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| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Airy Ai Function</title> | |
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| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="../airy.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.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="bi.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.airy.ai"></a><a class="link" href="ai.html" title="Airy Ai Function">Airy Ai Function</a> | |
| 28 | </h3></div></div></div> | |
| 29 | <h5> | |
| 30 | <a name="math_toolkit.airy.ai.h0"></a> | |
| 31 | <span class="phrase"><a name="math_toolkit.airy.ai.synopsis"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.synopsis">Synopsis</a> | |
| 32 | </h5> | |
| 33 | <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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 34 | </pre> | |
| 35 | <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> | |
| 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> | |
| 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">airy_ai</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">);</span> | |
| 39 | ||
| 40 | <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> <span class="identifier">Policy</span><span class="special">></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">airy_ai</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&);</span> | |
| 42 | ||
| 43 | <span class="special">}}</span> <span class="comment">// namespaces</span> | |
| 44 | </pre> | |
| 45 | <h5> | |
| 46 | <a name="math_toolkit.airy.ai.h1"></a> | |
| 47 | <span class="phrase"><a name="math_toolkit.airy.ai.description"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.description">Description</a> | |
| 48 | </h5> | |
| 49 | <p> | |
| 50 | The function <a class="link" href="ai.html" title="Airy Ai Function">airy_ai</a> calculates | |
| 51 | the Airy function Ai which is the first solution to the differential equation: | |
| 52 | </p> | |
| 53 | <p> | |
| 54 | <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span> | |
| 55 | </p> | |
| 56 | <p> | |
| 57 | See Weisstein, Eric W. "Airy Functions." From MathWorld--A Wolfram | |
| 58 | Web Resource. <a href="http://mathworld.wolfram.com/AiryFunctions.html" target="_top">http://mathworld.wolfram.com/AiryFunctions.html</a>; | |
| 59 | </p> | |
| 60 | <p> | |
| 61 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 62 | be used to control the behaviour of the function: how it handles errors, | |
| 63 | 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 | |
| 64 | documentation for more details</a>. | |
| 65 | </p> | |
| 66 | <p> | |
| 67 | The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span> | |
| 68 | changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while | |
| 69 | for positive <span class="emphasis"><em>x</em></span> the value tends to zero: | |
| 70 | </p> | |
| 71 | <p> | |
| 72 | <span class="inlinemediaobject"><img src="../../../graphs/airy_ai.svg" align="middle"></span> | |
| 73 | </p> | |
| 74 | <h5> | |
| 75 | <a name="math_toolkit.airy.ai.h2"></a> | |
| 76 | <span class="phrase"><a name="math_toolkit.airy.ai.accuracy"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.accuracy">Accuracy</a> | |
| 77 | </h5> | |
| 78 | <p> | |
| 79 | This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> and <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> - refer to those | |
| 80 | functions for detailed accuracy information. | |
| 81 | </p> | |
| 82 | <p> | |
| 83 | In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x | |
| 84 | > 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x < | |
| 85 | 0</em></span>. | |
| 86 | </p> | |
| 87 | <h5> | |
| 88 | <a name="math_toolkit.airy.ai.h3"></a> | |
| 89 | <span class="phrase"><a name="math_toolkit.airy.ai.testing"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.testing">Testing</a> | |
| 90 | </h5> | |
| 91 | <p> | |
| 92 | Since this function is implemented in terms of other special functions, there | |
| 93 | are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com/" target="_top">Wolfram | |
| 94 | Airy Functions</a>. | |
| 95 | </p> | |
| 96 | <h5> | |
| 97 | <a name="math_toolkit.airy.ai.h4"></a> | |
| 98 | <span class="phrase"><a name="math_toolkit.airy.ai.implementation"></a></span><a class="link" href="ai.html#math_toolkit.airy.ai.implementation">Implementation</a> | |
| 99 | </h5> | |
| 100 | <p> | |
| 101 | This function is implemented in terms of the Bessel functions using the relations: | |
| 102 | </p> | |
| 103 | <p> | |
| 104 | <span class="inlinemediaobject"><img src="../../../equations/airy_ai.svg"></span> | |
| 105 | </p> | |
| 106 | </div> | |
| 107 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 108 | <td align="left"></td> | |
| 109 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 110 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 111 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 112 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 113 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 114 | 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>) | |
| 115 | </p> | |
| 116 | </div></td> | |
| 117 | </tr></table> | |
| 118 | <hr> | |
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| 120 | <a accesskey="p" href="../airy.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../airy.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="bi.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 121 | </div> | |
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| 123 | </html> |