ceph/src/boost/libs/math/doc/html/math_toolkit/airy/ai.html

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  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">&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">airy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</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">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">&lt;</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">&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">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">&amp;);</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&#160;15.&#160;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&#160;15.&#160;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 &#949;) for <span class="emphasis"><em>x
  84         &gt; 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x &lt;
  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 &#169; 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 R&#229;de, 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>)
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