<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Error Function Inverses</title> <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../index.html" title="Math Toolkit"> <link rel="up" href="../sf_erf.html" title="Error Functions"> <link rel="prev" href="error_function.html" title="Error Functions"> <link rel="next" href="../sf_poly.html" title="Polynomials"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a 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name="math_toolkit.special.sf_erf.error_inv.synopsis"></a><h5> <a name="id1108928"></a> <a class="link" href="error_inv.html#math_toolkit.special.sf_erf.error_inv.synopsis">Synopsis</a> </h5> <p> </p> <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">erf</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> </pre> <p> </p> <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> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">);</span> <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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">);</span> <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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> The return type of these functions is computed using the <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <a name="math_toolkit.special.sf_erf.error_inv.description"></a><h5> <a name="id1109311"></a> <a class="link" href="error_inv.html#math_toolkit.special.sf_erf.error_inv.description">Description</a> </h5> <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> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> <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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erf_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns the <a href="http://functions.wolfram.com/GammaBetaErf/InverseErf/" target="_top">inverse error function</a> of z, that is a value x such that: </p> <pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">erf</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> </pre> <p> <span class="inlinemediaobject"><img src="../../../../graphs/erf_inv.png" align="middle"></span> </p> <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> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> <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="Policies">Policy</a><span class="special">></span> <a class="link" href="../../main_overview/result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">erfc_inv</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> </pre> <p> Returns the inverse of the complement of the error function of z, that is a value x such that: </p> <pre class="programlisting"><span class="identifier">p</span> <span class="special">=</span> <span class="identifier">erfc</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> </pre> <p> <span class="inlinemediaobject"><img src="../../../../graphs/erfc_inv.png" align="middle"></span> </p> <a name="math_toolkit.special.sf_erf.error_inv.accuracy"></a><h5> <a name="id1109699"></a> <a class="link" href="error_inv.html#math_toolkit.special.sf_erf.error_inv.accuracy">Accuracy</a> </h5> <p> For types up to and including 80-bit long doubles the approximations used are accurate to less than ~ 2 epsilon. For higher precision types these functions have the same accuracy as the <a class="link" href="error_function.html" title="Error Functions">forward error functions</a>. </p> <a name="math_toolkit.special.sf_erf.error_inv.testing"></a><h5> <a name="id1109720"></a> <a class="link" href="error_inv.html#math_toolkit.special.sf_erf.error_inv.testing">Testing</a> </h5> <p> There are two sets of tests: </p> <div class="itemizedlist"><ul type="disc"> <li> Basic sanity checks attempt to "round-trip" from <span class="emphasis"><em>x</em></span> to <span class="emphasis"><em>p</em></span> and back again. These tests have quite generous tolerances: in general both the error functions and their inverses change so rapidly in some places that round tripping to more than a couple of significant digits isn't possible. This is especially true when <span class="emphasis"><em>p</em></span> is very near one: in this case there isn't enough "information content" in the input to the inverse function to get back where you started. </li> <li> Accuracy checks using high-precision test values. These measure the accuracy of the result, given <span class="emphasis"><em>exact</em></span> input values. </li> </ul></div> <a name="math_toolkit.special.sf_erf.error_inv.implementation"></a><h5> <a name="id1109770"></a> <a class="link" href="error_inv.html#math_toolkit.special.sf_erf.error_inv.implementation">Implementation</a> </h5> <p> These functions use a rational approximation <a class="link" href="../../backgrounders/implementation.html#math_toolkit.backgrounders.implementation.rational_approximations_used">devised by JM</a> to calculate an initial approximation to the result that is accurate to ~10<sup>-19</sup>, then only if that has insufficient accuracy compared to the epsilon for T, do we clean up the result using <a href="http://en.wikipedia.org/wiki/Simple_rational_approximation" target="_top">Halley iteration</a>. </p> <p> Constructing rational approximations to the erf/erfc functions is actually surprisingly hard, especially at high precision. For this reason no attempt has been made to achieve 10<sup>-34 </sup> accuracy suitable for use with 128-bit reals. </p> <p> In the following discussion, <span class="emphasis"><em>p</em></span> is the value passed to erf_inv, and <span class="emphasis"><em>q</em></span> is the value passed to erfc_inv, so that <span class="emphasis"><em>p = 1 - q</em></span> and <span class="emphasis"><em>q = 1 - p</em></span> and in both cases we want to solve for the same result <span class="emphasis"><em>x</em></span>. </p> <p> For <span class="emphasis"><em>p < 0.5</em></span> the inverse erf function is reasonably smooth and the approximation: </p> <pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">p</span><span class="special">(</span><span class="identifier">p</span> <span class="special">+</span> <span class="number">10</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">p</span><span class="special">))</span> </pre> <p> Gives a good result for a constant Y, and R(p) optimised for low absolute error compared to |Y|. </p> <p> For q < 0.5 things get trickier, over the interval <span class="emphasis"><em>0.5 > q > 0.25</em></span> the following approximation works well: </p> <pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(-</span><span class="number">2l</span><span class="identifier">og</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span> </pre> <p> While for q < 0.25, let </p> <pre class="programlisting"><span class="identifier">z</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(-</span><span class="identifier">log</span><span class="special">(</span><span class="identifier">q</span><span class="special">))</span> </pre> <p> Then the result is given by: </p> <pre class="programlisting"><span class="identifier">x</span> <span class="special">=</span> <span class="identifier">z</span><span class="special">(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">z</span> <span class="special">-</span> <span class="identifier">B</span><span class="special">))</span> </pre> <p> As before Y is a constant and the rational function R is optimised for low absolute error compared to |Y|. B is also a constant: it is the smallest value of <span class="emphasis"><em>z</em></span> for which each approximation is valid. There are several approximations of this form each of which reaches a little further into the tail of the erfc function (at <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code> precision the extended exponent range compared to <code class="computeroutput"><span class="keyword">double</span></code> means that the tail goes on for a very long way indeed). </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="error_function.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_erf.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_poly.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>