="serif_italic"><span class="emphasis"><em>rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i-1)</em></span></span>
  53         </p></blockquote></div>
  54 <p>
  55         Note that both <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span> can be negative
  56         as well as positive.
  57       </p>
  58 <p>
  59         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  60         be used to control the behaviour of the function: how it handles errors,
  61         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  62         documentation for more details</a>.
  63       </p>
  64 <p>
  65         May return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
  66         if the result is too large to represent in type T.
  67       </p>
  68 <p>
  69         The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  70         type calculation rules</em></span></a>: the type of the result is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, otherwise
  71         the type of the result is T.
  72       </p>
  73 <h5>
  74 <a name="math_toolkit.factorials.sf_rising_factorial.h0"></a>
  75         <span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.accuracy"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.accuracy">Accuracy</a>
  76       </h5>
  77 <p>
  78         The accuracy will be the same as the <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>
  79         function.
  80       </p>
  81 <h5>
  82 <a name="math_toolkit.factorials.sf_rising_factorial.h1"></a>
  83         <span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.testing"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.testing">Testing</a>
  84       </h5>
  85 <p>
  86         The spot tests for the rising factorials use data generated by <a href="https://functions.wolfram.com" target="_top">functions.wolfram.com</a>.
  87       </p>
  88 <h5>
  89 <a name="math_toolkit.factorials.sf_rising_factorial.h2"></a>
  90         <span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.implementation"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.implementation">Implementation</a>
  91       </h5>
  92 <p>
  93         Rising and factorials are implemented as ratios of gamma functions using
  94         <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>.
  95         Optimisations for small integer arguments are handled internally by that
  96         function.
  97       </p>
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