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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Falling Factorial</title> | |
| 5 | <link rel="stylesheet" href="../../math.css" type="text/css"> | |
| 6 | <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> | |
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| 8 | <link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients"> | |
| 9 | <link rel="prev" href="sf_rising_factorial.html" title="Rising Factorial"> | |
| 10 | <link rel="next" href="sf_binomial.html" title="Binomial Coefficients"> | |
| 11 | </head> | |
| 12 | <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> | |
| 13 | <table cellpadding="2" width="100%"><tr> | |
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| 20 | </tr></table> | |
| 21 | <hr> | |
| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="sf_rising_factorial.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../factorials.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sf_binomial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 24 | </div> | |
| 25 | <div class="section"> | |
| 26 | <div class="titlepage"><div><div><h3 class="title"> | |
| 27 | <a name="math_toolkit.factorials.sf_falling_factorial"></a><a class="link" href="sf_falling_factorial.html" title="Falling Factorial">Falling | |
| 28 | Factorial</a> | |
| 29 | </h3></div></div></div> | |
| 30 | <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">factorials</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 31 | </pre> | |
| 32 | <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> | |
| 33 | ||
| 34 | <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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">falling_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">i</span><span class="special">);</span> | |
| 36 | ||
| 37 | <span class="keyword">template</span> <span class="special"><</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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></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">falling_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> | |
| 39 | ||
| 40 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
| 41 | </pre> | |
| 42 | <p> | |
| 43 | Returns the falling factorial of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span>: | |
| 44 | </p> | |
| 45 | <p> | |
| 46 | falling_factorial(x, i) = x(x-1)(x-2)(x-3)...(x-i+1) | |
| 47 | </p> | |
| 48 | <p> | |
| 49 | Note that this function is only defined for positive <span class="emphasis"><em>i</em></span>, | |
| 50 | hence the <code class="computeroutput"><span class="keyword">unsigned</span></code> second argument. | |
| 51 | Argument <span class="emphasis"><em>x</em></span> can be either positive or negative however. | |
| 52 | </p> | |
| 53 | <p> | |
| 54 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 55 | be used to control the behaviour of the function: how it handles errors, | |
| 56 | what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
| 57 | documentation for more details</a>. | |
| 58 | </p> | |
| 59 | <p> | |
| 60 | May return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a> | |
| 61 | if the result is too large to represent in type T. | |
| 62 | </p> | |
| 63 | <p> | |
| 64 | 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 | |
| 65 | 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 | |
| 66 | the type of the result is T. | |
| 67 | </p> | |
| 68 | <h5> | |
| 69 | <a name="math_toolkit.factorials.sf_falling_factorial.h0"></a> | |
| 70 | <span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.accuracy"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.accuracy">Accuracy</a> | |
| 71 | </h5> | |
| 72 | <p> | |
| 73 | 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> | |
| 74 | function. | |
| 75 | </p> | |
| 76 | <h5> | |
| 77 | <a name="math_toolkit.factorials.sf_falling_factorial.h1"></a> | |
| 78 | <span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.testing"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.testing">Testing</a> | |
| 79 | </h5> | |
| 80 | <p> | |
| 81 | The spot tests for the falling factorials use data generated by functions.wolfram.com. | |
| 82 | </p> | |
| 83 | <h5> | |
| 84 | <a name="math_toolkit.factorials.sf_falling_factorial.h2"></a> | |
| 85 | <span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.implementation"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.implementation">Implementation</a> | |
| 86 | </h5> | |
| 87 | <p> | |
| 88 | Rising and falling factorials are implemented as ratios of gamma functions | |
| 89 | using <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>. | |
| 90 | Optimisations for small integer arguments are handled internally by that | |
| 91 | function. | |
| 92 | </p> | |
| 93 | </div> | |
| 94 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 95 | <td align="left"></td> | |
| 96 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 97 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 98 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 99 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 100 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 101 | 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>) | |
| 102 | </p> | |
| 103 | </div></td> | |
| 104 | </tr></table> | |
| 105 | <hr> | |
| 106 | <div class="spirit-nav"> | |
| 107 | <a accesskey="p" href="sf_rising_factorial.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../factorials.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sf_binomial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 108 | </div> | |
| 109 | </body> | |
| 110 | </html> |