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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.sf_erf.error_inv"></a><a class="link" href="error_inv.html" title="Error Function Inverses">Error Function Inverses</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_inv.h0"></a> |
| <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.synopsis"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.synopsis">Synopsis</a> |
| </h5> |
| <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> |
| <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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="../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> |
| The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_inv.h1"></a> |
| <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.description"></a></span><a class="link" href="error_inv.html#math_toolkit.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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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.svg" 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="../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_inv.h2"></a> |
| <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.accuracy"></a></span><a class="link" href="error_inv.html#math_toolkit.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> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_inv.h3"></a> |
| <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.testing"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.testing">Testing</a> |
| </h5> |
| <p> |
| There are two sets of tests: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| 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 class="listitem"> |
| 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> |
| <h5> |
| <a name="math_toolkit.sf_erf.error_inv.h4"></a> |
| <span class="phrase"><a name="math_toolkit.sf_erf.error_inv.implementation"></a></span><a class="link" href="error_inv.html#math_toolkit.sf_erf.error_inv.implementation">Implementation</a> |
| </h5> |
| <p> |
| These functions use a rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>) |
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