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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.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> |
| </h4></div></div></div> |
| <a name="math_toolkit.special.bessel.mbessel.synopsis"></a><h5> |
| <a name="id1127045"></a> |
| <a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.synopsis">Synopsis</a> |
| </h5> |
| <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</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> |
| <a name="math_toolkit.special.bessel.mbessel.description"></a><h5> |
| <a name="id1127412"></a> |
| <a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.description">Description</a> |
| </h5> |
| <p> |
| 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: |
| </p> |
| <p> |
| cyl_bessel_i(v, x) = I<sub>v</sub>(x) |
| </p> |
| <p> |
| cyl_bessel_k(v, x) = K<sub>v</sub>(x) |
| </p> |
| <p> |
| where: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel2.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel3.png"></span> |
| </p> |
| <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> when T1 and T2 are different types. |
| The functions are also optimised for the relatively common case that T1 |
| is an integer. |
| </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> |
| <p> |
| The functions return the result of <a class="link" href="../../main_overview/error_handling.html#domain_error">domain_error</a> |
| whenever the result is undefined or complex. For <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a> |
| this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> |
| <span class="number">0</span></code> and v is not an integer, or when |
| <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span> |
| <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span> |
| <span class="special">!=</span> <span class="number">0</span></code>. |
| For <a class="link" href="bessel.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> |
| this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><=</span> |
| <span class="number">0</span></code>. |
| </p> |
| <p> |
| The following graph illustrates the exponential behaviour of I<sub>v</sub>. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/cyl_bessel_i.png" align="middle"></span> |
| </p> |
| <p> |
| The following graph illustrates the exponential decay of K<sub>v</sub>. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../graphs/cyl_bessel_k.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.special.bessel.mbessel.testing"></a><h5> |
| <a name="id1127684"></a> |
| <a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.testing">Testing</a> |
| </h5> |
| <p> |
| 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). |
| </p> |
| <a name="math_toolkit.special.bessel.mbessel.accuracy"></a><h5> |
| <a name="id1127706"></a> |
| <a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| 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="../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. All values are relative errors in units of epsilon. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.special.bessel.mbessel.errors_rates_in_cyl_bessel_i"></a><p class="title"><b>Table 38. Errors Rates in cyl_bessel_i</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in cyl_bessel_i"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| I<sub>v</sub> |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=10 Mean=3.4 GSL Peak=6000 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64 / G++ 3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11 Mean=3 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux AMD64 / G++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=11 Mean=4 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HP-UX / HP aCC 6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=15 Mean=4 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><div class="table"> |
| <a name="math_toolkit.special.bessel.mbessel.errors_rates_in_cyl_bessel_k"></a><p class="title"><b>Table 39. Errors Rates in cyl_bessel_k</b></p> |
| <div class="table-contents"><table class="table" summary="Errors Rates in cyl_bessel_k"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| K<sub>v</sub> |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 / Visual C++ 8.0 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=9 Mean=2 |
| </p> |
| <p> |
| GSL Peak=9 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Red Hat Linux IA64 / G++ 3.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=10 Mean=2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| SUSE Linux AMD64 / G++ 4.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=10 Mean=2 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HP-UX / HP aCC 6 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=12 Mean=5 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><a name="math_toolkit.special.bessel.mbessel.implementation"></a><h5> |
| <a name="id1128049"></a> |
| <a class="link" href="mbessel.html#math_toolkit.special.bessel.mbessel.implementation">Implementation</a> |
| </h5> |
| <p> |
| The following are handled as special cases first: |
| </p> |
| <p> |
| When computing I<sub>v</sub> ​ for <span class="emphasis"><em>x < 0</em></span>, 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 <span class="emphasis"><em>x > 0</em></span>. |
| </p> |
| <p> |
| For I<sub>v</sub> ​ with v equal to 0, 1 or 0.5 are handled as special cases. |
| </p> |
| <p> |
| The 0 and 1 cases use minimax rational approximations on finite and infinite |
| intervals. The coefficients are from: |
| </p> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| J.M. Blair and C.A. Edwards, <span class="emphasis"><em>Stable rational minimax approximations |
| to the modified Bessel functions I_0(x) and I_1(x)</em></span>, Atomic |
| Energy of Canada Limited Report 4928, Chalk River, 1974. |
| </li> |
| <li> |
| S. Moshier, <span class="emphasis"><em>Methods and Programs for Mathematical Functions</em></span>, |
| Ellis Horwood Ltd, Chichester, 1989. |
| </li> |
| </ul></div> |
| <p> |
| While the 0.5 case is a simple trigonometric function: |
| </p> |
| <p> |
| I<sub>0.5</sub>(x) = sqrt(2 / πx) * sinh(x) |
| </p> |
| <p> |
| For K<sub>v</sub> ​ with <span class="emphasis"><em>v</em></span> an integer, the result is calculated |
| using the recurrence relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel5.png"></span> |
| </p> |
| <p> |
| starting from K<sub>0</sub> ​ and K<sub>1</sub> ​ 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. |
| </p> |
| <p> |
| In the general case, we first normalize ν ​ to [<code class="literal">0, [inf</code>]) |
| with the help of the reflection formulae: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel9.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel10.png"></span> |
| </p> |
| <p> |
| 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 K<sub>μ</sub>(x) |
| and K<sub>μ+1</sub>(x), and use them to obtain I<sub>ν</sub>(x) and K<sub>ν</sub>(x). |
| </p> |
| <p> |
| The algorithm is proposed by Temme in N.M. Temme, <span class="emphasis"><em>On the numerical |
| evaluation of the modified bessel function of the third kind</em></span>, |
| Journal of Computational Physics, vol 19, 324 (1975), which needs two continued |
| fractions as well as the Wronskian: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel11.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel12.png"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel8.png"></span> |
| </p> |
| <p> |
| The continued fractions are computed using the modified Lentz's method |
| (W.J. Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations |
| using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)). |
| Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we |
| need different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>. |
| </p> |
| <p> |
| <span class="emphasis"><em>x > v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations |
| to converge, CF2 converges rapidly. |
| </p> |
| <p> |
| <span class="emphasis"><em>x <= v</em></span>, CF1 converges rapidly, CF2 fails to converge |
| when <span class="emphasis"><em>x</em></span> <code class="literal">-></code> 0. |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> > 2), both |
| continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>). |
| K<sub>μ</sub> ​ and K<sub>μ+1</sub> ​ |
| can be calculated by |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel13.png"></span> |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel14.png"></span> |
| </p> |
| <p> |
| <span class="emphasis"><em>S</em></span> is also a series that is summed along with CF2, |
| see I.J. Thompson and A.R. Barnett, <span class="emphasis"><em>Modified Bessel functions |
| I_v and K_v of real order and complex argument to selected accuracy</em></span>, |
| Computer Physics Communications, vol 47, 245 (1987). |
| </p> |
| <p> |
| When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> <= 2), |
| CF2 convergence may fail (but CF1 works very well). The solution here is |
| Temme's series: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel15.png"></span> |
| </p> |
| <p> |
| where |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../equations/mbessel16.png"></span> |
| </p> |
| <p> |
| f<sub>k</sub> ​ and h<sub>k</sub> ​ |
| are also computed by recursions (involving gamma functions), but |
| the formulas are a little complicated, readers are referred to N.M. Temme, |
| <span class="emphasis"><em>On the numerical evaluation of the modified Bessel function of |
| the third kind</em></span>, Journal of Computational Physics, vol 19, 324 |
| (1975). Note: Temme's series converge only for |μ| <= 1/2. |
| </p> |
| <p> |
| K<sub>ν</sub>(x) is then calculated from the forward recurrence, as is K<sub>ν+1</sub>(x). With |
| these two values and f<sub>ν</sub>, the Wronskian yields I<sub>ν</sub>(x) directly. |
| </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, 2010 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> |
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