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<title>Modified Bessel Functions of the First and Second Kinds</title>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<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
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of the First and Second Kinds</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.bessel.mbessel.h0"></a>
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<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>
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</h5>
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<p>
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<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>
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</p>
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<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>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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<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>
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<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>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
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</pre>
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<h5>
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<a name="math_toolkit.bessel.mbessel.h1"></a>
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<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>
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</h5>
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<p>
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The functions <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>
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and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> return
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the result of the modified Bessel functions of the first and second kind
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respectively:
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</p>
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<p>
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cyl_bessel_i(v, x) = I<sub>v</sub>(x)
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</p>
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<p>
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cyl_bessel_k(v, x) = K<sub>v</sub>(x)
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</p>
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<p>
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where:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
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</p>
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<p>
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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
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type calculation rules</em></span></a> when T1 and T2 are different types.
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The functions are also optimised for the relatively common case that T1 is
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an integer.
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</p>
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<p>
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The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
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be used to control the behaviour of the function: how it handles errors,
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what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
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documentation for more details</a>.
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</p>
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<p>
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The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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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>
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this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
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<span class="number">0</span></code> and v is not an integer, or when
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<code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span>
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<span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span>
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<span class="special">!=</span> <span class="number">0</span></code>.
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For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this
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occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><=</span>
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<span class="number">0</span></code>.
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</p>
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<p>
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The following graph illustrates the exponential behaviour of I<sub>v</sub>.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_i.svg" align="middle"></span>
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</p>
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<p>
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The following graph illustrates the exponential decay of K<sub>v</sub>.
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_k.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.bessel.mbessel.h2"></a>
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<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>
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</h5>
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<p>
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There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
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and a much larger set of tests computed using a simplified version of this
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implementation (with all the special case handling removed).
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</p>
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<h5>
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<a name="math_toolkit.bessel.mbessel.h3"></a>
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<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>
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</h5>
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<p>
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The following tables show how the accuracy of these functions varies on various
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platforms, along with comparison to other libraries. Note that only results
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for the widest floating-point type on the system are given, as narrower types
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have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
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error</a>. All values are relative errors in units of epsilon. Note that
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our test suite includes some fairly extreme inputs which results in most
|
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of the worst problem cases in other libraries:
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</p>
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<div class="table">
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<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table 6.44. Error rates for cyl_bessel_i (integer orders)</b></p>
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<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i (integer orders)">
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<colgroup>
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<col>
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<col>
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<col>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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</th>
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<th>
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<p>
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Microsoft Visual C++ version 12.0<br> Win32<br> double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> long double
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</p>
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</th>
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<th>
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<p>
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GNU C++ version 5.1.0<br> linux<br> double
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</p>
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</th>
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<th>
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<p>
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Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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Bessel I0: Mathworld Data (Integer Version)
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</p>
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</td>
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<td>
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<p>
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<span class="blue">Max = 0.877ε (Mean = 0.549ε)</span>
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</p>
|
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</td>
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<td>
|
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<p>
|
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<span class="blue">Max = 4.57ε (Mean = 2.1ε)</span><br> <br>
|
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(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 8.49ε (Mean = 3.46ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_I0_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)
|
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</p>
|
|
</td>
|
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<td>
|
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<p>
|
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<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
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1.16:</em></span> Max = 0.79ε (Mean = 0.482ε))<br> (<span class="emphasis"><em>Rmath
|
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3.0.2:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_0_2_Bessel_I0_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 2.55e+43ε (Mean = 8.06e+42ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_I0_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span>
|
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</p>
|
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</td>
|
|
<td>
|
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<p>
|
|
<span class="blue">Max = 4.54ε (Mean = 2.11ε)</span>
|
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</p>
|
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</td>
|
|
</tr>
|
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<tr>
|
|
<td>
|
|
<p>
|
|
Bessel I1: Mathworld Data (Integer Version)
|
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</p>
|
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</td>
|
|
<td>
|
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<p>
|
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<span class="blue">Max = 0.885ε (Mean = 0.55ε)</span>
|
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</p>
|
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</td>
|
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<td>
|
|
<p>
|
|
<span class="blue">Max = 7.83ε (Mean = 2.79ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 5ε (Mean = 2.15ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_I1_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)
|
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</p>
|
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</td>
|
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<td>
|
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<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
1.16:</em></span> Max = 0.82ε (Mean = 0.456ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_0_2_Bessel_I1_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 1.28e+43ε (Mean = 4.05e+42ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_I1_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 6.52ε (Mean = 2.25ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
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Bessel In: Mathworld Data (Integer Version)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 430ε (Mean = 163ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_In_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
|
|
1.16:</em></span> Max = 5.15ε (Mean = 2.13ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__GSL_1_16_Bessel_In_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Rmath_3_0_2_Bessel_In_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 3.67e+177ε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_In_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 463ε (Mean = 140ε)</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 6.45. 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>
|
|
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel I0: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.877ε (Mean = 0.549ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.57ε (Mean = 2.1ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 8.49ε (Mean = 3.46ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_I0_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
|
|
1.16:</em></span> Max = 270ε (Mean = 91.6ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_I0_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Rmath_3_0_2_Bessel_I0_Mathworld_Data">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 2.55e+43ε (Mean = 8.06e+42ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_I0_Mathworld_Data">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.54ε (Mean = 2.11ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel I1: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.885ε (Mean = 0.55ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 7.83ε (Mean = 2.79ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 5ε (Mean = 2.15ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_I1_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
|
|
1.16:</em></span> Max = 128ε (Mean = 41ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_I1_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Rmath_3_0_2_Bessel_I1_Mathworld_Data">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 1.28e+43ε (Mean = 4.05e+42ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_I1_Mathworld_Data">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 6.52ε (Mean = 2.25ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel In: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.46ε (Mean = 1.32ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.33ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 430ε (Mean = 163ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_In_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
|
|
1.16:</em></span> Max = 2.31ε (Mean = 0.838ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_In_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Rmath_3_0_2_Bessel_In_Mathworld_Data">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = 3.67e+177ε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_In_Mathworld_Data">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 463ε (Mean = 140ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Iv: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 2.97ε (Mean = 1.33ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.12ε (Mean = 1.85ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 616ε (Mean = 221ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_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
|
|
1.16:</em></span> Max = 5.95ε (Mean = 2.08ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 1e+04ε (Mean = 3.18e+03ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_Iv_Mathworld_Data">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.12ε (Mean = 1.95ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel In: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 9.67ε (Mean = 1.89ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 6.79ε (Mean = 1.15ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 645ε (Mean = 132ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
|
1.16:</em></span> Max = 261ε (Mean = 53.2ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_In_Random_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 7.37ε (Mean = 2.4ε))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
|
|
= 4.22e+06ε (Mean = 2.26e+05ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 176ε (Mean = 39.2ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Iv: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 7.46ε (Mean = 1.54ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 8.35ε (Mean = 1.49ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.05e+03ε (Mean =
|
|
224ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_Random_Data">And
|
|
other failures.</a>)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 0.661ε (Mean = 0.0441ε)</span><br>
|
|
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 6.18e+03ε (Mean = 1.55e+03ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Random_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
<span class="red">Max = 4.28e+08ε (Mean = 2.85e+07ε))</span><br>
|
|
(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 6e+30ε (Mean
|
|
= 4e+29ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_Iv_Random_Data">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 283ε (Mean = 88.4ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Iv: Mathworld Data (large values)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.67ε (Mean = 1.64ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 14.7ε (Mean = 6.57ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 118ε (Mean = 57.2ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_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
|
|
1.16:</em></span> Max = 37ε (Mean = 18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Mathworld_Data_large_values_">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</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_5_1_0_linux_double_cyl_bessel_i_Rmath_3_0_2_Bessel_Iv_Mathworld_Data_large_values_">And
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
Max = 73.7ε (Mean = 58.5ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 14.7ε (Mean = 6.59ε)</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 6.46. 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>
|
|
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel K0: Mathworld Data (Integer Version)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.55ε (Mean = 0.837ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.16ε (Mean = 1.46ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/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
|
|
1.16:</em></span> Max = 1.2ε (Mean = 0.733ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> Max = 0.833ε (Mean = 0.601ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
Max = 1.1e+06ε (Mean = 3.68e+05ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.16ε (Mean = 1.49ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel K1: Mathworld Data (Integer Version)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1ε (Mean = 0.573ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.02ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/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
|
|
1.16:</em></span> Max = 0.626ε (Mean = 0.333ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> Max = 0.894ε (Mean = 0.516ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
Max = 5.38e+05ε (Mean = 1.79e+05ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.02ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kn: Mathworld Data (Integer Version)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.48ε (Mean = 2.14ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 12.9ε (Mean = 4.91ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k_integer_orders___tr1_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
|
|
1.16:</em></span> Max = 168ε (Mean = 59.5ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> Max = 8.48ε (Mean = 2.98ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
|
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_integer_orders__Cephes_Bessel_Kn_Mathworld_Data_Integer_Version_">And
|
|
other failures.</a>)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.48ε (Mean = 1.98ε)</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 6.47. 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>
|
|
Microsoft Visual C++ version 12.0<br> Win32<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
GNU C++ version 5.1.0<br> linux<br> double
|
|
</p>
|
|
</th>
|
|
<th>
|
|
<p>
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
|
</p>
|
|
</th>
|
|
</tr></thead>
|
|
<tbody>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel K0: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.55ε (Mean = 0.837ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.16ε (Mean = 1.46ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/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
|
|
1.16:</em></span> Max = 6.04ε (Mean = 2.16ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> Max = 0.833ε (Mean = 0.601ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.16ε (Mean = 1.49ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel K1: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1ε (Mean = 0.573ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.02ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/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
|
|
1.16:</em></span> Max = 6.26ε (Mean = 2.21ε))<br> (<span class="emphasis"><em>Rmath
|
|
3.0.2:</em></span> Max = 0.894ε (Mean = 0.516ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 1.8ε (Mean = 1.02ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kn: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.63ε (Mean = 1.46ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.48ε (Mean = 2.14ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 12.9ε (Mean = 4.91ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_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
|
|
1.16:</em></span> Max = 3.36ε (Mean = 1.43ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kn_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 8.48ε (Mean = 2.98ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.48ε (Mean = 1.98ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kv: Mathworld Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.78ε (Mean = 2.2ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.58ε (Mean = 2.44ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 13ε (Mean = 4.81ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_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
|
|
1.16:</em></span> Max = 5.47ε (Mean = 2.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Mathworld_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 3.15ε (Mean = 1.35ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 3.58ε (Mean = 2.29ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kv: Mathworld Data (large values)
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 59.8ε (Mean = 26.9ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 42.3ε (Mean = 21ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 42.3ε (Mean = 19.8ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_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
|
|
1.16:</em></span> Max = 308ε (Mean = 142ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Mathworld_Data_large_values_">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 84.6ε (Mean = 37.8ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 43.1ε (Mean = 21.3ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kn: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 7.47ε (Mean = 1.4ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.55ε (Mean = 1.09ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/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 1.16:</em></span> Max = 8.71ε (Mean = 1.76ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kn_Random_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 7.47ε (Mean = 1.34ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 4.55ε (Mean = 1.21ε)</span>
|
|
</p>
|
|
</td>
|
|
</tr>
|
|
<tr>
|
|
<td>
|
|
<p>
|
|
Bessel Kv: Random Data
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 8.33ε (Mean = 1.62ε)</span>
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 7.88ε (Mean = 1.48ε)</span><br> <br>
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 13.6ε (Mean = 2.68ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_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 1.16:</em></span> Max = 9.71ε (Mean = 1.47ε)
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Random_Data">And
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
|
Max = 7.37ε (Mean = 1.49ε))
|
|
</p>
|
|
</td>
|
|
<td>
|
|
<p>
|
|
<span class="blue">Max = 7.88ε (Mean = 1.49ε)</span>
|
|
</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.svg"></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.svg"></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.svg"></span>
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></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.svg"></span>
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel12.svg"></span>
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></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.svg"></span>
|
|
</p>
|
|
<p>
|
|
where
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel14.svg"></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.svg"></span>
|
|
</p>
|
|
<p>
|
|
where
|
|
</p>
|
|
<p>
|
|
<span class="inlinemediaobject"><img src="../../../equations/mbessel16.svg"></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 © 2006-2010, 2012-2014 Nikhar Agrawal,
|
|
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, 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>)
|
|
</p>
|
|
</div></td>
|
|
</tr></table>
|
|
<hr>
|
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