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1 | <html> |
| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>acosh</title> | |
| 5 | <link rel="stylesheet" href="../../math.css" type="text/css"> | |
| 6 | <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> | |
| 7 | <link rel="home" href="../../index.html" title="Math Toolkit 2.5.1"> | |
| 8 | <link rel="up" href="../inv_hyper.html" title="Inverse Hyperbolic Functions"> | |
| 9 | <link rel="prev" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview"> | |
| 10 | <link rel="next" href="asinh.html" title="asinh"> | |
| 11 | </head> | |
| 12 | <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> | |
| 13 | <table cellpadding="2" width="100%"><tr> | |
| 14 | <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> | |
| 15 | <td align="center"><a href="../../../../../../index.html">Home</a></td> | |
| 16 | <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> | |
| 17 | <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> | |
| 18 | <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> | |
| 19 | <td align="center"><a href="../../../../../../more/index.htm">More</a></td> | |
| 20 | </tr></table> | |
| 21 | <hr> | |
| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="inv_hyper_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.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="asinh.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.inv_hyper.acosh"></a><a class="link" href="acosh.html" title="acosh">acosh</a> | |
| 28 | </h3></div></div></div> | |
| 29 | <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">acosh</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> | |
| 30 | </pre> | |
| 31 | <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> | |
| 32 | <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">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</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> <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> | |
| 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">acosh</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">x</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> | |
| 36 | </pre> | |
| 37 | <p> | |
| 38 | Computes the reciprocal of (the restriction to the range of <code class="literal">[0;+∞[</code>) | |
| 39 | <a class="link" href="inv_hyper_over.html" title="Inverse Hyperbolic Functions Overview">the hyperbolic cosine | |
| 40 | function</a>, at x. Values returned are positive. | |
| 41 | </p> | |
| 42 | <p> | |
| 43 | If x is in the range <code class="literal">]-∞;+1[</code> then returns the result of | |
| 44 | <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. | |
| 45 | </p> | |
| 46 | <p> | |
| 47 | The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result | |
| 48 | type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and T otherwise. | |
| 49 | </p> | |
| 50 | <p> | |
| 51 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
| 52 | be used to control the behaviour of the function: how it handles errors, | |
| 53 | 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 | |
| 54 | documentation for more details</a>. | |
| 55 | </p> | |
| 56 | <p> | |
| 57 | <span class="inlinemediaobject"><img src="../../../graphs/acosh.svg" align="middle"></span> | |
| 58 | </p> | |
| 59 | <h5> | |
| 60 | <a name="math_toolkit.inv_hyper.acosh.h0"></a> | |
| 61 | <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.accuracy"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.accuracy">Accuracy</a> | |
| 62 | </h5> | |
| 63 | <p> | |
| 64 | Generally accuracy is to within 1 or 2 epsilon across all supported platforms. | |
| 65 | </p> | |
| 66 | <h5> | |
| 67 | <a name="math_toolkit.inv_hyper.acosh.h1"></a> | |
| 68 | <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.testing"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.testing">Testing</a> | |
| 69 | </h5> | |
| 70 | <p> | |
| 71 | This function is tested using a combination of random test values designed | |
| 72 | to give full function coverage computed at high precision using the "naive" | |
| 73 | formula: | |
| 74 | </p> | |
| 75 | <p> | |
| 76 | <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span> | |
| 77 | </p> | |
| 78 | <p> | |
| 79 | along with a selection of sanity check values computed using functions.wolfram.com | |
| 80 | to at least 50 decimal digits. | |
| 81 | </p> | |
| 82 | <h5> | |
| 83 | <a name="math_toolkit.inv_hyper.acosh.h2"></a> | |
| 84 | <span class="phrase"><a name="math_toolkit.inv_hyper.acosh.implementation"></a></span><a class="link" href="acosh.html#math_toolkit.inv_hyper.acosh.implementation">Implementation</a> | |
| 85 | </h5> | |
| 86 | <p> | |
| 87 | For sufficiently large x, we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/06/01/0001/" target="_top">approximation</a>: | |
| 88 | </p> | |
| 89 | <p> | |
| 90 | <span class="inlinemediaobject"><img src="../../../equations/acosh2.svg"></span> | |
| 91 | </p> | |
| 92 | <p> | |
| 93 | For x sufficiently close to 1 we can use the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/06/01/04/01/0001/" target="_top">approximation</a>: | |
| 94 | </p> | |
| 95 | <p> | |
| 96 | <span class="inlinemediaobject"><img src="../../../equations/acosh4.svg"></span> | |
| 97 | </p> | |
| 98 | <p> | |
| 99 | Otherwise for x close to 1 we can use the following rearrangement of the | |
| 100 | primary definition to preserve accuracy: | |
| 101 | </p> | |
| 102 | <p> | |
| 103 | <span class="inlinemediaobject"><img src="../../../equations/acosh3.svg"></span> | |
| 104 | </p> | |
| 105 | <p> | |
| 106 | Otherwise the <a href="http://functions.wolfram.com/ElementaryFunctions/ArcCosh/02/" target="_top">primary | |
| 107 | definition</a> is used: | |
| 108 | </p> | |
| 109 | <p> | |
| 110 | <span class="inlinemediaobject"><img src="../../../equations/acosh1.svg"></span> | |
| 111 | </p> | |
| 112 | </div> | |
| 113 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
| 114 | <td align="left"></td> | |
| 115 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 116 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 117 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 118 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
| 119 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
| 120 | 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>) | |
| 121 | </p> | |
| 122 | </div></td> | |
| 123 | </tr></table> | |
| 124 | <hr> | |
| 125 | <div class="spirit-nav"> | |
| 126 | <a accesskey="p" href="inv_hyper_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../inv_hyper.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="asinh.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 127 | </div> | |
| 128 | </body> | |
| 129 | </html> |