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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 22. 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 22. 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 22. 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 22. 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>
<div class="blockquote"><blockquote class="blockquote"><p>
cyl_bessel_i(v, x) = I<sub>v</sub>(x)
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
cyl_bessel_k(v, x) = K<sub>v</sub>(x)
</p></blockquote></div>
<p>
where:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
</p></blockquote></div>
<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 22. 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 22. 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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_i.svg" align="middle"></span>
</p></blockquote></div>
<p>
The following graph illustrates the exponential decay of K<sub>v</sub>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_k.svg" align="middle"></span>
</p></blockquote></div>
<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 comparison to other libraries. 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. Note that
our test suite includes some fairly extreme inputs which results in most
of the worst problem cases in other libraries:
</p>
<div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table 8.44. Error rates for cyl_bessel_i (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i (integer orders)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Bessel I0: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.79ε (Mean = 0.482ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.52ε (Mean = 0.622ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I0_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95ε (Mean = 0.738ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 8.49ε (Mean = 3.46ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I0_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95ε (Mean = 0.661ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.762ε (Mean = 0.329ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel I1: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.82ε (Mean = 0.456ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.53ε (Mean = 0.483ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_I1_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 5ε (Mean = 2.15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_I1_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.767ε (Mean = 0.398ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel In: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.15ε (Mean = 2.13ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__GSL_2_1_Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.73ε (Mean = 0.601ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_2_3_Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 430ε (Mean = 163ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i_integer_orders___cmath__Bessel_In_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 463ε (Mean = 140ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table 8.45. Error rates for cyl_bessel_i</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Bessel I0: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 270ε (Mean = 91.6ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I0_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.52ε (Mean = 0.622ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I0_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95ε (Mean = 0.738ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 8.49ε (Mean = 3.46ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I0_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.95ε (Mean = 0.661ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.762ε (Mean = 0.329ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel I1: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 128ε (Mean = 41ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_I1_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.53ε (Mean = 0.483ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_I1_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 5ε (Mean = 2.15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_I1_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.64ε (Mean = 0.202ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.767ε (Mean = 0.398ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel In: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.31ε (Mean = 0.838ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 1.73ε (Mean = 0.601ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_In_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 430ε (Mean = 163ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_In_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 463ε (Mean = 140ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Iv: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.95ε (Mean = 2.08ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 3.53ε (Mean = 1.39ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.12ε (Mean = 1.85ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 616ε (Mean = 221ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.12ε (Mean = 1.95ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.97ε (Mean = 1.24ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel In: Random Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 261ε (Mean = 53.2ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_In_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.37ε (Mean = 2.4ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.62ε (Mean = 1.06ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 645ε (Mean = 132ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 176ε (Mean = 39.1ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 9.67ε (Mean = 1.88ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Iv: Random Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.661ε (Mean = 0.0441ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 6.18e+03ε (Mean = 1.55e+03ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = 4.28e+08ε (Mean = 2.85e+07ε))</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.35ε (Mean = 1.62ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 1.05e+03ε (Mean = 224ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Random_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 283ε (Mean = 88.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.46ε (Mean = 1.71ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Iv: Mathworld Data (large values)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 37ε (Mean = 18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_GSL_2_1_Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = 3.77e+168ε (Mean = 2.39e+168ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_i_Rmath_3_2_3_Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 14.7ε (Mean = 6.66ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 118ε (Mean = 57.2ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_i__cmath__Bessel_Iv_Mathworld_Data_large_values_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 14.7ε (Mean = 6.59ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.67ε (Mean = 1.64ε)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table 8.46. Error rates for cyl_bessel_k (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k (integer orders)">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Bessel K0: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 9.33ε (Mean = 3.25ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.2ε (Mean = 0.733ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833ε (Mean = 0.601ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.552ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel K1: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 8.94ε (Mean = 3.19ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.626ε (Mean = 0.333ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894ε (Mean = 0.516ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.39ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kn: Mathworld Data (Integer Version)
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 12.9ε (Mean = 4.91ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k_integer_orders___cmath__Bessel_Kn_Mathworld_Data_Integer_Version_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 168ε (Mean = 59.5ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 8.48ε (Mean = 2.98ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table 8.47. Error rates for cyl_bessel_k</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
Bessel K0: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 9.33ε (Mean = 3.25ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.04ε (Mean = 2.16ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.833ε (Mean = 0.601ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.436ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833ε (Mean = 0.552ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel K1: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 8.94ε (Mean = 3.19ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 6.26ε (Mean = 2.21ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.894ε (Mean = 0.516ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.329ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.786ε (Mean = 0.39ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kn: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 12.9ε (Mean = 4.91ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kn_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 3.36ε (Mean = 1.43ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kn_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 8.48ε (Mean = 2.98ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6ε (Mean = 1.21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kv: Mathworld Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.58ε (Mean = 2.39ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 13ε (Mean = 4.81ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.47ε (Mean = 2.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 3.15ε (Mean = 1.35ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.21ε (Mean = 2.53ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.78ε (Mean = 2.19ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kv: Mathworld Data (large values)
</p>
</td>
<td>
<p>
<span class="blue">Max = 42.3ε (Mean = 21ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 42.3ε (Mean = 19.8ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Mathworld_Data_large_values_">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 308ε (Mean = 142ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Mathworld_Data_large_values_">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 84.6ε (Mean = 37.8ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 42.3ε (Mean = 21ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 59.8ε (Mean = 26.9ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kn: Random Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.55ε (Mean = 1.12ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 13.9ε (Mean = 2.91ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.764ε (Mean = 0.0348ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.71ε (Mean = 1.76ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kn_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.47ε (Mean = 1.34ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.55ε (Mean = 1.12ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 9.34ε (Mean = 1.7ε)</span>
</p>
</td>
</tr>
<tr>
<td>
<p>
Bessel Kv: Random Data
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.88ε (Mean = 1.48ε)</span><br> <br>
(<span class="emphasis"><em><cmath>:</em></span> Max = 13.6ε (Mean = 2.68ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_k__cmath__Bessel_Kv_Random_Data">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.507ε (Mean = 0.0313ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 9.71ε (Mean = 1.47ε)
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_k_GSL_2_1_Bessel_Kv_Random_Data">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 7.37ε (Mean = 1.49ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.88ε (Mean = 1.47ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.33ε (Mean = 1.62ε)</span>
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
The following error plot are based on an exhaustive search of the functions
domain for I0, I1, K0, and K1, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i0__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i0__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i0____float128.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i1__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i1__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/i1____float128.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k0__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k0__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k0____float128.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k1__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k1__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/k1____float128.svg" align="middle"></span>
</p></blockquote></div>
<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 polynomial approximations on finite and infinite intervals.
The approximating forms are based on <a href="http://www.advanpix.com/2015/11/11/rational-approximations-for-the-modified-bessel-function-of-the-first-kind-i0-computations-double-precision/" target="_top">"Rational
Approximations for the Modified Bessel Function of the First Kind - I<sub>0</sub>(x)
for Computations with Double Precision"</a> by Pavel Holoborodko,
extended by us to deal with up to 128-bit precision (with different approximations
for each target precision).
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel21.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel20.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel17.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel18.svg"></span>
</p></blockquote></div>
<p>
Similarly we have:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel22.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel23.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel24.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/bessel25.svg"></span>
</p></blockquote></div>
<p>
The 0.5 case is a simple trigonometric function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
I<sub>0.5</sub>(x) = sqrt(2 / πx) * sinh(x)
</p></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
</p></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel17.svg"></span>
</p></blockquote></div>
<p>
In the general case, we first normalize ν to [<code class="literal">0, [inf]</code>)
with the help of the reflection formulae:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
</p></blockquote></div>
<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
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
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),
</p></blockquote></div>
<p>
which needs two continued fractions as well as the Wronskian:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel11.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel12.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
</p></blockquote></div>
<p>
The continued fractions are computed using the modified Lentz's method
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
(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)).
</p></blockquote></div>
<p>
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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel13.svg"></span>
</p></blockquote></div>
<p>
where
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel14.svg"></span>
</p></blockquote></div>
<p>
<span class="emphasis"><em>S</em></span> is also a series that is summed along with CF2, see
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
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></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel15.svg"></span>
</p></blockquote></div>
<p>
where
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/mbessel16.svg"></span>
</p></blockquote></div>
<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
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
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).
</p></blockquote></div>
<p>
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>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div>
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