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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <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
  28       the First and Second Kinds</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.bessel.bessel_first.h0"></a>
  32         <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>
  33       </h5>
  34 <p>
  35         <code class="computeroutput"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span></code>
  36       </p>
  37 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
  38 <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>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</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">&gt;</span>
  44 <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>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48 </pre>
  49 <h5>
  50 <a name="math_toolkit.bessel.bessel_first.h1"></a>
  51         <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>
  52       </h5>
  53 <p>
  54         The functions <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_bessel_j</a>
  55         and <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> return
  56         the result of the Bessel functions of the first and second kinds respectively:
  57       </p>
  58 <div class="blockquote"><blockquote class="blockquote"><p>
  59           <span class="serif_italic">cyl_bessel_j(v, x) = J<sub>v</sub>(x)</span>
  60         </p></blockquote></div>
  61 <div class="blockquote"><blockquote class="blockquote"><p>
  62           <span class="serif_italic">cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x)</span>
  63         </p></blockquote></div>
  64 <p>
  65         where:
  66       </p>
  67 <div class="blockquote"><blockquote class="blockquote"><p>
  68           <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
  69
  70         </p></blockquote></div>
  71 <div class="blockquote"><blockquote class="blockquote"><p>
  72           <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
  73
  74         </p></blockquote></div>
  75 <p>
  76         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
  77         type calculation rules</em></span></a> when T1 and T2 are different types.
  78         The functions are also optimised for the relatively common case that T1 is
  79         an integer.
  80       </p>
  81 <p>
  82         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  83         be used to control the behaviour of the function: how it handles errors,
  84         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  85         documentation for more details</a>.
  86       </p>
  87 <p>
  88         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  89         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>
  90         this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
  91         <span class="number">0</span></code> and v is not an integer, or when
  92         <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span>
  93         <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span>
  94         <span class="special">!=</span> <span class="number">0</span></code>.
  95         For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this
  96         occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;=</span>
  97         <span class="number">0</span></code>.
  98       </p>
  99 <p>
 100         The following graph illustrates the cyclic nature of J<sub>v</sub>:
 101       </p>
 102 <div class="blockquote"><blockquote class="blockquote"><p>
 103           <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span>
 104
 105         </p></blockquote></div>
 106 <p>
 107         The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large
 108         <span class="emphasis"><em>x</em></span>, but tends to -&#8734; for small <span class="emphasis"><em>x</em></span>:
 109       </p>
 110 <div class="blockquote"><blockquote class="blockquote"><p>
 111           <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
 112
 113         </p></blockquote></div>
 114 <h5>
 115 <a name="math_toolkit.bessel.bessel_first.h2"></a>
 116         <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>
 117       </h5>
 118 <p>
 119         There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
 120         and a much larger set of tests computed using a simplified version of this
 121         implementation (with all the special case handling removed).
 122       </p>
 123 <h5>
 124 <a name="math_toolkit.bessel.bessel_first.h3"></a>
 125         <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>
 126       </h5>
 127 <p>
 128         The following tables show how the accuracy of these functions varies on various
 129         platforms, along with comparisons to other libraries. Note that the cyclic
 130         nature of these functions means that they have an infinite number of irrational
 131         roots: in general these functions have arbitrarily large <span class="emphasis"><em>relative</em></span>
 132         errors when the arguments are sufficiently close to a root. Of course the
 133         absolute error in such cases is always small. Note that only results for
 134         the widest floating-point type on the system are given as narrower types
 135         have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
 136         error</a>. All values are relative errors in units of epsilon. Most of
 137         the gross errors exhibited by other libraries occur for very large arguments
 138         - you will need to drill down into the actual program output if you need
 139         more information on this.
 140       </p>
 141 <div class="table">
 142 <a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table&#160;8.40.&#160;Error rates for cyl_bessel_j (integer orders)</b></p>
 143 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j (integer orders)">
 144 <colgroup>
 145 <col>
 146 <col>
 147 <col>
 148 <col>
 149 <col>
 150 </colgroup>
 151 <thead><tr>
 152 <th>
 153               </th>
 154 <th>
 155                 <p>
 156                   GNU C++ version 7.1.0<br> linux<br> long double
 157                 </p>
 158               </th>
 159 <th>
 160                 <p>
 161                   GNU C++ version 7.1.0<br> linux<br> double
 162                 </p>
 163               </th>
 164 <th>
 165                 <p>
 166                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 167                 </p>
 168               </th>
 169 <th>
 170                 <p>
 171                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 172                 </p>
 173               </th>
 174 </tr></thead>
 175 <tbody>
 176 <tr>
 177 <td>
 178                 <p>
 179                   Bessel J0: Mathworld Data (Integer Version)
 180                 </p>
 181               </td>
 182 <td>
 183                 <p>
 184                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span><br> <br>
 185                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5.04&#949; (Mean = 1.78&#949;) <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
 186                   other failures.</a>)
 187                 </p>
 188               </td>
 189 <td>
 190                 <p>
 191                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 192                   2.1:</em></span> Max = 1.12&#949; (Mean = 0.488&#949;))<br> (<span class="emphasis"><em>Rmath
 193                   3.2.3:</em></span> Max = 0.629&#949; (Mean = 0.223&#949;) <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
 194                   other failures.</a>)
 195                 </p>
 196               </td>
 197 <td>
 198                 <p>
 199                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span>
 200                 </p>
 201               </td>
 202 <td>
 203                 <p>
 204                   <span class="blue">Max = 2.52&#949; (Mean = 1.2&#949;)</span><br> <br>
 205                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.89&#949; (Mean = 0.988&#949;))
 206                 </p>
 207               </td>
 208 </tr>
 209 <tr>
 210 <td>
 211                 <p>
 212                   Bessel J0: Mathworld Data (Tricky cases) (Integer Version)
 213                 </p>
 214               </td>
 215 <td>
 216                 <p>
 217                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#949;)</span><br>
 218                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 4.79e+08&#949; (Mean
 219                   = 1.96e+08&#949;))
 220                 </p>
 221               </td>
 222 <td>
 223                 <p>
 224                   <span class="blue">Max = 8e+04&#949; (Mean = 3.27e+04&#949;)</span><br>
 225                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1e+07&#949; (Mean = 4.11e+06&#949;))<br>
 226                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.04e+07&#949; (Mean = 4.29e+06&#949;))
 227                 </p>
 228               </td>
 229 <td>
 230                 <p>
 231                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#949;)</span>
 232                 </p>
 233               </td>
 234 <td>
 235                 <p>
 236                   <span class="blue">Max = 1e+07&#949; (Mean = 4.09e+06&#949;)</span><br>
 237                   <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> <span class="red">Max
 238                   = 2.54e+08&#949; (Mean = 1.04e+08&#949;))</span>
 239                 </p>
 240               </td>
 241 </tr>
 242 <tr>
 243 <td>
 244                 <p>
 245                   Bessel J1: Mathworld Data (Integer Version)
 246                 </p>
 247               </td>
 248 <td>
 249                 <p>
 250                   <span class="blue">Max = 3.59&#949; (Mean = 1.33&#949;)</span><br> <br>
 251                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 6.1&#949; (Mean = 2.95&#949;) <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
 252                   other failures.</a>)
 253                 </p>
 254               </td>
 255 <td>
 256                 <p>
 257                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 258                   2.1:</em></span> Max = 1.89&#949; (Mean = 0.721&#949;))<br> (<span class="emphasis"><em>Rmath
 259                   3.2.3:</em></span> Max = 0.946&#949; (Mean = 0.39&#949;) <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
 260                   other failures.</a>)
 261                 </p>
 262               </td>
 263 <td>
 264                 <p>
 265                   <span class="blue">Max = 1.44&#949; (Mean = 0.637&#949;)</span>
 266                 </p>
 267               </td>
 268 <td>
 269                 <p>
 270                   <span class="blue">Max = 1.73&#949; (Mean = 0.976&#949;)</span><br> <br>
 271                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 11.4&#949; (Mean = 4.15&#949;))
 272                 </p>
 273               </td>
 274 </tr>
 275 <tr>
 276 <td>
 277                 <p>
 278                   Bessel J1: Mathworld Data (tricky cases) (Integer Version)
 279                 </p>
 280               </td>
 281 <td>
 282                 <p>
 283                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span><br>
 284                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.15e+06&#949; (Mean
 285                   = 1.58e+06&#949;))
 286                 </p>
 287               </td>
 288 <td>
 289                 <p>
 290                   <span class="blue">Max = 106&#949; (Mean = 47.5&#949;)</span><br> <br>
 291                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.26e+06&#949; (Mean = 6.28e+05&#949;))<br>
 292                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.93e+06&#949; (Mean = 1.7e+06&#949;))
 293                 </p>
 294               </td>
 295 <td>
 296                 <p>
 297                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span>
 298                 </p>
 299               </td>
 300 <td>
 301                 <p>
 302                   <span class="blue">Max = 3.23e+04&#949; (Mean = 1.45e+04&#949;)</span><br>
 303                   <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.44e+07&#949; (Mean
 304                   = 6.5e+06&#949;))
 305                 </p>
 306               </td>
 307 </tr>
 308 <tr>
 309 <td>
 310                 <p>
 311                   Bessel JN: Mathworld Data (Integer Version)
 312                 </p>
 313               </td>
 314 <td>
 315                 <p>
 316                   <span class="blue">Max = 6.85&#949; (Mean = 3.35&#949;)</span><br> <br>
 317                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.13e+19&#949; (Mean
 318                   = 5.16e+18&#949;) <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
 319                   other failures.</a>)</span>
 320                 </p>
 321               </td>
 322 <td>
 323                 <p>
 324                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 325                   2.1:</em></span> Max = 6.9e+05&#949; (Mean = 2.53e+05&#949;) <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
 326                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 327                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <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
 328                   other failures.</a>)</span>
 329                 </p>
 330               </td>
 331 <td>
 332                 <p>
 333                   <span class="blue">Max = 463&#949; (Mean = 112&#949;)</span>
 334                 </p>
 335               </td>
 336 <td>
 337                 <p>
 338                   <span class="blue">Max = 14.7&#949; (Mean = 5.4&#949;)</span><br> <br>
 339                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> <span class="red">Max =
 340                   +INF&#949; (Mean = +INF&#949;) <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
 341                   other failures.</a>)</span>
 342                 </p>
 343               </td>
 344 </tr>
 345 </tbody>
 346 </table></div>
 347 </div>
 348 <br class="table-break"><div class="table">
 349 <a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j"></a><p class="title"><b>Table&#160;8.41.&#160;Error rates for cyl_bessel_j</b></p>
 350 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j">
 351 <colgroup>
 352 <col>
 353 <col>
 354 <col>
 355 <col>
 356 <col>
 357 </colgroup>
 358 <thead><tr>
 359 <th>
 360               </th>
 361 <th>
 362                 <p>
 363                   GNU C++ version 7.1.0<br> linux<br> long double
 364                 </p>
 365               </th>
 366 <th>
 367                 <p>
 368                   GNU C++ version 7.1.0<br> linux<br> double
 369                 </p>
 370               </th>
 371 <th>
 372                 <p>
 373                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 374                 </p>
 375               </th>
 376 <th>
 377                 <p>
 378                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 379                 </p>
 380               </th>
 381 </tr></thead>
 382 <tbody>
 383 <tr>
 384 <td>
 385                 <p>
 386                   Bessel J0: Mathworld Data
 387                 </p>
 388               </td>
 389 <td>
 390                 <p>
 391                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span><br> <br>
 392                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 5.04&#949; (Mean = 1.78&#949;) <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
 393                   other failures.</a>)
 394                 </p>
 395               </td>
 396 <td>
 397                 <p>
 398                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 399                   2.1:</em></span> Max = 0.629&#949; (Mean = 0.223&#949;) <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
 400                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 401                   Max = 0.629&#949; (Mean = 0.223&#949;) <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
 402                   other failures.</a>)
 403                 </p>
 404               </td>
 405 <td>
 406                 <p>
 407                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span>
 408                 </p>
 409               </td>
 410 <td>
 411                 <p>
 412                   <span class="blue">Max = 2.52&#949; (Mean = 1.2&#949;)</span>
 413                 </p>
 414               </td>
 415 </tr>
 416 <tr>
 417 <td>
 418                 <p>
 419                   Bessel J0: Mathworld Data (Tricky cases)
 420                 </p>
 421               </td>
 422 <td>
 423                 <p>
 424                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#949;)</span><br>
 425                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 4.79e+08&#949; (Mean
 426                   = 1.96e+08&#949;))
 427                 </p>
 428               </td>
 429 <td>
 430                 <p>
 431                   <span class="blue">Max = 8e+04&#949; (Mean = 3.27e+04&#949;)</span><br>
 432                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 6.5e+07&#949; (Mean = 2.66e+07&#949;))<br>
 433                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.04e+07&#949; (Mean = 4.29e+06&#949;))
 434                 </p>
 435               </td>
 436 <td>
 437                 <p>
 438                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#949;)</span>
 439                 </p>
 440               </td>
 441 <td>
 442                 <p>
 443                   <span class="blue">Max = 1e+07&#949; (Mean = 4.09e+06&#949;)</span>
 444                 </p>
 445               </td>
 446 </tr>
 447 <tr>
 448 <td>
 449                 <p>
 450                   Bessel J1: Mathworld Data
 451                 </p>
 452               </td>
 453 <td>
 454                 <p>
 455                   <span class="blue">Max = 3.59&#949; (Mean = 1.33&#949;)</span><br> <br>
 456                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 6.1&#949; (Mean = 2.95&#949;) <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
 457                   other failures.</a>)
 458                 </p>
 459               </td>
 460 <td>
 461                 <p>
 462                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 463                   2.1:</em></span> Max = 6.62&#949; (Mean = 2.35&#949;) <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
 464                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 465                   Max = 0.946&#949; (Mean = 0.39&#949;) <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
 466                   other failures.</a>)
 467                 </p>
 468               </td>
 469 <td>
 470                 <p>
 471                   <span class="blue">Max = 1.44&#949; (Mean = 0.637&#949;)</span>
 472                 </p>
 473               </td>
 474 <td>
 475                 <p>
 476                   <span class="blue">Max = 1.73&#949; (Mean = 0.976&#949;)</span>
 477                 </p>
 478               </td>
 479 </tr>
 480 <tr>
 481 <td>
 482                 <p>
 483                   Bessel J1: Mathworld Data (tricky cases)
 484                 </p>
 485               </td>
 486 <td>
 487                 <p>
 488                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span><br>
 489                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.15e+06&#949; (Mean
 490                   = 1.58e+06&#949;))
 491                 </p>
 492               </td>
 493 <td>
 494                 <p>
 495                   <span class="blue">Max = 106&#949; (Mean = 47.5&#949;)</span><br> <br>
 496                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 8.75e+05&#949; (Mean = 5.32e+05&#949;))<br>
 497                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.93e+06&#949; (Mean = 1.7e+06&#949;))
 498                 </p>
 499               </td>
 500 <td>
 501                 <p>
 502                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span>
 503                 </p>
 504               </td>
 505 <td>
 506                 <p>
 507                   <span class="blue">Max = 3.23e+04&#949; (Mean = 1.45e+04&#949;)</span>
 508                 </p>
 509               </td>
 510 </tr>
 511 <tr>
 512 <td>
 513                 <p>
 514                   Bessel JN: Mathworld Data
 515                 </p>
 516               </td>
 517 <td>
 518                 <p>
 519                   <span class="blue">Max = 6.85&#949; (Mean = 3.35&#949;)</span><br> <br>
 520                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.13e+19&#949; (Mean
 521                   = 5.16e+18&#949;) <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
 522                   other failures.</a>)</span>
 523                 </p>
 524               </td>
 525 <td>
 526                 <p>
 527                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 528                   2.1:</em></span> Max = 6.9e+05&#949; (Mean = 2.15e+05&#949;) <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
 529                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 530                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <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
 531                   other failures.</a>)</span>
 532                 </p>
 533               </td>
 534 <td>
 535                 <p>
 536                   <span class="blue">Max = 463&#949; (Mean = 112&#949;)</span>
 537                 </p>
 538               </td>
 539 <td>
 540                 <p>
 541                   <span class="blue">Max = 14.7&#949; (Mean = 5.4&#949;)</span>
 542                 </p>
 543               </td>
 544 </tr>
 545 <tr>
 546 <td>
 547                 <p>
 548                   Bessel J: Mathworld Data
 549                 </p>
 550               </td>
 551 <td>
 552                 <p>
 553                   <span class="blue">Max = 14.7&#949; (Mean = 4.11&#949;)</span><br> <br>
 554                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 3.49e+05&#949; (Mean = 8.09e+04&#949;)
 555                   <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
 556                   other failures.</a>)
 557                 </p>
 558               </td>
 559 <td>
 560                 <p>
 561                   <span class="blue">Max = 10&#949; (Mean = 2.24&#949;)</span><br> <br>
 562                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.39e+05&#949; (Mean = 5.37e+04&#949;)
 563                   <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
 564                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 565                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <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
 566                   other failures.</a>)</span>
 567                 </p>
 568               </td>
 569 <td>
 570                 <p>
 571                   <span class="blue">Max = 14.7&#949; (Mean = 4.22&#949;)</span>
 572                 </p>
 573               </td>
 574 <td>
 575                 <p>
 576                   <span class="blue">Max = 14.9&#949; (Mean = 3.89&#949;)</span>
 577                 </p>
 578               </td>
 579 </tr>
 580 <tr>
 581 <td>
 582                 <p>
 583                   Bessel J: Mathworld Data (large values)
 584                 </p>
 585               </td>
 586 <td>
 587                 <p>
 588                   <span class="blue">Max = 607&#949; (Mean = 305&#949;)</span><br> <br>
 589                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 34.9&#949; (Mean = 17.4&#949;) <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
 590                   other failures.</a>)
 591                 </p>
 592               </td>
 593 <td>
 594                 <p>
 595                   <span class="blue">Max = 0.536&#949; (Mean = 0.268&#949;)</span><br> <br>
 596                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 4.91e+03&#949; (Mean = 2.46e+03&#949;)
 597                   <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
 598                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 599                   Max = 5.9&#949; (Mean = 3.76&#949;))
 600                 </p>
 601               </td>
 602 <td>
 603                 <p>
 604                   <span class="blue">Max = 607&#949; (Mean = 305&#949;)</span>
 605                 </p>
 606               </td>
 607 <td>
 608                 <p>
 609                   <span class="blue">Max = 9.31&#949; (Mean = 5.52&#949;)</span>
 610                 </p>
 611               </td>
 612 </tr>
 613 <tr>
 614 <td>
 615                 <p>
 616                   Bessel JN: Random Data
 617                 </p>
 618               </td>
 619 <td>
 620                 <p>
 621                   <span class="blue">Max = 50.8&#949; (Mean = 3.69&#949;)</span><br> <br>
 622                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.12e+03&#949; (Mean = 88.7&#949;))
 623                 </p>
 624               </td>
 625 <td>
 626                 <p>
 627                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 628                   2.1:</em></span> Max = 75.7&#949; (Mean = 5.36&#949;))<br> (<span class="emphasis"><em>Rmath
 629                   3.2.3:</em></span> Max = 3.93&#949; (Mean = 1.22&#949;))
 630                 </p>
 631               </td>
 632 <td>
 633                 <p>
 634                   <span class="blue">Max = 99.6&#949; (Mean = 22&#949;)</span>
 635                 </p>
 636               </td>
 637 <td>
 638                 <p>
 639                   <span class="blue">Max = 17.5&#949; (Mean = 1.46&#949;)</span>
 640                 </p>
 641               </td>
 642 </tr>
 643 <tr>
 644 <td>
 645                 <p>
 646                   Bessel J: Random Data
 647                 </p>
 648               </td>
 649 <td>
 650                 <p>
 651                   <span class="blue">Max = 11.4&#949; (Mean = 1.68&#949;)</span><br> <br>
 652                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 501&#949; (Mean = 52.3&#949;))
 653                 </p>
 654               </td>
 655 <td>
 656                 <p>
 657                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 658                   2.1:</em></span> Max = 15.5&#949; (Mean = 3.33&#949;) <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
 659                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 660                   Max = 6.74&#949; (Mean = 1.3&#949;))
 661                 </p>
 662               </td>
 663 <td>
 664                 <p>
 665                   <span class="blue">Max = 260&#949; (Mean = 34&#949;)</span>
 666                 </p>
 667               </td>
 668 <td>
 669                 <p>
 670                   <span class="blue">Max = 9.24&#949; (Mean = 1.17&#949;)</span>
 671                 </p>
 672               </td>
 673 </tr>
 674 <tr>
 675 <td>
 676                 <p>
 677                   Bessel J: Random Data (Tricky large values)
 678                 </p>
 679               </td>
 680 <td>
 681                 <p>
 682                   <span class="blue">Max = 785&#949; (Mean = 94.2&#949;)</span><br> <br>
 683                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 5.01e+17&#949; (Mean
 684                   = 6.23e+16&#949;))</span>
 685                 </p>
 686               </td>
 687 <td>
 688                 <p>
 689                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 690                   2.1:</em></span> Max = 2.48e+05&#949; (Mean = 5.11e+04&#949;))<br> (<span class="emphasis"><em>Rmath
 691                   3.2.3:</em></span> Max = 71.6&#949; (Mean = 11.7&#949;))
 692                 </p>
 693               </td>
 694 <td>
 695                 <p>
 696                   <span class="blue">Max = 785&#949; (Mean = 97.4&#949;)</span>
 697                 </p>
 698               </td>
 699 <td>
 700                 <p>
 701                   <span class="blue">Max = 59.2&#949; (Mean = 8.67&#949;)</span>
 702                 </p>
 703               </td>
 704 </tr>
 705 </tbody>
 706 </table></div>
 707 </div>
 708 <br class="table-break"><div class="table">
 709 <a name="math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_"></a><p class="title"><b>Table&#160;8.42.&#160;Error rates for cyl_neumann (integer orders)</b></p>
 710 <div class="table-contents"><table class="table" summary="Error rates for cyl_neumann (integer orders)">
 711 <colgroup>
 712 <col>
 713 <col>
 714 <col>
 715 <col>
 716 <col>
 717 </colgroup>
 718 <thead><tr>
 719 <th>
 720               </th>
 721 <th>
 722                 <p>
 723                   GNU C++ version 7.1.0<br> linux<br> long double
 724                 </p>
 725               </th>
 726 <th>
 727                 <p>
 728                   GNU C++ version 7.1.0<br> linux<br> double
 729                 </p>
 730               </th>
 731 <th>
 732                 <p>
 733                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 734                 </p>
 735               </th>
 736 <th>
 737                 <p>
 738                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 739                 </p>
 740               </th>
 741 </tr></thead>
 742 <tbody>
 743 <tr>
 744 <td>
 745                 <p>
 746                   Y0: Mathworld Data (Integer Version)
 747                 </p>
 748               </td>
 749 <td>
 750                 <p>
 751                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span><br> <br>
 752                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.05e+05&#949; (Mean = 6.87e+04&#949;))
 753                 </p>
 754               </td>
 755 <td>
 756                 <p>
 757                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 758                   2.1:</em></span> Max = 6.46&#949; (Mean = 2.38&#949;))<br> (<span class="emphasis"><em>Rmath
 759                   3.2.3:</em></span> Max = 167&#949; (Mean = 56.5&#949;))
 760                 </p>
 761               </td>
 762 <td>
 763                 <p>
 764                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span>
 765                 </p>
 766               </td>
 767 <td>
 768                 <p>
 769                   <span class="blue">Max = 4.61&#949; (Mean = 2.29&#949;)</span><br> <br>
 770                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 5.37e+03&#949; (Mean = 1.81e+03&#949;))
 771                 </p>
 772               </td>
 773 </tr>
 774 <tr>
 775 <td>
 776                 <p>
 777                   Y1: Mathworld Data (Integer Version)
 778                 </p>
 779               </td>
 780 <td>
 781                 <p>
 782                   <span class="blue">Max = 6.33&#949; (Mean = 2.25&#949;)</span><br> <br>
 783                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.71e+03&#949; (Mean = 4.08e+03&#949;))
 784                 </p>
 785               </td>
 786 <td>
 787                 <p>
 788                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 789                   2.1:</em></span> Max = 1.51&#949; (Mean = 0.839&#949;))<br> (<span class="emphasis"><em>Rmath
 790                   3.2.3:</em></span> Max = 193&#949; (Mean = 64.4&#949;))
 791                 </p>
 792               </td>
 793 <td>
 794                 <p>
 795                   <span class="blue">Max = 6.33&#949; (Mean = 2.29&#949;)</span>
 796                 </p>
 797               </td>
 798 <td>
 799                 <p>
 800                   <span class="blue">Max = 4.75&#949; (Mean = 1.72&#949;)</span><br> <br>
 801                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.86e+04&#949; (Mean = 6.2e+03&#949;))
 802                 </p>
 803               </td>
 804 </tr>
 805 <tr>
 806 <td>
 807                 <p>
 808                   Yn: Mathworld Data (Integer Version)
 809                 </p>
 810               </td>
 811 <td>
 812                 <p>
 813                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span><br> <br>
 814                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.2e+20&#949; (Mean
 815                   = 6.97e+19&#949;) <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
 816                   other failures.</a>)</span>
 817                 </p>
 818               </td>
 819 <td>
 820                 <p>
 821                   <span class="blue">Max = 0.993&#949; (Mean = 0.314&#949;)</span><br> <br>
 822                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.41e+05&#949; (Mean = 7.62e+04&#949;))<br>
 823                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.24e+04&#949; (Mean = 4e+03&#949;))
 824                 </p>
 825               </td>
 826 <td>
 827                 <p>
 828                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span>
 829                 </p>
 830               </td>
 831 <td>
 832                 <p>
 833                   <span class="blue">Max = 35&#949; (Mean = 11.9&#949;)</span><br> <br>
 834                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.49e+05&#949; (Mean = 8.14e+04&#949;))
 835                 </p>
 836               </td>
 837 </tr>
 838 </tbody>
 839 </table></div>
 840 </div>
 841 <br class="table-break"><div class="table">
 842 <a name="math_toolkit.bessel.bessel_first.table_cyl_neumann"></a><p class="title"><b>Table&#160;8.43.&#160;Error rates for cyl_neumann</b></p>
 843 <div class="table-contents"><table class="table" summary="Error rates for cyl_neumann">
 844 <colgroup>
 845 <col>
 846 <col>
 847 <col>
 848 <col>
 849 <col>
 850 </colgroup>
 851 <thead><tr>
 852 <th>
 853               </th>
 854 <th>
 855                 <p>
 856                   GNU C++ version 7.1.0<br> linux<br> long double
 857                 </p>
 858               </th>
 859 <th>
 860                 <p>
 861                   GNU C++ version 7.1.0<br> linux<br> double
 862                 </p>
 863               </th>
 864 <th>
 865                 <p>
 866                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 867                 </p>
 868               </th>
 869 <th>
 870                 <p>
 871                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 872                 </p>
 873               </th>
 874 </tr></thead>
 875 <tbody>
 876 <tr>
 877 <td>
 878                 <p>
 879                   Y0: Mathworld Data
 880                 </p>
 881               </td>
 882 <td>
 883                 <p>
 884                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span><br> <br>
 885                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.05e+05&#949; (Mean = 6.87e+04&#949;))
 886                 </p>
 887               </td>
 888 <td>
 889                 <p>
 890                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 891                   2.1:</em></span> Max = 60.9&#949; (Mean = 20.4&#949;))<br> (<span class="emphasis"><em>Rmath
 892                   3.2.3:</em></span> Max = 167&#949; (Mean = 56.5&#949;))
 893                 </p>
 894               </td>
 895 <td>
 896                 <p>
 897                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span>
 898                 </p>
 899               </td>
 900 <td>
 901                 <p>
 902                   <span class="blue">Max = 4.61&#949; (Mean = 2.29&#949;)</span>
 903                 </p>
 904               </td>
 905 </tr>
 906 <tr>
 907 <td>
 908                 <p>
 909                   Y1: Mathworld Data
 910                 </p>
 911               </td>
 912 <td>
 913                 <p>
 914                   <span class="blue">Max = 6.33&#949; (Mean = 2.25&#949;)</span><br> <br>
 915                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 9.71e+03&#949; (Mean = 4.08e+03&#949;))
 916                 </p>
 917               </td>
 918 <td>
 919                 <p>
 920                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 921                   2.1:</em></span> Max = 23.4&#949; (Mean = 8.1&#949;))<br> (<span class="emphasis"><em>Rmath
 922                   3.2.3:</em></span> Max = 193&#949; (Mean = 64.4&#949;))
 923                 </p>
 924               </td>
 925 <td>
 926                 <p>
 927                   <span class="blue">Max = 6.33&#949; (Mean = 2.29&#949;)</span>
 928                 </p>
 929               </td>
 930 <td>
 931                 <p>
 932                   <span class="blue">Max = 4.75&#949; (Mean = 1.72&#949;)</span>
 933                 </p>
 934               </td>
 935 </tr>
 936 <tr>
 937 <td>
 938                 <p>
 939                   Yn: Mathworld Data
 940                 </p>
 941               </td>
 942 <td>
 943                 <p>
 944                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span><br> <br>
 945                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 2.2e+20&#949; (Mean
 946                   = 6.97e+19&#949;) <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
 947                   other failures.</a>)</span>
 948                 </p>
 949               </td>
 950 <td>
 951                 <p>
 952                   <span class="blue">Max = 0.993&#949; (Mean = 0.314&#949;)</span><br> <br>
 953                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 2.41e+05&#949; (Mean = 7.62e+04&#949;)
 954                   <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
 955                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 956                   Max = 1.24e+04&#949; (Mean = 4e+03&#949;))
 957                 </p>
 958               </td>
 959 <td>
 960                 <p>
 961                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span>
 962                 </p>
 963               </td>
 964 <td>
 965                 <p>
 966                   <span class="blue">Max = 35&#949; (Mean = 11.9&#949;)</span>
 967                 </p>
 968               </td>
 969 </tr>
 970 <tr>
 971 <td>
 972                 <p>
 973                   Yv: Mathworld Data
 974                 </p>
 975               </td>
 976 <td>
 977                 <p>
 978                   <span class="blue">Max = 10.7&#949; (Mean = 4.93&#949;)</span><br> <br>
 979                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max = 3.49e+15&#949; (Mean
 980                   = 1.05e+15&#949;) <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
 981                   other failures.</a>)</span>
 982                 </p>
 983               </td>
 984 <td>
 985                 <p>
 986                   <span class="blue">Max = 10&#949; (Mean = 3.02&#949;)</span><br> <br>
 987                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.07e+05&#949; (Mean = 3.22e+04&#949;)
 988                   <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
 989                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
 990                   Max = 243&#949; (Mean = 73.9&#949;))
 991                 </p>
 992               </td>
 993 <td>
 994                 <p>
 995                   <span class="blue">Max = 10.7&#949; (Mean = 5.1&#949;)</span>
 996                 </p>
 997               </td>
 998 <td>
 999                 <p>
1000                   <span class="blue">Max = 7.89&#949; (Mean = 3.27&#949;)</span>
1001                 </p>
1002               </td>
1003 </tr>
1004 <tr>
1005 <td>
1006                 <p>
1007                   Yv: Mathworld Data (large values)
1008                 </p>
1009               </td>
1010 <td>
1011                 <p>
1012                   <span class="blue">Max = 1.7&#949; (Mean = 1.33&#949;)</span><br> <br>
1013                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 43.2&#949; (Mean = 16.3&#949;) <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
1014                   other failures.</a>)
1015                 </p>
1016               </td>
1017 <td>
1018                 <p>
1019                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1020                   2.1:</em></span> Max = 60.8&#949; (Mean = 23&#949;) <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
1021                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
1022                   Max = 0.682&#949; (Mean = 0.335&#949;))
1023                 </p>
1024               </td>
1025 <td>
1026                 <p>
1027                   <span class="blue">Max = 1.7&#949; (Mean = 1.33&#949;)</span>
1028                 </p>
1029               </td>
1030 <td>
1031                 <p>
1032                   <span class="blue">Max = 0.682&#949; (Mean = 0.423&#949;)</span>
1033                 </p>
1034               </td>
1035 </tr>
1036 <tr>
1037 <td>
1038                 <p>
1039                   Y0 and Y1: Random Data
1040                 </p>
1041               </td>
1042 <td>
1043                 <p>
1044                   <span class="blue">Max = 10.8&#949; (Mean = 3.04&#949;)</span><br> <br>
1045                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.59e+03&#949; (Mean = 500&#949;))
1046                 </p>
1047               </td>
1048 <td>
1049                 <p>
1050                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1051                   2.1:</em></span> Max = 34.4&#949; (Mean = 8.9&#949;))<br> (<span class="emphasis"><em>Rmath
1052                   3.2.3:</em></span> Max = 83&#949; (Mean = 14.2&#949;))
1053                 </p>
1054               </td>
1055 <td>
1056                 <p>
1057                   <span class="blue">Max = 10.8&#949; (Mean = 3.04&#949;)</span>
1058                 </p>
1059               </td>
1060 <td>
1061                 <p>
1062                   <span class="blue">Max = 4.17&#949; (Mean = 1.24&#949;)</span>
1063                 </p>
1064               </td>
1065 </tr>
1066 <tr>
1067 <td>
1068                 <p>
1069                   Yn: Random Data
1070                 </p>
1071               </td>
1072 <td>
1073                 <p>
1074                   <span class="blue">Max = 338&#949; (Mean = 27.5&#949;)</span><br> <br>
1075                   (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 4.01e+03&#949; (Mean = 348&#949;))
1076                 </p>
1077               </td>
1078 <td>
1079                 <p>
1080                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1081                   2.1:</em></span> Max = 500&#949; (Mean = 47.8&#949;))<br> (<span class="emphasis"><em>Rmath
1082                   3.2.3:</em></span> Max = 691&#949; (Mean = 67.9&#949;))
1083                 </p>
1084               </td>
1085 <td>
1086                 <p>
1087                   <span class="blue">Max = 338&#949; (Mean = 27.5&#949;)</span>
1088                 </p>
1089               </td>
1090 <td>
1091                 <p>
1092                   <span class="blue">Max = 117&#949; (Mean = 10.2&#949;)</span>
1093                 </p>
1094               </td>
1095 </tr>
1096 <tr>
1097 <td>
1098                 <p>
1099                   Yv: Random Data
1100                 </p>
1101               </td>
1102 <td>
1103                 <p>
1104                   <span class="blue">Max = 2.08e+03&#949; (Mean = 149&#949;)</span><br>
1105                   <br> (<span class="emphasis"><em>&lt;cmath&gt;:</em></span> <span class="red">Max
1106                   = +INF&#949; (Mean = +INF&#949;) <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
1107                   other failures.</a>)</span>
1108                 </p>
1109               </td>
1110 <td>
1111                 <p>
1112                   <span class="blue">Max = 1.53&#949; (Mean = 0.102&#949;)</span><br> <br>
1113                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.41e+06&#949; (Mean = 7.67e+04&#949;))<br>
1114                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.79e+05&#949; (Mean = 9.64e+03&#949;))
1115                 </p>
1116               </td>
1117 <td>
1118                 <p>
1119                   <span class="blue">Max = 2.08e+03&#949; (Mean = 149&#949;)</span>
1120                 </p>
1121               </td>
1122 <td>
1123                 <p>
1124                   <span class="blue">Max = 1.23e+03&#949; (Mean = 69.9&#949;)</span>
1125                 </p>
1126               </td>
1127 </tr>
1128 </tbody>
1129 </table></div>
1130 </div>
1131 <br class="table-break"><p>
1132         Note that for large <span class="emphasis"><em>x</em></span> these functions are largely dependent
1133         on the accuracy of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span></code> and
1134         <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span></code> functions.
1135       </p>
1136 <p>
1137         Comparison to GSL and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
1138         is interesting: both <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
1139         and this library optimise the integer order case - leading to identical results
1140         - simply using the general case is for the most part slightly more accurate
1141         though, as noted by the better accuracy of GSL in the integer argument cases.
1142         This implementation tends to perform much better when the arguments become
1143         large, <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> in particular
1144         produces some remarkably inaccurate results with some of the test data (no
1145         significant figures correct), and even GSL performs badly with some inputs
1146         to J<sub>v</sub>. Note that by way of double-checking these results, the worst performing
1147         <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> and GSL cases were
1148         recomputed using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
1149         and the result checked against our test data: no errors in the test data
1150         were found.
1151       </p>
1152 <p>
1153         The following error plot are based on an exhaustive search of the functions
1154         domain for J0 and Y0, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
1155         precision, other compilers and precisions are very similar - the plots simply
1156         illustrate the relatively large errors as you approach a zero, and the very
1157         low errors elsewhere.
1158       </p>
1159 <div class="blockquote"><blockquote class="blockquote"><p>
1160           <span class="inlinemediaobject"><img src="../../../graphs/j0__double.svg" align="middle"></span>
1161
1162         </p></blockquote></div>
1163 <div class="blockquote"><blockquote class="blockquote"><p>
1164           <span class="inlinemediaobject"><img src="../../../graphs/y0__double.svg" align="middle"></span>
1165
1166         </p></blockquote></div>
1167 <h5>
1168 <a name="math_toolkit.bessel.bessel_first.h4"></a>
1169         <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>
1170       </h5>
1171 <p>
1172         The implementation is mostly about filtering off various special cases:
1173       </p>
1174 <p>
1175         When <span class="emphasis"><em>x</em></span> is negative, then the order <span class="emphasis"><em>v</em></span>
1176         must be an integer or the result is a domain error. If the order is an integer
1177         then the function is odd for odd orders and even for even orders, so we reflect
1178         to <span class="emphasis"><em>x &gt; 0</em></span>.
1179       </p>
1180 <p>
1181         When the order <span class="emphasis"><em>v</em></span> is negative then the reflection formulae
1182         can be used to move to <span class="emphasis"><em>v &gt; 0</em></span>:
1183       </p>
1184 <div class="blockquote"><blockquote class="blockquote"><p>
1185           <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
1186
1187         </p></blockquote></div>
1188 <div class="blockquote"><blockquote class="blockquote"><p>
1189           <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
1190
1191         </p></blockquote></div>
1192 <p>
1193         Note that if the order is an integer, then these formulae reduce to:
1194       </p>
1195 <div class="blockquote"><blockquote class="blockquote"><p>
1196           <span class="serif_italic">J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub></span>
1197         </p></blockquote></div>
1198 <div class="blockquote"><blockquote class="blockquote"><p>
1199           <span class="serif_italic">Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub></span>
1200         </p></blockquote></div>
1201 <p>
1202         However, in general, a negative order implies that we will need to compute
1203         both J and Y.
1204       </p>
1205 <p>
1206         When <span class="emphasis"><em>x</em></span> is large compared to the order <span class="emphasis"><em>v</em></span>
1207         then the asymptotic expansions for large <span class="emphasis"><em>x</em></span> in M. Abramowitz
1208         and I.A. Stegun, <span class="emphasis"><em>Handbook of Mathematical Functions</em></span>
1209         9.2.19 are used (these were found to be more reliable than those in A&amp;S
1210         9.2.5).
1211       </p>
1212 <p>
1213         When the order <span class="emphasis"><em>v</em></span> is an integer the method first relates
1214         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
1215         (Miller's algorithm) depending upon which is stable. The values for J<sub>0</sub>, J<sub>1</sub>,
1216         Y<sub>0</sub> and Y<sub>1</sub> are calculated using the rational minimax approximations on root-bracketing
1217         intervals for small <span class="emphasis"><em>|x|</em></span> and Hankel asymptotic expansion
1218         for large <span class="emphasis"><em>|x|</em></span>. The coefficients are from:
1219       </p>
1220 <div class="blockquote"><blockquote class="blockquote"><p>
1221           W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package
1222           of Special Function Routines and Test Drivers</em></span>, ACM Transactions
1223           on Mathematical Software, vol 19, 22 (1993).
1224         </p></blockquote></div>
1225 <p>
1226         and
1227       </p>
1228 <div class="blockquote"><blockquote class="blockquote"><p>
1229           J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
1230           &amp; Sons, New York, 1968.
1231         </p></blockquote></div>
1232 <p>
1233         These approximations are accurate to around 19 decimal digits: therefore
1234         these methods are not used when type T has more than 64 binary digits.
1235       </p>
1236 <p>
1237         When <span class="emphasis"><em>x</em></span> is smaller than machine epsilon then the following
1238         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>,
1239         <a href="http://functions.wolfram.com/03.03.06.0038.01" target="_top">http://functions.wolfram.com/03.03.06.0038.01</a>,
1240         <a href="http://functions.wolfram.com/03.03.06.0039.01" target="_top">http://functions.wolfram.com/03.03.06.0039.01</a>
1241         and <a href="http://functions.wolfram.com/03.03.06.0040.01" target="_top">http://functions.wolfram.com/03.03.06.0040.01</a>):
1242       </p>
1243 <div class="blockquote"><blockquote class="blockquote"><p>
1244           <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
1245
1246         </p></blockquote></div>
1247 <div class="blockquote"><blockquote class="blockquote"><p>
1248           <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
1249
1250         </p></blockquote></div>
1251 <div class="blockquote"><blockquote class="blockquote"><p>
1252           <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
1253
1254         </p></blockquote></div>
1255 <div class="blockquote"><blockquote class="blockquote"><p>
1256           <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
1257
1258         </p></blockquote></div>
1259 <p>
1260         When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span> and
1261         <span class="emphasis"><em>v</em></span> is not an integer, then the following series approximation
1262         can be used for Y<sub>v</sub>(x), this is also an area where other approximations are
1263         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>):
1264       </p>
1265 <div class="blockquote"><blockquote class="blockquote"><p>
1266           <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
1267
1268         </p></blockquote></div>
1269 <p>
1270         When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
1271         J<sub>v</sub>x is best computed directly from the series:
1272       </p>
1273 <div class="blockquote"><blockquote class="blockquote"><p>
1274           <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
1275
1276         </p></blockquote></div>
1277 <p>
1278         In the general case we compute J<sub>v</sub> and Y<sub>v</sub> simultaneously.
1279       </p>
1280 <p>
1281         To get the initial values, let &#956; = &#957; - floor(&#957; + 1/2), then &#956; is the fractional part
1282         of &#957; such that |&#956;| &lt;= 1/2 (we need this for convergence later). The idea
1283         is to calculate J<sub>&#956;</sub>(x), J<sub>&#956;+1</sub>(x), Y<sub>&#956;</sub>(x), Y<sub>&#956;+1</sub>(x) and use them to obtain J<sub>&#957;</sub>(x), Y<sub>&#957;</sub>(x).
1284       </p>
1285 <p>
1286         The algorithm is called Steed's method, which needs two continued fractions
1287         as well as the Wronskian:
1288       </p>
1289 <div class="blockquote"><blockquote class="blockquote"><p>
1290           <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
1291
1292         </p></blockquote></div>
1293 <div class="blockquote"><blockquote class="blockquote"><p>
1294           <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
1295
1296         </p></blockquote></div>
1297 <div class="blockquote"><blockquote class="blockquote"><p>
1298           <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
1299
1300         </p></blockquote></div>
1301 <p>
1302         See: F.S. Acton, <span class="emphasis"><em>Numerical Methods that Work</em></span>, The Mathematical
1303         Association of America, Washington, 1997.
1304       </p>
1305 <p>
1306         The continued fractions are computed using the modified Lentz's method (W.J.
1307         Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
1308         using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
1309         Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need
1310         different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>:
1311       </p>
1312 <div class="blockquote"><blockquote class="blockquote"><p>
1313           <span class="emphasis"><em>x &gt; v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations
1314           to converge, CF2 converges rapidly
1315         </p></blockquote></div>
1316 <div class="blockquote"><blockquote class="blockquote"><p>
1317           <span class="emphasis"><em>x &lt;= v</em></span>, CF1 converges rapidly, CF2 fails to converge
1318           when <span class="emphasis"><em>x</em></span> <code class="literal">-&gt;</code> 0
1319         </p></blockquote></div>
1320 <p>
1321         When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> &gt; 2), both
1322         continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
1323         J<sub>&#956;</sub>, J<sub>&#956;+1</sub>, Y<sub>&#956;</sub>, Y<sub>&#956;+1</sub> can be calculated by
1324       </p>
1325 <div class="blockquote"><blockquote class="blockquote"><p>
1326           <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
1327
1328         </p></blockquote></div>
1329 <p>
1330         where
1331       </p>
1332 <div class="blockquote"><blockquote class="blockquote"><p>
1333           <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
1334
1335         </p></blockquote></div>
1336 <p>
1337         J<sub>&#957;</sub> and Y<sub>&#956;</sub> are then calculated using backward (Miller's algorithm) and forward
1338         recurrence respectively.
1339       </p>
1340 <p>
1341         When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> &lt;= 2), CF2
1342         convergence may fail (but CF1 works very well). The solution here is Temme's
1343         series:
1344       </p>
1345 <div class="blockquote"><blockquote class="blockquote"><p>
1346           <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span>
1347
1348         </p></blockquote></div>
1349 <p>
1350         where
1351       </p>
1352 <div class="blockquote"><blockquote class="blockquote"><p>
1353           <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
1354
1355         </p></blockquote></div>
1356 <p>
1357         g<sub>k</sub> and h<sub>k</sub>
1358 are also computed by recursions (involving gamma functions), but
1359         the formulas are a little complicated, readers are refered to N.M. Temme,
1360         <span class="emphasis"><em>On the numerical evaluation of the ordinary Bessel function of
1361         the second kind</em></span>, Journal of Computational Physics, vol 21, 343
1362         (1976). Note Temme's series converge only for |&#956;| &lt;= 1/2.
1363       </p>
1364 <p>
1365         As the previous case, Y<sub>&#957;</sub> is calculated from the forward recurrence, so is Y<sub>&#957;+1</sub>.
1366         With these two values and f<sub>&#957;</sub>, the Wronskian yields J<sub>&#957;</sub>(x) directly without backward
1367         recurrence.
1368       </p>
1369 </div>
1370 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1371 <td align="left"></td>
1372 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
1373       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
1374       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
1375       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
1376       Daryle Walker and Xiaogang Zhang<p>
1377         Distributed under the Boost Software License, Version 1.0. (See accompanying
1378         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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