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26 | <div class="titlepage"><div><div><h3 class="title"> | |
27 | <a name="math_toolkit.airy.bi"></a><a class="link" href="bi.html" title="Airy Bi Function">Airy Bi Function</a> | |
28 | </h3></div></div></div> | |
29 | <h5> | |
30 | <a name="math_toolkit.airy.bi.h0"></a> | |
31 | <span class="phrase"><a name="math_toolkit.airy.bi.synopsis"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.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_bi</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_bi</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.bi.h1"></a> | |
47 | <span class="phrase"><a name="math_toolkit.airy.bi.description"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.description">Description</a> | |
48 | </h5> | |
49 | <p> | |
50 | The function <a class="link" href="bi.html" title="Airy Bi Function">airy_bi</a> calculates | |
51 | the Airy function Bi which is the second solution to the differential equation: | |
52 | </p> | |
53 | <p> | |
54 | <span class="inlinemediaobject"><img src="../../../equations/airy.svg"></span> | |
55 | </p> | |
56 | <p> | |
57 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
58 | be used to control the behaviour of the function: how it handles errors, | |
59 | 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 | |
60 | documentation for more details</a>. | |
61 | </p> | |
62 | <p> | |
63 | The following graph illustrates how this function changes as <span class="emphasis"><em>x</em></span> | |
64 | changes: for negative <span class="emphasis"><em>x</em></span> the function is cyclic, while | |
65 | for positive <span class="emphasis"><em>x</em></span> the value tends to infinity: | |
66 | </p> | |
67 | <p> | |
68 | <span class="inlinemediaobject"><img src="../../../graphs/airy_bi.svg" align="middle"></span> | |
69 | </p> | |
70 | <h5> | |
71 | <a name="math_toolkit.airy.bi.h2"></a> | |
72 | <span class="phrase"><a name="math_toolkit.airy.bi.accuracy"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.accuracy">Accuracy</a> | |
73 | </h5> | |
74 | <p> | |
75 | This function is implemented entirely in terms of the Bessel functions <a class="link" href="../bessel/mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> and <a class="link" href="../bessel/bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> | |
76 | - refer to those functions for detailed accuracy information. | |
77 | </p> | |
78 | <p> | |
79 | In general though, the relative error is low (less than 100 ε) for <span class="emphasis"><em>x | |
80 | > 0</em></span> while only the absolute error is low for <span class="emphasis"><em>x < | |
81 | 0</em></span>. | |
82 | </p> | |
83 | <h5> | |
84 | <a name="math_toolkit.airy.bi.h3"></a> | |
85 | <span class="phrase"><a name="math_toolkit.airy.bi.testing"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.testing">Testing</a> | |
86 | </h5> | |
87 | <p> | |
88 | Since this function is implemented in terms of other special functions, there | |
89 | are only a few basic sanity checks, using test values from <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>. | |
90 | </p> | |
91 | <h5> | |
92 | <a name="math_toolkit.airy.bi.h4"></a> | |
93 | <span class="phrase"><a name="math_toolkit.airy.bi.implementation"></a></span><a class="link" href="bi.html#math_toolkit.airy.bi.implementation">Implementation</a> | |
94 | </h5> | |
95 | <p> | |
96 | This function is implemented in terms of the Bessel functions using the relations: | |
97 | </p> | |
98 | <p> | |
99 | <span class="inlinemediaobject"><img src="../../../equations/airy_bi.svg"></span> | |
100 | </p> | |
101 | </div> | |
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104 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
105 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
106 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
107 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
108 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
109 | 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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