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<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Modified Bessel Functions of the First and Second Kinds</title> <link rel="stylesheet" href="../../math.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.78.1"> <link rel="home" href="../../index.html" title="Math Toolkit 2.1.0"> <link rel="up" href="../bessel.html" title="Bessel Functions"> <link rel="prev" href="bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds"> <link rel="next" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds"> </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 href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="bessel_root.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="sph_bessel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section"> <div class="titlepage"><div><div><h3 class="title"> <a name="math_toolkit.bessel.mbessel"></a><a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">Modified Bessel Functions of the First and Second Kinds</a> </h3></div></div></div> <h5> <a name="math_toolkit.bessel.mbessel.h0"></a> <a name="math_toolkit.bessel.mbessel.synopsis"></a><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.synopsis">Synopsis</a> </h5> <code class="computeroutput">#include <boost/math/special_functions/bessel.hpp></code> <pre class="programlisting">template <class T1, class T2> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_i(T1 v, T2 x); template <class T1, class T2, class <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_i(T1 v, T2 x, const <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>&); template <class T1, class T2> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_k(T1 v, T2 x); template <class T1, class T2, class <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>> <a class="link" href="../result_type.html" title="Calculation of the Type of the Result">calculated-result-type</a> cyl_bessel_k(T1 v, T2 x, const <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a>&); </pre> <h5> <a name="math_toolkit.bessel.mbessel.h1"></a> <a name="math_toolkit.bessel.mbessel.description"></a><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.description">Description</a> </h5> The functions <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a> and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> return the result of the modified Bessel functions of the first and second kind respectively: cyl_bessel_i(v, x) = Iv(x) cyl_bessel_k(v, x) = Kv(x) where: <img src="../../../equations/mbessel2.png"> <img src="../../../equations/mbessel3.png"> 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">result type calculation rules</a> when T1 and T2 are different types. The functions are also optimised for the relatively common case that T1 is an integer. 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>. The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> whenever the result is undefined or complex. For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> this occurs when <code class="computeroutput">x < 0</code> and v is not an integer, or when <code class="computeroutput">x == 0</code> and <code class="computeroutput">v != 0</code>. For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this occurs when <code class="computeroutput">x <= 0</code>. The following graph illustrates the exponential behaviour of Iv. <img src="../../../graphs/cyl_bessel_i.png" align="middle"> The following graph illustrates the exponential decay of Kv. <img src="../../../graphs/cyl_bessel_k.png" align="middle"> <h5> <a name="math_toolkit.bessel.mbessel.h2"></a> <a name="math_toolkit.bessel.mbessel.testing"></a><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.testing">Testing</a> </h5> There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>, and a much larger set of tests computed using a simplified version of this implementation (with all the special case handling removed). <h5> <a name="math_toolkit.bessel.mbessel.h3"></a> <a name="math_toolkit.bessel.mbessel.accuracy"></a><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.accuracy">Accuracy</a> </h5> The following tables show how the accuracy of these functions varies on various platforms, along with a comparison to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> library. Note that only results for the widest floating-point type on the system are given, as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. All values are relative errors in units of epsilon. <div class="table"> <a name="math_toolkit.bessel.mbessel.errors_rates_in_cyl_bessel_i"></a>Table 6.23. Errors Rates in cyl_bessel_i <div class="table-contents"><table class="table" summary="Errors Rates in cyl_bessel_i"> <colgroup> <col> <col> <col> </colgroup> <thead><tr> <th> Significand Size </th> <th> Platform and Compiler </th> <th> Iv </th> </tr></thead> <tbody> <tr> <td> 53 </td> <td> Win32 / Visual C++ 8.0 </td> <td> Peak=10 Mean=3.4 GSL Peak=6000 </td> </tr> <tr> <td> 64 </td> <td> Red Hat Linux IA64 / G++ 3.4 </td> <td> Peak=11 Mean=3 </td> </tr> <tr> <td> 64 </td> <td> SUSE Linux AMD64 / G++ 4.1 </td> <td> Peak=11 Mean=4 </td> </tr> <tr> <td> 113 </td> <td> HP-UX / HP aCC 6 </td> <td> Peak=15 Mean=4 </td> </tr> </tbody> </table></div> </div> <div class="table"> <a name="math_toolkit.bessel.mbessel.errors_rates_in_cyl_bessel_k"></a>Table 6.24. Errors Rates in cyl_bessel_k <div class="table-contents"><table class="table" summary="Errors Rates in cyl_bessel_k"> <colgroup> <col> <col> <col> </colgroup> <thead><tr> <th> Significand Size </th> <th> Platform and Compiler </th> <th> Kv </th> </tr></thead> <tbody> <tr> <td> 53 </td> <td> Win32 / Visual C++ 8.0 </td> <td> Peak=9 Mean=2 GSL Peak=9 </td> </tr> <tr> <td> 64 </td> <td> Red Hat Linux IA64 / G++ 3.4 </td> <td> Peak=10 Mean=2 </td> </tr> <tr> <td> 64 </td> <td> SUSE Linux AMD64 / G++ 4.1 </td> <td> Peak=10 Mean=2 </td> </tr> <tr> <td> 113 </td> <td> HP-UX / HP aCC 6 </td> <td> Peak=12 Mean=5 </td> </tr> </tbody> </table></div> </div> <h5> <a name="math_toolkit.bessel.mbessel.h4"></a> <a name="math_toolkit.bessel.mbessel.implementation"></a><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.implementation">Implementation</a> </h5> The following are handled as special cases first: When computing Iv   for x < 0, then ν   must be an integer or a domain error occurs. If ν   is an integer, then the function is odd if ν   is odd and even if ν   is even, and we can reflect to x > 0. For Iv   with v equal to 0, 1 or 0.5 are handled as special cases. The 0 and 1 cases use minimax rational approximations on finite and infinite intervals. The coefficients are from: <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> <li class="listitem"> J.M. Blair and C.A. Edwards, Stable rational minimax approximations to the modified Bessel functions I_0(x) and I_1(x), Atomic Energy of Canada Limited Report 4928, Chalk River, 1974. </li> <li class="listitem"> S. Moshier, Methods and Programs for Mathematical Functions, Ellis Horwood Ltd, Chichester, 1989. </li> </ul></div> While the 0.5 case is a simple trigonometric function: I0.5(x) = sqrt(2 / πx) * sinh(x) For Kv   with v an integer, the result is calculated using the recurrence relation: <img src="../../../equations/mbessel5.png"> starting from K0   and K1   which are calculated using rational the approximations above. These rational approximations are accurate to around 19 digits, and are therefore only used when T has no more than 64 binary digits of precision. When x is small compared to v, Ivx   is best computed directly from the series: <img src="../../../equations/mbessel17.png"> In the general case, we first normalize ν   to [<code class="literal">0, [inf]</code>) with the help of the reflection formulae: <img src="../../../equations/mbessel9.png"> <img src="../../../equations/mbessel10.png"> Let μ   = ν - floor(ν + 1/2), then μ   is the fractional part of ν   such that |μ| <= 1/2 (we need this for convergence later). The idea is to calculate Kμ(x) and Kμ+1(x), and use them to obtain Iν(x) and Kν(x). The algorithm is proposed by Temme in N.M. Temme, On the numerical evaluation of the modified bessel function of the third kind, Journal of Computational Physics, vol 19, 324 (1975), which needs two continued fractions as well as the Wronskian: <img src="../../../equations/mbessel11.png"> <img src="../../../equations/mbessel12.png"> <img src="../../../equations/mbessel8.png"> The continued fractions are computed using the modified Lentz's method (W.J. Lentz, Generating Bessel functions in Mie scattering calculations using continued fractions, Applied Optics, vol 15, 668 (1976)). Their convergence rates depend on x, therefore we need different strategies for large x and small x. x > v, CF1 needs O(x) iterations to converge, CF2 converges rapidly. x <= v, CF1 converges rapidly, CF2 fails to converge when x <code class="literal">-></code> 0. When x is large (x > 2), both continued fractions converge (CF1 may be slow for really large x). Kμ   and Kμ+1   can be calculated by <img src="../../../equations/mbessel13.png"> where <img src="../../../equations/mbessel14.png"> S is also a series that is summed along with CF2, see I.J. Thompson and A.R. Barnett, Modified Bessel functions I_v and K_v of real order and complex argument to selected accuracy, Computer Physics Communications, vol 47, 245 (1987). When x is small (x <= 2), CF2 convergence may fail (but CF1 works very well). The solution here is Temme's series: <img src="../../../equations/mbessel15.png"> where <img src="../../../equations/mbessel16.png"> fk   and hk   are also computed by recursions (involving gamma functions), but the formulas are a little complicated, readers are referred to N.M. Temme, On the numerical evaluation of the modified Bessel function of the third kind, Journal of Computational Physics, vol 19, 324 (1975). Note: Temme's series converge only for |μ| <= 1/2. Kν(x) is then calculated from the forward recurrence, as is Kν+1(x). With these two values and fν, the Wronskian yields Iν(x) directly. </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 © 2006-2010, 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 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>) </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="bessel_root.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.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="sph_bessel.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>