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- <div class="titlepage"><div><div><h3 class="title">
- <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
- the First and Second Kinds</a>
- </h3></div></div></div>
- <h5>
- <a name="math_toolkit.bessel.bessel_first.h0"></a>
- <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>
- </h5>
- <p>
- <code class="computeroutput"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">bessel</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></code>
- </p>
- <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
- <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_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">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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
- <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_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">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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- </pre>
- <h5>
- <a name="math_toolkit.bessel.bessel_first.h1"></a>
- <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>
- </h5>
- <p>
- The functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
- and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> return
- the result of the Bessel functions of the first and second kinds respectively:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic">cyl_bessel_j(v, x) = J<sub>v</sub>(x)</span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic">cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x)</span>
- </p></blockquote></div>
- <p>
- where:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
- </p></blockquote></div>
- <p>
- The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
- type calculation rules</em></span></a> when T1 and T2 are different types.
- The functions are also optimised for the relatively common case that T1 is
- an integer.
- </p>
- <p>
- The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
- be used to control the behaviour of the function: how it handles errors,
- what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
- documentation for more details</a>.
- </p>
- <p>
- The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
- whenever the result is undefined or complex. For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
- this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span>
- <span class="number">0</span></code> and v is not an integer, or when
- <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span>
- <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span>
- <span class="special">!=</span> <span class="number">0</span></code>.
- For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this
- occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special"><=</span>
- <span class="number">0</span></code>.
- </p>
- <p>
- The following graph illustrates the cyclic nature of J<sub>v</sub>:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span>
- </p></blockquote></div>
- <p>
- The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large
- <span class="emphasis"><em>x</em></span>, but tends to -∞ for small <span class="emphasis"><em>x</em></span>:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
- </p></blockquote></div>
- <h5>
- <a name="math_toolkit.bessel.bessel_first.h2"></a>
- <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>
- </h5>
- <p>
- There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
- and a much larger set of tests computed using a simplified version of this
- implementation (with all the special case handling removed).
- </p>
- <h5>
- <a name="math_toolkit.bessel.bessel_first.h3"></a>
- <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>
- </h5>
- <p>
- The following tables show how the accuracy of these functions varies on various
- platforms, along with comparisons to other libraries. Note that the cyclic
- nature of these functions means that they have an infinite number of irrational
- roots: in general these functions have arbitrarily large <span class="emphasis"><em>relative</em></span>
- 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
- have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
- error</a>. All values are relative errors in units of epsilon. Most of
- the gross errors exhibited by other libraries occur for very large arguments
- - you will need to drill down into the actual program output if you need
- more information on this.
- </p>
- <div class="table">
- <a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table 8.40. Error rates for cyl_bessel_j (integer orders)</b></p>
- <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j (integer orders)">
- <colgroup>
- <col>
- <col>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- Bessel J0: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.55ε (Mean = 2.86ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 5.04ε (Mean = 1.78ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j_integer_orders___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
- 2.1:</em></span> Max = 1.12ε (Mean = 0.488ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 0.629ε (Mean = 0.223ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_2_3_Bessel_J0_Mathworld_Data_Integer_Version_">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.55ε (Mean = 2.86ε)</span>
- </p>
- </td>
- <td>
- <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>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J0: Mathworld Data (Tricky cases) (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span><br>
- <br> (<span class="emphasis"><em><cmath>:</em></span> Max = 4.79e+08ε (Mean
- = 1.96e+08ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 8e+04ε (Mean = 3.27e+04ε)</span><br>
- <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1e+07ε (Mean = 4.11e+06ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.04e+07ε (Mean = 4.29e+06ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1e+07ε (Mean = 4.09e+06ε)</span><br>
- <br> (<span class="emphasis"><em><math.h>:</em></span> <span class="red">Max
- = 2.54e+08ε (Mean = 1.04e+08ε))</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J1: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.59ε (Mean = 1.33ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 6.1ε (Mean = 2.95ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j_integer_orders___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
- 2.1:</em></span> Max = 1.89ε (Mean = 0.721ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 0.946ε (Mean = 0.39ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_2_3_Bessel_J1_Mathworld_Data_Integer_Version_">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.44ε (Mean = 0.637ε)</span>
- </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>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J1: Mathworld Data (tricky cases) (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span><br>
- <br> (<span class="emphasis"><em><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 2.1:</em></span> Max = 1.26e+06ε (Mean = 6.28e+05ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.93e+06ε (Mean = 1.7e+06ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.23e+04ε (Mean = 1.45e+04ε)</span><br>
- <br> (<span class="emphasis"><em><math.h>:</em></span> Max = 1.44e+07ε (Mean
- = 6.5e+06ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel JN: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.85ε (Mean = 3.35ε)</span><br> <br>
- (<span class="emphasis"><em><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_7_1_0_linux_long_double_cyl_bessel_j_integer_orders___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
- 2.1:</em></span> Max = 6.9e+05ε (Mean = 2.53e+05ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_integer_orders__GSL_2_1_Bessel_JN_Mathworld_Data_Integer_Version_">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_2_3_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>
- <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_14_1_Win32_double_cyl_bessel_j_integer_orders___math_h__Bessel_JN_Mathworld_Data_Integer_Version_">And
- other failures.</a>)</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 8.41. 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>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- Bessel J0: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.55ε (Mean = 2.86ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 5.04ε (Mean = 1.78ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j__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
- 2.1:</em></span> Max = 0.629ε (Mean = 0.223ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_J0_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 0.629ε (Mean = 0.223ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_Rmath_3_2_3_Bessel_J0_Mathworld_Data">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.55ε (Mean = 2.86ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.52ε (Mean = 1.2ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J0: Mathworld Data (Tricky cases)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span><br>
- <br> (<span class="emphasis"><em><cmath>:</em></span> Max = 4.79e+08ε (Mean
- = 1.96e+08ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 8e+04ε (Mean = 3.27e+04ε)</span><br>
- <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 6.5e+07ε (Mean = 2.66e+07ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.04e+07ε (Mean = 4.29e+06ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.64e+08ε (Mean = 6.69e+07ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1e+07ε (Mean = 4.09e+06ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J1: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.59ε (Mean = 1.33ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 6.1ε (Mean = 2.95ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j__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
- 2.1:</em></span> Max = 6.62ε (Mean = 2.35ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_J1_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 0.946ε (Mean = 0.39ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_Rmath_3_2_3_Bessel_J1_Mathworld_Data">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.44ε (Mean = 0.637ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.73ε (Mean = 0.976ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J1: Mathworld Data (tricky cases)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span><br>
- <br> (<span class="emphasis"><em><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 2.1:</em></span> Max = 8.75e+05ε (Mean = 5.32e+05ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.93e+06ε (Mean = 1.7e+06ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.18e+05ε (Mean = 9.76e+04ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 3.23e+04ε (Mean = 1.45e+04ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel JN: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.85ε (Mean = 3.35ε)</span><br> <br>
- (<span class="emphasis"><em><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_7_1_0_linux_long_double_cyl_bessel_j__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
- 2.1:</em></span> Max = 6.9e+05ε (Mean = 2.15e+05ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_JN_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_Rmath_3_2_3_Bessel_JN_Mathworld_Data">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 463ε (Mean = 112ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 14.7ε (Mean = 5.4ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 14.7ε (Mean = 4.11ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 3.49e+05ε (Mean = 8.09e+04ε)
- <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j__cmath__Bessel_J_Mathworld_Data">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10ε (Mean = 2.24ε)</span><br> <br>
- (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.39e+05ε (Mean = 5.37e+04ε)
- <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_J_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_Rmath_3_2_3_Bessel_J_Mathworld_Data">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 14.7ε (Mean = 4.22ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 14.9ε (Mean = 3.89ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J: Mathworld Data (large values)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 607ε (Mean = 305ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 34.9ε (Mean = 17.4ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_bessel_j__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 2.1:</em></span> Max = 4.91e+03ε (Mean = 2.46e+03ε)
- <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_J_Mathworld_Data_large_values_">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 5.9ε (Mean = 3.76ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 607ε (Mean = 305ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 9.31ε (Mean = 5.52ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel JN: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 50.8ε (Mean = 3.69ε)</span><br> <br>
- (<span class="emphasis"><em><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
- 2.1:</em></span> Max = 75.7ε (Mean = 5.36ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 3.93ε (Mean = 1.22ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 99.6ε (Mean = 22ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 17.5ε (Mean = 1.46ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 11.4ε (Mean = 1.68ε)</span><br> <br>
- (<span class="emphasis"><em><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
- 2.1:</em></span> Max = 15.5ε (Mean = 3.33ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_bessel_j_GSL_2_1_Bessel_J_Random_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 6.74ε (Mean = 1.3ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 260ε (Mean = 34ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 9.24ε (Mean = 1.17ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Bessel J: Random Data (Tricky large values)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 785ε (Mean = 94.2ε)</span><br> <br>
- (<span class="emphasis"><em><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
- 2.1:</em></span> Max = 2.48e+05ε (Mean = 5.11e+04ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 71.6ε (Mean = 11.7ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 785ε (Mean = 97.4ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 59.2ε (Mean = 8.67ε)</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 8.42. 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>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- Y0: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 5.53ε (Mean = 2.4ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 2.05e+05ε (Mean = 6.87e+04ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 6.46ε (Mean = 2.38ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 167ε (Mean = 56.5ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 5.53ε (Mean = 2.4ε)</span>
- </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+03ε (Mean = 1.81e+03ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Y1: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.33ε (Mean = 2.25ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 9.71e+03ε (Mean = 4.08e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 1.51ε (Mean = 0.839ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 193ε (Mean = 64.4ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.33ε (Mean = 2.29ε)</span>
- </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+04ε (Mean = 6.2e+03ε))
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yn: Mathworld Data (Integer Version)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 55.2ε (Mean = 17.8ε)</span><br> <br>
- (<span class="emphasis"><em><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_7_1_0_linux_long_double_cyl_neumann_integer_orders___cmath__Yn_Mathworld_Data_Integer_Version_">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0.993ε (Mean = 0.314ε)</span><br> <br>
- (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.41e+05ε (Mean = 7.62e+04ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.24e+04ε (Mean = 4e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 55.2ε (Mean = 17.8ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 35ε (Mean = 11.9ε)</span><br> <br>
- (<span class="emphasis"><em><math.h>:</em></span> Max = 2.49e+05ε (Mean = 8.14e+04ε))
- </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 8.43. 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>
- GNU C++ version 7.1.0<br> linux<br> long double
- </p>
- </th>
- <th>
- <p>
- GNU C++ version 7.1.0<br> linux<br> double
- </p>
- </th>
- <th>
- <p>
- Sun compiler version 0x5150<br> Sun Solaris<br> long double
- </p>
- </th>
- <th>
- <p>
- Microsoft Visual C++ version 14.1<br> Win32<br> double
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- Y0: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 5.53ε (Mean = 2.4ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 2.05e+05ε (Mean = 6.87e+04ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 60.9ε (Mean = 20.4ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 167ε (Mean = 56.5ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 5.53ε (Mean = 2.4ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 4.61ε (Mean = 2.29ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Y1: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.33ε (Mean = 2.25ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 9.71e+03ε (Mean = 4.08e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 23.4ε (Mean = 8.1ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 193ε (Mean = 64.4ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 6.33ε (Mean = 2.29ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 4.75ε (Mean = 1.72ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yn: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 55.2ε (Mean = 17.8ε)</span><br> <br>
- (<span class="emphasis"><em><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_7_1_0_linux_long_double_cyl_neumann__cmath__Yn_Mathworld_Data">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0.993ε (Mean = 0.314ε)</span><br> <br>
- (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.41e+05ε (Mean = 7.62e+04ε)
- <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_neumann_GSL_2_1_Yn_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 1.24e+04ε (Mean = 4e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 55.2ε (Mean = 17.8ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 35ε (Mean = 11.9ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yv: Mathworld Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10.7ε (Mean = 4.93ε)</span><br> <br>
- (<span class="emphasis"><em><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_7_1_0_linux_long_double_cyl_neumann__cmath__Yv_Mathworld_Data">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10ε (Mean = 3.02ε)</span><br> <br>
- (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.07e+05ε (Mean = 3.22e+04ε)
- <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_neumann_GSL_2_1_Yv_Mathworld_Data">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 243ε (Mean = 73.9ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10.7ε (Mean = 5.1ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 7.89ε (Mean = 3.27ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yv: Mathworld Data (large values)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.7ε (Mean = 1.33ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 43.2ε (Mean = 16.3ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_neumann__cmath__Yv_Mathworld_Data_large_values_">And
- other failures.</a>)
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 60.8ε (Mean = 23ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_cyl_neumann_GSL_2_1_Yv_Mathworld_Data_large_values_">And
- other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
- Max = 0.682ε (Mean = 0.335ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.7ε (Mean = 1.33ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0.682ε (Mean = 0.423ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Y0 and Y1: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10.8ε (Mean = 3.04ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 2.59e+03ε (Mean = 500ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 34.4ε (Mean = 8.9ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 83ε (Mean = 14.2ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 10.8ε (Mean = 3.04ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 4.17ε (Mean = 1.24ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yn: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 338ε (Mean = 27.5ε)</span><br> <br>
- (<span class="emphasis"><em><cmath>:</em></span> Max = 4.01e+03ε (Mean = 348ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
- 2.1:</em></span> Max = 500ε (Mean = 47.8ε))<br> (<span class="emphasis"><em>Rmath
- 3.2.3:</em></span> Max = 691ε (Mean = 67.9ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 338ε (Mean = 27.5ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 117ε (Mean = 10.2ε)</span>
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- Yv: Random Data
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.08e+03ε (Mean = 149ε)</span><br>
- <br> (<span class="emphasis"><em><cmath>:</em></span> <span class="red">Max
- = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_long_double_cyl_neumann__cmath__Yv_Random_Data">And
- other failures.</a>)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.53ε (Mean = 0.102ε)</span><br> <br>
- (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.41e+06ε (Mean = 7.67e+04ε))<br>
- (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.79e+05ε (Mean = 9.64e+03ε))
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 2.08e+03ε (Mean = 149ε)</span>
- </p>
- </td>
- <td>
- <p>
- <span class="blue">Max = 1.23e+03ε (Mean = 69.9ε)</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>
- <p>
- The following error plot are based on an exhaustive search of the functions
- domain for J0 and Y0, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
- precision, other compilers and precisions are very similar - the plots simply
- illustrate the relatively large errors as you approach a zero, and the very
- low errors elsewhere.
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/j0__double.svg" align="middle"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../graphs/y0__double.svg" align="middle"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
- </p></blockquote></div>
- <p>
- Note that if the order is an integer, then these formulae reduce to:
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic">J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="serif_italic">Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><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></blockquote></div>
- <p>
- and
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
- & Sons, New York, 1968.
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
- </p></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><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></blockquote></div>
- <div class="blockquote"><blockquote class="blockquote"><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></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
- </p></blockquote></div>
- <p>
- where
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
- </p></blockquote></div>
- <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>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span>
- </p></blockquote></div>
- <p>
- where
- </p>
- <div class="blockquote"><blockquote class="blockquote"><p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
- </p></blockquote></div>
- <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 © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
- </div></td>
- </tr></table>
- <hr>
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