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<html>
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<title>Modified Bessel Functions of the First and Second Kinds</title>
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<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>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.synopsis"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.synopsis">Synopsis</a>
</h5>
<p>
<code class="computeroutput"><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">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>
</p>
<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="../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="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">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="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">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</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">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="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">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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<h5>
<a name="math_toolkit.bessel.mbessel.h1"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.description"></a></span><a class="link" href="mbessel.html#math_toolkit.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="../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>
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>
<p>
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"><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_first.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>
<h5>
<a name="math_toolkit.bessel.mbessel.h2"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.testing"></a></span><a class="link" href="mbessel.html#math_toolkit.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>
<h5>
<a name="math_toolkit.bessel.mbessel.h3"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.accuracy"></a></span><a class="link" href="mbessel.html#math_toolkit.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="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
zero error</a>. All values are relative errors in units of epsilon.
</p>
<div class="table">
<a name="math_toolkit.bessel.mbessel.errors_rates_in_cyl_bessel_i"></a><p class="title"><b>Table 6.23. 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.bessel.mbessel.errors_rates_in_cyl_bessel_k"></a><p class="title"><b>Table 6.24. 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"><h5>
<a name="math_toolkit.bessel.mbessel.h4"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.implementation"></a></span><a class="link" href="mbessel.html#math_toolkit.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 class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
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 class="listitem">
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>
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
I<sub>v</sub>x   is best computed directly from the series:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel17.png"></span>
</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>
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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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