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<td align="center"><a href="../../../../../../index.html">Home</a></td>
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<a accesskey="p" href="bessel_over.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../bessel.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bessel_root.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
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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.bessel_first"></a><a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">Bessel Functions of
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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.bessel_first.h0"></a>
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<span class="phrase"><a name="math_toolkit.bessel.bessel_first.synopsis"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.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_j</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_j</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_neumann</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_neumann</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.bessel_first.h1"></a>
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<span class="phrase"><a name="math_toolkit.bessel.bessel_first.description"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.description">Description</a>
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</h5>
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<p>
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The functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
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and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> return
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the result of the Bessel functions of the first and second kinds respectively:
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</p>
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<p>
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cyl_bessel_j(v, x) = J<sub>v</sub>(x)
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</p>
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<p>
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cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<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/bessel2.svg"></span>
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../equations/bessel3.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 cyclic nature of J<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_j.svg" align="middle"></span>
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</p>
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<p>
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The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large
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<span class="emphasis"><em>x</em></span>, but tends to -∞   for small <span class="emphasis"><em>x</em></span>:
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</p>
|
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<p>
|
||
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<span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
|
||
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</p>
|
||
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<h5>
|
||
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<a name="math_toolkit.bessel.bessel_first.h2"></a>
|
||
|
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<span class="phrase"><a name="math_toolkit.bessel.bessel_first.testing"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.testing">Testing</a>
|
||
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</h5>
|
||
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<p>
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||
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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.bessel_first.h3"></a>
|
||
|
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<span class="phrase"><a name="math_toolkit.bessel.bessel_first.accuracy"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.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 comparisons to other libraries. Note that the cyclic
|
||
|
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nature of these functions means that they have an infinite number of irrational
|
||
|
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roots: in general these functions have arbitrarily large <span class="emphasis"><em>relative</em></span>
|
||
|
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errors when the arguments are sufficiently close to a root. Of course the
|
||
|
|
absolute error in such cases is always small. Note that only results 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. Most of
|
||
|
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the gross errors exhibited by other libraries occur for very large arguments
|
||
|
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- you will need to drill down into the actual program output if you need
|
||
|
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more information on this.
|
||
|
|
</p>
|
||
|
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<div class="table">
|
||
|
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<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table 6.40. Error rates for cyl_bessel_j (integer orders)</b></p>
|
||
|
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<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j (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>
|
||
|
|
</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>
|
||
|
|
</th>
|
||
|
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</tr></thead>
|
||
|
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<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J0: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
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<p>
|
||
|
|
<span class="blue">Max = 2.52ε (Mean = 1.2ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.89ε (Mean = 0.988ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.55ε (Mean = 2.89ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 5.04ε (Mean = 1.78ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_J0_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 = 1.12ε (Mean = 0.488ε))<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_j_integer_orders__Rmath_3_0_2_Bessel_J0_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 1.12ε (Mean = 0.568ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.55ε (Mean = 2.86ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J0: Mathworld Data (Tricky cases) (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1e+007ε (Mean = 4.09e+006ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><math.h>:</em></span> <span class="red">Max
|
||
|
|
= 2.54e+008ε (Mean = 1.04e+008ε))</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.63e+08ε (Mean = 6.67e+07ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 4.79e+08ε (Mean
|
||
|
|
= 1.96e+08ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 7.98e+04ε (Mean = 3.26e+04ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1e+07ε (Mean = 4.11e+06ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.04e+07ε (Mean = 4.29e+06ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 2.54e+08ε (Mean
|
||
|
|
= 1.04e+08ε))</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J1: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.73ε (Mean = 0.976ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 11.4ε (Mean = 4.15ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.66ε (Mean = 1.38ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 6.1ε (Mean = 2.95ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_J1_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 = 1.89ε (Mean = 0.721ε))<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_j_integer_orders__Rmath_3_0_2_Bessel_J1_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 2.88ε (Mean = 1.12ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.44ε (Mean = 0.637ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J1: Mathworld Data (tricky cases) (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 3.23e+004ε (Mean = 1.45e+004ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><math.h>:</em></span> Max = 1.44e+007ε (Mean
|
||
|
|
= 6.5e+006ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.15e+06ε (Mean
|
||
|
|
= 1.58e+06ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 106ε (Mean = 47.5ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.26e+06ε (Mean = 6.28e+05ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.93e+06ε (Mean = 1.7e+06ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> Max = 9.56e+05ε (Mean = 4.99e+05ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel JN: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 14.7ε (Mean = 5.4ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> <span class="red">Max =
|
||
|
|
+INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_Microsoft_Visual_C_version_12_0_Win32_double_cyl_bessel_j_integer_orders___math_h__Bessel_JN_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.85ε (Mean = 3.41ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 2.13e+19ε (Mean = 5.16e+18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_JN_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 6.9e+05ε (Mean = 2.53e+05ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_integer_orders__GSL_1_16_Bessel_JN_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_j_integer_orders__Rmath_3_0_2_Bessel_JN_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span><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_j_integer_orders__Cephes_Bessel_JN_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 463ε (Mean = 112ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><div class="table">
|
||
|
|
<a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j"></a><p class="title"><b>Table 6.41. Error rates for cyl_bessel_j</b></p>
|
||
|
|
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j">
|
||
|
|
<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 J0: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.52ε (Mean = 1.2ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.55ε (Mean = 2.89ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 5.04ε (Mean = 1.78ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J0_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 = 0.629ε (Mean = 0.223ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J0_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_j_Rmath_3_0_2_Bessel_J0_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 1.12ε (Mean = 0.568ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.55ε (Mean = 2.86ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J0: Mathworld Data (Tricky cases)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1e+007ε (Mean = 4.09e+006ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.63e+08ε (Mean = 6.67e+07ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 4.79e+08ε (Mean
|
||
|
|
= 1.96e+08ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 7.98e+04ε (Mean = 3.26e+04ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 6.5e+07ε (Mean = 2.66e+07ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.04e+07ε (Mean = 4.29e+06ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 2.54e+08ε (Mean
|
||
|
|
= 1.04e+08ε))</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J1: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.73ε (Mean = 0.976ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.66ε (Mean = 1.38ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 6.1ε (Mean = 2.95ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J1_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 = 6.62ε (Mean = 2.35ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J1_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_j_Rmath_3_0_2_Bessel_J1_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 2.88ε (Mean = 1.12ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.44ε (Mean = 0.637ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J1: Mathworld Data (tricky cases)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 3.23e+004ε (Mean = 1.45e+004ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.15e+06ε (Mean
|
||
|
|
= 1.58e+06ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 106ε (Mean = 47.5ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.75e+05ε (Mean = 5.32e+05ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.93e+06ε (Mean = 1.7e+06ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> Max = 9.56e+05ε (Mean = 4.99e+05ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel JN: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 14.7ε (Mean = 5.4ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.85ε (Mean = 3.41ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 2.13e+19ε (Mean = 5.16e+18ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_JN_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 6.9e+05ε (Mean = 2.15e+05ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_JN_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_j_Rmath_3_0_2_Bessel_JN_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 5.53e+05ε (Mean = 1.9e+05ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 463ε (Mean = 112ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 14.9ε (Mean = 3.82ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 14.7ε (Mean = 4.05ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 3.49e+05ε (Mean =
|
||
|
|
7.89e+04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J_Mathworld_Data">And
|
||
|
|
other failures.</a>)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 10ε (Mean = 2.19ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.39e+05ε (Mean = 5.24e+04ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_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_j_Rmath_3_0_2_Bessel_J_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 5.47e+05ε (Mean = 1.3e+05ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 14.7ε (Mean = 4.12ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J: Mathworld Data (large values)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 9.31ε (Mean = 5.52ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 607ε (Mean = 305ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 34.9ε (Mean = 17.4ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J_Mathworld_Data_large_values_">And
|
||
|
|
other failures.</a>)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.536ε (Mean = 0.268ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 4.91e+03ε (Mean = 2.46e+03ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_Mathworld_Data_large_values_">And
|
||
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
||
|
|
Max = 35.9ε (Mean = 18.1ε))<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_j_Cephes_Bessel_J_Mathworld_Data_large_values_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 607ε (Mean = 305ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel JN: Random Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 17.5ε (Mean = 1.46ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 50.8ε (Mean = 4.15ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 1.12e+03ε (Mean =
|
||
|
|
88.7ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 75.7ε (Mean = 5.36ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 3.93ε (Mean = 1.22ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 91.4ε (Mean = 6.47ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 99.6ε (Mean = 22ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J: Random Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 9.24ε (Mean = 1.36ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 9.81ε (Mean = 1.59ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 501ε (Mean = 52.3ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 15.5ε (Mean = 3.33ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_Random_Data">And
|
||
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
||
|
|
Max = 6.74ε (Mean = 1.3ε))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
|
||
|
|
= 16.7ε (Mean = 2.5ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 260ε (Mean = 34ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Bessel J: Random Data (Tricky large values)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 59.2ε (Mean = 8.67ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 785ε (Mean = 94.2ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 5.01e+17ε (Mean = 6.23e+16ε))</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 2.48e+05ε (Mean = 5.11e+04ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 71.6ε (Mean = 11.7ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 2.48e+05ε (Mean = 3.02e+04ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 785ε (Mean = 97.4ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><div class="table">
|
||
|
|
<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_"></a><p class="title"><b>Table 6.42. Error rates for cyl_neumann (integer orders)</b></p>
|
||
|
|
<div class="table-contents"><table class="table" summary="Error rates for cyl_neumann (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> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
</tr></thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Y0: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 4.61ε (Mean = 2.29ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 5.37e+003ε (Mean = 1.81e+003ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 6.46ε (Mean = 2.38ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 167ε (Mean = 56.5ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 5.37e+03ε (Mean = 1.81e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 5.59ε (Mean = 2.54ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.05e+05ε (Mean =
|
||
|
|
6.87e+04ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 5.53ε (Mean = 2.4ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Y1: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 4.75ε (Mean = 1.72ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 1.86e+004ε (Mean = 6.2e+003ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 1.51ε (Mean = 0.839ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 193ε (Mean = 64.4ε))<br> (<span class="emphasis"><em>Cephes:</em></span>
|
||
|
|
Max = 1.86e+04ε (Mean = 6.2e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 12.7ε (Mean = 4.34ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 9.71e+03ε (Mean =
|
||
|
|
4.08e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.33ε (Mean = 2.29ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yn: Mathworld Data (Integer Version)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 35ε (Mean = 11.8ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><math.h>:</em></span> Max = 2.49e+005ε (Mean = 8.14e+004ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.993ε (Mean = 0.314ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.41e+05ε (Mean = 7.62e+04ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.24e+04ε (Mean = 4e+03ε))<br>
|
||
|
|
(<span class="emphasis"><em>Cephes:</em></span> Max = 2.49e+05ε (Mean = 8.14e+04ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 55.2ε (Mean = 17.7ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 2.2e+20ε (Mean = 6.97e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann_integer_orders___tr1_cmath__Yn_Mathworld_Data_Integer_Version_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 55.2ε (Mean = 17.8ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><div class="table">
|
||
|
|
<a name="math_toolkit.bessel.bessel_first.table_cyl_neumann"></a><p class="title"><b>Table 6.43. Error rates for cyl_neumann</b></p>
|
||
|
|
<div class="table-contents"><table class="table" summary="Error rates for cyl_neumann">
|
||
|
|
<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> double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
GNU C++ version 5.1.0<br> linux<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
<th>
|
||
|
|
<p>
|
||
|
|
Sun compiler version 0x5130<br> Sun Solaris<br> long double
|
||
|
|
</p>
|
||
|
|
</th>
|
||
|
|
</tr></thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Y0: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 4.61ε (Mean = 2.29ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 60.9ε (Mean = 20.4ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 167ε (Mean = 56.5ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 5.59ε (Mean = 2.54ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.05e+05ε (Mean =
|
||
|
|
6.87e+04ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 5.53ε (Mean = 2.4ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Y1: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 4.75ε (Mean = 1.72ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 23.4ε (Mean = 8.1ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 193ε (Mean = 64.4ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 12.7ε (Mean = 4.34ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 9.71e+03ε (Mean =
|
||
|
|
4.08e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 6.33ε (Mean = 2.29ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yn: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 35ε (Mean = 11.8ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.993ε (Mean = 0.314ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.41e+05ε (Mean = 7.62e+04ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yn_Mathworld_Data">And
|
||
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
||
|
|
Max = 1.24e+04ε (Mean = 4e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 55.2ε (Mean = 17.7ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 2.2e+20ε (Mean = 6.97e+19ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yn_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 55.2ε (Mean = 17.8ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yv: Mathworld Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 7.89ε (Mean = 3.27ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 10ε (Mean = 3.02ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.07e+05ε (Mean = 3.22e+04ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yv_Mathworld_Data">And
|
||
|
|
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
|
||
|
|
Max = 1.05e+03ε (Mean = 326ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 10.7ε (Mean = 4.92ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> <span class="red">Max
|
||
|
|
= 3.49e+15ε (Mean = 1.05e+15ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yv_Mathworld_Data">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 10.7ε (Mean = 5.1ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yv: Mathworld Data (large values)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0.682ε (Mean = 0.35ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 60.8ε (Mean = 23ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yv_Mathworld_Data_large_values_">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_neumann_Rmath_3_0_2_Yv_Mathworld_Data_large_values_">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.57ε (Mean = 1.17ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 43.2ε (Mean = 16.3ε)
|
||
|
|
<a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yv_Mathworld_Data_large_values_">And
|
||
|
|
other failures.</a>)
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.57ε (Mean = 1.24ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Y0 and Y1: Random Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 4.17ε (Mean = 1.24ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 34.4ε (Mean = 8.9ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 83ε (Mean = 14.2ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 11.8ε (Mean = 3.28ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 2.59e+03ε (Mean =
|
||
|
|
500ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 10.8ε (Mean = 3.04ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yn: Random Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 117ε (Mean = 10.2ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
|
||
|
|
1.16:</em></span> Max = 500ε (Mean = 47.8ε))<br> (<span class="emphasis"><em>Rmath
|
||
|
|
3.0.2:</em></span> Max = 691ε (Mean = 67.9ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 338ε (Mean = 28.2ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em><tr1/cmath>:</em></span> Max = 4.01e+03ε (Mean =
|
||
|
|
348ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 338ε (Mean = 27.5ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
Yv: Random Data
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.23e+003ε (Mean = 69.9ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 1.53ε (Mean = 0.102ε)</span><br> <br>
|
||
|
|
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.41e+06ε (Mean = 7.67e+04ε))<br>
|
||
|
|
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.79e+05ε (Mean = 9.64e+03ε))
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.08e+03ε (Mean = 149ε)</span><br>
|
||
|
|
<br> (<span class="emphasis"><em><tr1/cmath>:</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_long_double_cyl_neumann__tr1_cmath__Yv_Random_Data">And
|
||
|
|
other failures.</a>)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
<td>
|
||
|
|
<p>
|
||
|
|
<span class="blue">Max = 2.08e+03ε (Mean = 149ε)</span>
|
||
|
|
</p>
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table></div>
|
||
|
|
</div>
|
||
|
|
<br class="table-break"><p>
|
||
|
|
Note that for large <span class="emphasis"><em>x</em></span> these functions are largely dependent
|
||
|
|
on the accuracy of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span></code> and
|
||
|
|
<code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span></code> functions.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Comparison to GSL and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
|
||
|
|
is interesting: both <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
|
||
|
|
and this library optimise the integer order case - leading to identical results
|
||
|
|
- simply using the general case is for the most part slightly more accurate
|
||
|
|
though, as noted by the better accuracy of GSL in the integer argument cases.
|
||
|
|
This implementation tends to perform much better when the arguments become
|
||
|
|
large, <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> in particular
|
||
|
|
produces some remarkably inaccurate results with some of the test data (no
|
||
|
|
significant figures correct), and even GSL performs badly with some inputs
|
||
|
|
to J<sub>v</sub>. Note that by way of double-checking these results, the worst performing
|
||
|
|
<a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> and GSL cases were
|
||
|
|
recomputed using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
|
||
|
|
and the result checked against our test data: no errors in the test data
|
||
|
|
were found.
|
||
|
|
</p>
|
||
|
|
<h5>
|
||
|
|
<a name="math_toolkit.bessel.bessel_first.h4"></a>
|
||
|
|
<span class="phrase"><a name="math_toolkit.bessel.bessel_first.implementation"></a></span><a class="link" href="bessel_first.html#math_toolkit.bessel.bessel_first.implementation">Implementation</a>
|
||
|
|
</h5>
|
||
|
|
<p>
|
||
|
|
The implementation is mostly about filtering off various special cases:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When <span class="emphasis"><em>x</em></span> is negative, then the order <span class="emphasis"><em>v</em></span>
|
||
|
|
must be an integer or the result is a domain error. If the order is an integer
|
||
|
|
then the function is odd for odd orders and even for even orders, so we reflect
|
||
|
|
to <span class="emphasis"><em>x > 0</em></span>.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When the order <span class="emphasis"><em>v</em></span> is negative then the reflection formulae
|
||
|
|
can be used to move to <span class="emphasis"><em>v > 0</em></span>:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Note that if the order is an integer, then these formulae reduce to:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
However, in general, a negative order implies that we will need to compute
|
||
|
|
both J and Y.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When <span class="emphasis"><em>x</em></span> is large compared to the order <span class="emphasis"><em>v</em></span>
|
||
|
|
then the asymptotic expansions for large <span class="emphasis"><em>x</em></span> in M. Abramowitz
|
||
|
|
and I.A. Stegun, <span class="emphasis"><em>Handbook of Mathematical Functions</em></span>
|
||
|
|
9.2.19 are used (these were found to be more reliable than those in A&S
|
||
|
|
9.2.5).
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When the order <span class="emphasis"><em>v</em></span> is an integer the method first relates
|
||
|
|
the result to J<sub>0</sub>, J<sub>1</sub>, Y<sub>0</sub>   and Y<sub>1</sub>   using either forwards or backwards recurrence
|
||
|
|
(Miller's algorithm) depending upon which is stable. The values for J<sub>0</sub>, J<sub>1</sub>,
|
||
|
|
Y<sub>0</sub>   and Y<sub>1</sub>   are calculated using the rational minimax approximations on root-bracketing
|
||
|
|
intervals for small <span class="emphasis"><em>|x|</em></span> and Hankel asymptotic expansion
|
||
|
|
for large <span class="emphasis"><em>|x|</em></span>. The coefficients are from:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package
|
||
|
|
of Special Function Routines and Test Drivers</em></span>, ACM Transactions
|
||
|
|
on Mathematical Software, vol 19, 22 (1993).
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
and
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
|
||
|
|
& Sons, New York, 1968.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
These approximations are accurate to around 19 decimal digits: therefore
|
||
|
|
these methods are not used when type T has more than 64 binary digits.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When <span class="emphasis"><em>x</em></span> is smaller than machine epsilon then the following
|
||
|
|
approximations for Y<sub>0</sub>(x), Y<sub>1</sub>(x), Y<sub>2</sub>(x) and Y<sub>n</sub>(x) can be used (see: <a href="http://functions.wolfram.com/03.03.06.0037.01" target="_top">http://functions.wolfram.com/03.03.06.0037.01</a>,
|
||
|
|
<a href="http://functions.wolfram.com/03.03.06.0038.01" target="_top">http://functions.wolfram.com/03.03.06.0038.01</a>,
|
||
|
|
<a href="http://functions.wolfram.com/03.03.06.0039.01" target="_top">http://functions.wolfram.com/03.03.06.0039.01</a>
|
||
|
|
and <a href="http://functions.wolfram.com/03.03.06.0040.01" target="_top">http://functions.wolfram.com/03.03.06.0040.01</a>):
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span> and
|
||
|
|
<span class="emphasis"><em>v</em></span> is not an integer, then the following series approximation
|
||
|
|
can be used for Y<sub>v</sub>(x), this is also an area where other approximations are
|
||
|
|
often too slow to converge to be used (see <a href="http://functions.wolfram.com/03.03.06.0034.01" target="_top">http://functions.wolfram.com/03.03.06.0034.01</a>):
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
|
||
|
|
J<sub>v</sub>x   is best computed directly from the series:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
In the general case we compute J<sub>v</sub>   and Y<sub>v</sub>   simultaneously.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
To get the initial values, 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 J<sub>μ</sub>(x), J<sub>μ+1</sub>(x), Y<sub>μ</sub>(x), Y<sub>μ+1</sub>(x) and use them to obtain J<sub>ν</sub>(x), Y<sub>ν</sub>(x).
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
The algorithm is called Steed's method, which needs two continued fractions
|
||
|
|
as well as the Wronskian:
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
See: F.S. Acton, <span class="emphasis"><em>Numerical Methods that Work</em></span>, The Mathematical
|
||
|
|
Association of America, Washington, 1997.
|
||
|
|
</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>).
|
||
|
|
J<sub>μ</sub>, J<sub>μ+1</sub>, Y<sub>μ</sub>, Y<sub>μ+1</sub> can be calculated by
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
where
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
J<sub>ν</sub> and Y<sub>μ</sub> are then calculated using backward (Miller's algorithm) and forward
|
||
|
|
recurrence respectively.
|
||
|
|
</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/bessel15.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
where
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
<span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
g<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 refered to N.M. Temme,
|
||
|
|
<span class="emphasis"><em>On the numerical evaluation of the ordinary Bessel function of
|
||
|
|
the second kind</em></span>, Journal of Computational Physics, vol 21, 343
|
||
|
|
(1976). Note Temme's series converge only for |μ| <= 1/2.
|
||
|
|
</p>
|
||
|
|
<p>
|
||
|
|
As the previous case, Y<sub>ν</sub>   is calculated from the forward recurrence, so is Y<sub>ν+1</sub>.
|
||
|
|
With these two values and f<sub>ν</sub>, the Wronskian yields J<sub>ν</sub>(x) directly without backward
|
||
|
|
recurrence.
|
||
|
|
</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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|
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