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  24 </div>
  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;15.&#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;15.&#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;15.&#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;15.&#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 <p>
  59         cyl_bessel_j(v, x) = J<sub>v</sub>(x)
  60       </p>
  61 <p>
  62         cyl_neumann(v, x) = Y<sub>v</sub>(x) = N<sub>v</sub>(x)
  63       </p>
  64 <p>
  65         where:
  66       </p>
  67 <p>
  68         <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
  69       </p>
  70 <p>
  71         <span class="inlinemediaobject"><img src="../../../equations/bessel3.svg"></span>
  72       </p>
  73 <p>
  74         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
  75         type calculation rules</em></span></a> when T1 and T2 are different types.
  76         The functions are also optimised for the relatively common case that T1 is
  77         an integer.
  78       </p>
  79 <p>
  80         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  81         be used to control the behaviour of the function: how it handles errors,
  82         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  83         documentation for more details</a>.
  84       </p>
  85 <p>
  86         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  87         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>
  88         this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
  89         <span class="number">0</span></code> and v is not an integer, or when
  90         <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span>
  91         <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span>
  92         <span class="special">!=</span> <span class="number">0</span></code>.
  93         For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this
  94         occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;=</span>
  95         <span class="number">0</span></code>.
  96       </p>
  97 <p>
  98         The following graph illustrates the cyclic nature of J<sub>v</sub>:
  99       </p>
 100 <p>
 101         <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_j.svg" align="middle"></span>
 102       </p>
 103 <p>
 104         The following graph shows the behaviour of Y<sub>v</sub>: this is also cyclic for large
 105         <span class="emphasis"><em>x</em></span>, but tends to -&#8734; &#160; for small <span class="emphasis"><em>x</em></span>:
 106       </p>
 107 <p>
 108         <span class="inlinemediaobject"><img src="../../../graphs/cyl_neumann.svg" align="middle"></span>
 109       </p>
 110 <h5>
 111 <a name="math_toolkit.bessel.bessel_first.h2"></a>
 112         <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>
 113       </h5>
 114 <p>
 115         There are two sets of test values: spot values calculated using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
 116         and a much larger set of tests computed using a simplified version of this
 117         implementation (with all the special case handling removed).
 118       </p>
 119 <h5>
 120 <a name="math_toolkit.bessel.bessel_first.h3"></a>
 121         <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>
 122       </h5>
 123 <p>
 124         The following tables show how the accuracy of these functions varies on various
 125         platforms, along with comparisons to other libraries. Note that the cyclic
 126         nature of these functions means that they have an infinite number of irrational
 127         roots: in general these functions have arbitrarily large <span class="emphasis"><em>relative</em></span>
 128         errors when the arguments are sufficiently close to a root. Of course the
 129         absolute error in such cases is always small. Note that only results for
 130         the widest floating-point type on the system are given as narrower types
 131         have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
 132         error</a>. All values are relative errors in units of epsilon. Most of
 133         the gross errors exhibited by other libraries occur for very large arguments
 134         - you will need to drill down into the actual program output if you need
 135         more information on this.
 136       </p>
 137 <div class="table">
 138 <a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j_integer_orders_"></a><p class="title"><b>Table&#160;6.40.&#160;Error rates for cyl_bessel_j (integer orders)</b></p>
 139 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j (integer orders)">
 140 <colgroup>
 141 <col>
 142 <col>
 143 <col>
 144 <col>
 145 <col>
 146 </colgroup>
 147 <thead><tr>
 148 <th>
 149               </th>
 150 <th>
 151                 <p>
 152                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 153                 </p>
 154               </th>
 155 <th>
 156                 <p>
 157                   GNU C++ version 5.1.0<br> linux<br> long double
 158                 </p>
 159               </th>
 160 <th>
 161                 <p>
 162                   GNU C++ version 5.1.0<br> linux<br> double
 163                 </p>
 164               </th>
 165 <th>
 166                 <p>
 167                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 168                 </p>
 169               </th>
 170 </tr></thead>
 171 <tbody>
 172 <tr>
 173 <td>
 174                 <p>
 175                   Bessel J0: Mathworld Data (Integer Version)
 176                 </p>
 177               </td>
 178 <td>
 179                 <p>
 180                   <span class="blue">Max = 2.52&#949; (Mean = 1.2&#949;)</span><br> <br>
 181                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.89&#949; (Mean = 0.988&#949;))
 182                 </p>
 183               </td>
 184 <td>
 185                 <p>
 186                   <span class="blue">Max = 6.55&#949; (Mean = 2.89&#949;)</span><br> <br>
 187                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 5.04&#949; (Mean = 1.78&#949;)
 188                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_J0_Mathworld_Data_Integer_Version_">And
 189                   other failures.</a>)
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 195                   1.16:</em></span> Max = 1.12&#949; (Mean = 0.488&#949;))<br> (<span class="emphasis"><em>Rmath
 196                   3.0.2:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_0_2_Bessel_J0_Mathworld_Data_Integer_Version_">And
 197                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 198                   Max = 1.12&#949; (Mean = 0.568&#949;))
 199                 </p>
 200               </td>
 201 <td>
 202                 <p>
 203                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span>
 204                 </p>
 205               </td>
 206 </tr>
 207 <tr>
 208 <td>
 209                 <p>
 210                   Bessel J0: Mathworld Data (Tricky cases) (Integer Version)
 211                 </p>
 212               </td>
 213 <td>
 214                 <p>
 215                   <span class="blue">Max = 1e+007&#949; (Mean = 4.09e+006&#949;)</span><br>
 216                   <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> <span class="red">Max
 217                   = 2.54e+008&#949; (Mean = 1.04e+008&#949;))</span>
 218                 </p>
 219               </td>
 220 <td>
 221                 <p>
 222                   <span class="blue">Max = 1.63e+08&#949; (Mean = 6.67e+07&#949;)</span><br>
 223                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 4.79e+08&#949; (Mean
 224                   = 1.96e+08&#949;))
 225                 </p>
 226               </td>
 227 <td>
 228                 <p>
 229                   <span class="blue">Max = 7.98e+04&#949; (Mean = 3.26e+04&#949;)</span><br>
 230                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1e+07&#949; (Mean = 4.11e+06&#949;))<br>
 231                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.04e+07&#949; (Mean = 4.29e+06&#949;))<br>
 232                   (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 2.54e+08&#949; (Mean
 233                   = 1.04e+08&#949;))</span>
 234                 </p>
 235               </td>
 236 <td>
 237                 <p>
 238                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#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 = 1.73&#949; (Mean = 0.976&#949;)</span><br> <br>
 251                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 11.4&#949; (Mean = 4.15&#949;))
 252                 </p>
 253               </td>
 254 <td>
 255                 <p>
 256                   <span class="blue">Max = 2.66&#949; (Mean = 1.38&#949;)</span><br> <br>
 257                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 6.1&#949; (Mean = 2.95&#949;)
 258                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_J1_Mathworld_Data_Integer_Version_">And
 259                   other failures.</a>)
 260                 </p>
 261               </td>
 262 <td>
 263                 <p>
 264                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 265                   1.16:</em></span> Max = 1.89&#949; (Mean = 0.721&#949;))<br> (<span class="emphasis"><em>Rmath
 266                   3.0.2:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_0_2_Bessel_J1_Mathworld_Data_Integer_Version_">And
 267                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 268                   Max = 2.88&#949; (Mean = 1.12&#949;))
 269                 </p>
 270               </td>
 271 <td>
 272                 <p>
 273                   <span class="blue">Max = 1.44&#949; (Mean = 0.637&#949;)</span>
 274                 </p>
 275               </td>
 276 </tr>
 277 <tr>
 278 <td>
 279                 <p>
 280                   Bessel J1: Mathworld Data (tricky cases) (Integer Version)
 281                 </p>
 282               </td>
 283 <td>
 284                 <p>
 285                   <span class="blue">Max = 3.23e+004&#949; (Mean = 1.45e+004&#949;)</span><br>
 286                   <br> (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.44e+007&#949; (Mean
 287                   = 6.5e+006&#949;))
 288                 </p>
 289               </td>
 290 <td>
 291                 <p>
 292                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span><br>
 293                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.15e+06&#949; (Mean
 294                   = 1.58e+06&#949;))
 295                 </p>
 296               </td>
 297 <td>
 298                 <p>
 299                   <span class="blue">Max = 106&#949; (Mean = 47.5&#949;)</span><br> <br>
 300                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.26e+06&#949; (Mean = 6.28e+05&#949;))<br>
 301                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.93e+06&#949; (Mean = 1.7e+06&#949;))<br>
 302                   (<span class="emphasis"><em>Cephes:</em></span> Max = 9.56e+05&#949; (Mean = 4.99e+05&#949;))
 303                 </p>
 304               </td>
 305 <td>
 306                 <p>
 307                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span>
 308                 </p>
 309               </td>
 310 </tr>
 311 <tr>
 312 <td>
 313                 <p>
 314                   Bessel JN: Mathworld Data (Integer Version)
 315                 </p>
 316               </td>
 317 <td>
 318                 <p>
 319                   <span class="blue">Max = 14.7&#949; (Mean = 5.4&#949;)</span><br> <br>
 320                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> <span class="red">Max =
 321                   +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_Microsoft_Visual_C_version_12_0_Win32_double_cyl_bessel_j_integer_orders___math_h__Bessel_JN_Mathworld_Data_Integer_Version_">And
 322                   other failures.</a>)</span>
 323                 </p>
 324               </td>
 325 <td>
 326                 <p>
 327                   <span class="blue">Max = 6.85&#949; (Mean = 3.41&#949;)</span><br> <br>
 328                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
 329                   = 2.13e+19&#949; (Mean = 5.16e+18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j_integer_orders___tr1_cmath__Bessel_JN_Mathworld_Data_Integer_Version_">And
 330                   other failures.</a>)</span>
 331                 </p>
 332               </td>
 333 <td>
 334                 <p>
 335                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 336                   1.16:</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_5_1_0_linux_double_cyl_bessel_j_integer_orders__GSL_1_16_Bessel_JN_Mathworld_Data_Integer_Version_">And
 337                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 338                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_integer_orders__Rmath_3_0_2_Bessel_JN_Mathworld_Data_Integer_Version_">And
 339                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 340                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_integer_orders__Cephes_Bessel_JN_Mathworld_Data_Integer_Version_">And
 341                   other failures.</a>)</span>
 342                 </p>
 343               </td>
 344 <td>
 345                 <p>
 346                   <span class="blue">Max = 463&#949; (Mean = 112&#949;)</span>
 347                 </p>
 348               </td>
 349 </tr>
 350 </tbody>
 351 </table></div>
 352 </div>
 353 <br class="table-break"><div class="table">
 354 <a name="math_toolkit.bessel.bessel_first.table_cyl_bessel_j"></a><p class="title"><b>Table&#160;6.41.&#160;Error rates for cyl_bessel_j</b></p>
 355 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j">
 356 <colgroup>
 357 <col>
 358 <col>
 359 <col>
 360 <col>
 361 <col>
 362 </colgroup>
 363 <thead><tr>
 364 <th>
 365               </th>
 366 <th>
 367                 <p>
 368                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 369                 </p>
 370               </th>
 371 <th>
 372                 <p>
 373                   GNU C++ version 5.1.0<br> linux<br> long double
 374                 </p>
 375               </th>
 376 <th>
 377                 <p>
 378                   GNU C++ version 5.1.0<br> linux<br> double
 379                 </p>
 380               </th>
 381 <th>
 382                 <p>
 383                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 384                 </p>
 385               </th>
 386 </tr></thead>
 387 <tbody>
 388 <tr>
 389 <td>
 390                 <p>
 391                   Bessel J0: Mathworld Data
 392                 </p>
 393               </td>
 394 <td>
 395                 <p>
 396                   <span class="blue">Max = 2.52&#949; (Mean = 1.2&#949;)</span>
 397                 </p>
 398               </td>
 399 <td>
 400                 <p>
 401                   <span class="blue">Max = 6.55&#949; (Mean = 2.89&#949;)</span><br> <br>
 402                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 5.04&#949; (Mean = 1.78&#949;)
 403                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J0_Mathworld_Data">And
 404                   other failures.</a>)
 405                 </p>
 406               </td>
 407 <td>
 408                 <p>
 409                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 410                   1.16:</em></span> Max = 0.629&#949; (Mean = 0.223&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J0_Mathworld_Data">And
 411                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 412                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_Rmath_3_0_2_Bessel_J0_Mathworld_Data">And
 413                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 414                   Max = 1.12&#949; (Mean = 0.568&#949;))
 415                 </p>
 416               </td>
 417 <td>
 418                 <p>
 419                   <span class="blue">Max = 6.55&#949; (Mean = 2.86&#949;)</span>
 420                 </p>
 421               </td>
 422 </tr>
 423 <tr>
 424 <td>
 425                 <p>
 426                   Bessel J0: Mathworld Data (Tricky cases)
 427                 </p>
 428               </td>
 429 <td>
 430                 <p>
 431                   <span class="blue">Max = 1e+007&#949; (Mean = 4.09e+006&#949;)</span>
 432                 </p>
 433               </td>
 434 <td>
 435                 <p>
 436                   <span class="blue">Max = 1.63e+08&#949; (Mean = 6.67e+07&#949;)</span><br>
 437                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 4.79e+08&#949; (Mean
 438                   = 1.96e+08&#949;))
 439                 </p>
 440               </td>
 441 <td>
 442                 <p>
 443                   <span class="blue">Max = 7.98e+04&#949; (Mean = 3.26e+04&#949;)</span><br>
 444                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 6.5e+07&#949; (Mean = 2.66e+07&#949;))<br>
 445                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.04e+07&#949; (Mean = 4.29e+06&#949;))<br>
 446                   (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 2.54e+08&#949; (Mean
 447                   = 1.04e+08&#949;))</span>
 448                 </p>
 449               </td>
 450 <td>
 451                 <p>
 452                   <span class="blue">Max = 1.64e+08&#949; (Mean = 6.69e+07&#949;)</span>
 453                 </p>
 454               </td>
 455 </tr>
 456 <tr>
 457 <td>
 458                 <p>
 459                   Bessel J1: Mathworld Data
 460                 </p>
 461               </td>
 462 <td>
 463                 <p>
 464                   <span class="blue">Max = 1.73&#949; (Mean = 0.976&#949;)</span>
 465                 </p>
 466               </td>
 467 <td>
 468                 <p>
 469                   <span class="blue">Max = 2.66&#949; (Mean = 1.38&#949;)</span><br> <br>
 470                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 6.1&#949; (Mean = 2.95&#949;)
 471                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J1_Mathworld_Data">And
 472                   other failures.</a>)
 473                 </p>
 474               </td>
 475 <td>
 476                 <p>
 477                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 478                   1.16:</em></span> Max = 6.62&#949; (Mean = 2.35&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J1_Mathworld_Data">And
 479                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 480                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_Rmath_3_0_2_Bessel_J1_Mathworld_Data">And
 481                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 482                   Max = 2.88&#949; (Mean = 1.12&#949;))
 483                 </p>
 484               </td>
 485 <td>
 486                 <p>
 487                   <span class="blue">Max = 1.44&#949; (Mean = 0.637&#949;)</span>
 488                 </p>
 489               </td>
 490 </tr>
 491 <tr>
 492 <td>
 493                 <p>
 494                   Bessel J1: Mathworld Data (tricky cases)
 495                 </p>
 496               </td>
 497 <td>
 498                 <p>
 499                   <span class="blue">Max = 3.23e+004&#949; (Mean = 1.45e+004&#949;)</span>
 500                 </p>
 501               </td>
 502 <td>
 503                 <p>
 504                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span><br>
 505                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.15e+06&#949; (Mean
 506                   = 1.58e+06&#949;))
 507                 </p>
 508               </td>
 509 <td>
 510                 <p>
 511                   <span class="blue">Max = 106&#949; (Mean = 47.5&#949;)</span><br> <br>
 512                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.75e+05&#949; (Mean = 5.32e+05&#949;))<br>
 513                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.93e+06&#949; (Mean = 1.7e+06&#949;))<br>
 514                   (<span class="emphasis"><em>Cephes:</em></span> Max = 9.56e+05&#949; (Mean = 4.99e+05&#949;))
 515                 </p>
 516               </td>
 517 <td>
 518                 <p>
 519                   <span class="blue">Max = 2.18e+05&#949; (Mean = 9.76e+04&#949;)</span>
 520                 </p>
 521               </td>
 522 </tr>
 523 <tr>
 524 <td>
 525                 <p>
 526                   Bessel JN: Mathworld Data
 527                 </p>
 528               </td>
 529 <td>
 530                 <p>
 531                   <span class="blue">Max = 14.7&#949; (Mean = 5.4&#949;)</span>
 532                 </p>
 533               </td>
 534 <td>
 535                 <p>
 536                   <span class="blue">Max = 6.85&#949; (Mean = 3.41&#949;)</span><br> <br>
 537                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
 538                   = 2.13e+19&#949; (Mean = 5.16e+18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_JN_Mathworld_Data">And
 539                   other failures.</a>)</span>
 540                 </p>
 541               </td>
 542 <td>
 543                 <p>
 544                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 545                   1.16:</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_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_JN_Mathworld_Data">And
 546                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 547                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_Rmath_3_0_2_Bessel_JN_Mathworld_Data">And
 548                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 549                   Max = 5.53e+05&#949; (Mean = 1.9e+05&#949;))
 550                 </p>
 551               </td>
 552 <td>
 553                 <p>
 554                   <span class="blue">Max = 463&#949; (Mean = 112&#949;)</span>
 555                 </p>
 556               </td>
 557 </tr>
 558 <tr>
 559 <td>
 560                 <p>
 561                   Bessel J: Mathworld Data
 562                 </p>
 563               </td>
 564 <td>
 565                 <p>
 566                   <span class="blue">Max = 14.9&#949; (Mean = 3.82&#949;)</span>
 567                 </p>
 568               </td>
 569 <td>
 570                 <p>
 571                   <span class="blue">Max = 14.7&#949; (Mean = 4.05&#949;)</span><br> <br>
 572                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 3.49e+05&#949; (Mean =
 573                   7.89e+04&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J_Mathworld_Data">And
 574                   other failures.</a>)
 575                 </p>
 576               </td>
 577 <td>
 578                 <p>
 579                   <span class="blue">Max = 10&#949; (Mean = 2.19&#949;)</span><br> <br>
 580                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.39e+05&#949; (Mean = 5.24e+04&#949;)
 581                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_Mathworld_Data">And
 582                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 583                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_Rmath_3_0_2_Bessel_J_Mathworld_Data">And
 584                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 585                   Max = 5.47e+05&#949; (Mean = 1.3e+05&#949;))
 586                 </p>
 587               </td>
 588 <td>
 589                 <p>
 590                   <span class="blue">Max = 14.7&#949; (Mean = 4.12&#949;)</span>
 591                 </p>
 592               </td>
 593 </tr>
 594 <tr>
 595 <td>
 596                 <p>
 597                   Bessel J: Mathworld Data (large values)
 598                 </p>
 599               </td>
 600 <td>
 601                 <p>
 602                   <span class="blue">Max = 9.31&#949; (Mean = 5.52&#949;)</span>
 603                 </p>
 604               </td>
 605 <td>
 606                 <p>
 607                   <span class="blue">Max = 607&#949; (Mean = 305&#949;)</span><br> <br>
 608                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 34.9&#949; (Mean = 17.4&#949;)
 609                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_j__tr1_cmath__Bessel_J_Mathworld_Data_large_values_">And
 610                   other failures.</a>)
 611                 </p>
 612               </td>
 613 <td>
 614                 <p>
 615                   <span class="blue">Max = 0.536&#949; (Mean = 0.268&#949;)</span><br> <br>
 616                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 4.91e+03&#949; (Mean = 2.46e+03&#949;)
 617                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_Mathworld_Data_large_values_">And
 618                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 619                   Max = 35.9&#949; (Mean = 18.1&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_Cephes_Bessel_J_Mathworld_Data_large_values_">And
 620                   other failures.</a>)</span>
 621                 </p>
 622               </td>
 623 <td>
 624                 <p>
 625                   <span class="blue">Max = 607&#949; (Mean = 305&#949;)</span>
 626                 </p>
 627               </td>
 628 </tr>
 629 <tr>
 630 <td>
 631                 <p>
 632                   Bessel JN: Random Data
 633                 </p>
 634               </td>
 635 <td>
 636                 <p>
 637                   <span class="blue">Max = 17.5&#949; (Mean = 1.46&#949;)</span>
 638                 </p>
 639               </td>
 640 <td>
 641                 <p>
 642                   <span class="blue">Max = 50.8&#949; (Mean = 4.15&#949;)</span><br> <br>
 643                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.12e+03&#949; (Mean =
 644                   88.7&#949;))
 645                 </p>
 646               </td>
 647 <td>
 648                 <p>
 649                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 650                   1.16:</em></span> Max = 75.7&#949; (Mean = 5.36&#949;))<br> (<span class="emphasis"><em>Rmath
 651                   3.0.2:</em></span> Max = 3.93&#949; (Mean = 1.22&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 652                   Max = 91.4&#949; (Mean = 6.47&#949;))
 653                 </p>
 654               </td>
 655 <td>
 656                 <p>
 657                   <span class="blue">Max = 99.6&#949; (Mean = 22&#949;)</span>
 658                 </p>
 659               </td>
 660 </tr>
 661 <tr>
 662 <td>
 663                 <p>
 664                   Bessel J: Random Data
 665                 </p>
 666               </td>
 667 <td>
 668                 <p>
 669                   <span class="blue">Max = 9.24&#949; (Mean = 1.36&#949;)</span>
 670                 </p>
 671               </td>
 672 <td>
 673                 <p>
 674                   <span class="blue">Max = 9.81&#949; (Mean = 1.59&#949;)</span><br> <br>
 675                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 501&#949; (Mean = 52.3&#949;))
 676                 </p>
 677               </td>
 678 <td>
 679                 <p>
 680                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 681                   1.16:</em></span> Max = 15.5&#949; (Mean = 3.33&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_j_GSL_1_16_Bessel_J_Random_Data">And
 682                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 683                   Max = 6.74&#949; (Mean = 1.3&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
 684                   = 16.7&#949; (Mean = 2.5&#949;))
 685                 </p>
 686               </td>
 687 <td>
 688                 <p>
 689                   <span class="blue">Max = 260&#949; (Mean = 34&#949;)</span>
 690                 </p>
 691               </td>
 692 </tr>
 693 <tr>
 694 <td>
 695                 <p>
 696                   Bessel J: Random Data (Tricky large values)
 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 <td>
 705                 <p>
 706                   <span class="blue">Max = 785&#949; (Mean = 94.2&#949;)</span><br> <br>
 707                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
 708                   = 5.01e+17&#949; (Mean = 6.23e+16&#949;))</span>
 709                 </p>
 710               </td>
 711 <td>
 712                 <p>
 713                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 714                   1.16:</em></span> Max = 2.48e+05&#949; (Mean = 5.11e+04&#949;))<br> (<span class="emphasis"><em>Rmath
 715                   3.0.2:</em></span> Max = 71.6&#949; (Mean = 11.7&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 716                   Max = 2.48e+05&#949; (Mean = 3.02e+04&#949;))
 717                 </p>
 718               </td>
 719 <td>
 720                 <p>
 721                   <span class="blue">Max = 785&#949; (Mean = 97.4&#949;)</span>
 722                 </p>
 723               </td>
 724 </tr>
 725 </tbody>
 726 </table></div>
 727 </div>
 728 <br class="table-break"><div class="table">
 729 <a name="math_toolkit.bessel.bessel_first.table_cyl_neumann_integer_orders_"></a><p class="title"><b>Table&#160;6.42.&#160;Error rates for cyl_neumann (integer orders)</b></p>
 730 <div class="table-contents"><table class="table" summary="Error rates for cyl_neumann (integer orders)">
 731 <colgroup>
 732 <col>
 733 <col>
 734 <col>
 735 <col>
 736 <col>
 737 </colgroup>
 738 <thead><tr>
 739 <th>
 740               </th>
 741 <th>
 742                 <p>
 743                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 744                 </p>
 745               </th>
 746 <th>
 747                 <p>
 748                   GNU C++ version 5.1.0<br> linux<br> double
 749                 </p>
 750               </th>
 751 <th>
 752                 <p>
 753                   GNU C++ version 5.1.0<br> linux<br> long double
 754                 </p>
 755               </th>
 756 <th>
 757                 <p>
 758                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 759                 </p>
 760               </th>
 761 </tr></thead>
 762 <tbody>
 763 <tr>
 764 <td>
 765                 <p>
 766                   Y0: Mathworld Data (Integer Version)
 767                 </p>
 768               </td>
 769 <td>
 770                 <p>
 771                   <span class="blue">Max = 4.61&#949; (Mean = 2.29&#949;)</span><br> <br>
 772                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 5.37e+003&#949; (Mean = 1.81e+003&#949;))
 773                 </p>
 774               </td>
 775 <td>
 776                 <p>
 777                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 778                   1.16:</em></span> Max = 6.46&#949; (Mean = 2.38&#949;))<br> (<span class="emphasis"><em>Rmath
 779                   3.0.2:</em></span> Max = 167&#949; (Mean = 56.5&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 780                   Max = 5.37e+03&#949; (Mean = 1.81e+03&#949;))
 781                 </p>
 782               </td>
 783 <td>
 784                 <p>
 785                   <span class="blue">Max = 5.59&#949; (Mean = 2.54&#949;)</span><br> <br>
 786                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.05e+05&#949; (Mean =
 787                   6.87e+04&#949;))
 788                 </p>
 789               </td>
 790 <td>
 791                 <p>
 792                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span>
 793                 </p>
 794               </td>
 795 </tr>
 796 <tr>
 797 <td>
 798                 <p>
 799                   Y1: Mathworld Data (Integer Version)
 800                 </p>
 801               </td>
 802 <td>
 803                 <p>
 804                   <span class="blue">Max = 4.75&#949; (Mean = 1.72&#949;)</span><br> <br>
 805                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.86e+004&#949; (Mean = 6.2e+003&#949;))
 806                 </p>
 807               </td>
 808 <td>
 809                 <p>
 810                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 811                   1.16:</em></span> Max = 1.51&#949; (Mean = 0.839&#949;))<br> (<span class="emphasis"><em>Rmath
 812                   3.0.2:</em></span> Max = 193&#949; (Mean = 64.4&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 813                   Max = 1.86e+04&#949; (Mean = 6.2e+03&#949;))
 814                 </p>
 815               </td>
 816 <td>
 817                 <p>
 818                   <span class="blue">Max = 12.7&#949; (Mean = 4.34&#949;)</span><br> <br>
 819                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 9.71e+03&#949; (Mean =
 820                   4.08e+03&#949;))
 821                 </p>
 822               </td>
 823 <td>
 824                 <p>
 825                   <span class="blue">Max = 6.33&#949; (Mean = 2.29&#949;)</span>
 826                 </p>
 827               </td>
 828 </tr>
 829 <tr>
 830 <td>
 831                 <p>
 832                   Yn: Mathworld Data (Integer Version)
 833                 </p>
 834               </td>
 835 <td>
 836                 <p>
 837                   <span class="blue">Max = 35&#949; (Mean = 11.8&#949;)</span><br> <br>
 838                   (<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.49e+005&#949; (Mean = 8.14e+004&#949;))
 839                 </p>
 840               </td>
 841 <td>
 842                 <p>
 843                   <span class="blue">Max = 0.993&#949; (Mean = 0.314&#949;)</span><br> <br>
 844                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.41e+05&#949; (Mean = 7.62e+04&#949;))<br>
 845                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.24e+04&#949; (Mean = 4e+03&#949;))<br>
 846                   (<span class="emphasis"><em>Cephes:</em></span> Max = 2.49e+05&#949; (Mean = 8.14e+04&#949;))
 847                 </p>
 848               </td>
 849 <td>
 850                 <p>
 851                   <span class="blue">Max = 55.2&#949; (Mean = 17.7&#949;)</span><br> <br>
 852                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
 853                   = 2.2e+20&#949; (Mean = 6.97e+19&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann_integer_orders___tr1_cmath__Yn_Mathworld_Data_Integer_Version_">And
 854                   other failures.</a>)</span>
 855                 </p>
 856               </td>
 857 <td>
 858                 <p>
 859                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span>
 860                 </p>
 861               </td>
 862 </tr>
 863 </tbody>
 864 </table></div>
 865 </div>
 866 <br class="table-break"><div class="table">
 867 <a name="math_toolkit.bessel.bessel_first.table_cyl_neumann"></a><p class="title"><b>Table&#160;6.43.&#160;Error rates for cyl_neumann</b></p>
 868 <div class="table-contents"><table class="table" summary="Error rates for cyl_neumann">
 869 <colgroup>
 870 <col>
 871 <col>
 872 <col>
 873 <col>
 874 <col>
 875 </colgroup>
 876 <thead><tr>
 877 <th>
 878               </th>
 879 <th>
 880                 <p>
 881                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 882                 </p>
 883               </th>
 884 <th>
 885                 <p>
 886                   GNU C++ version 5.1.0<br> linux<br> double
 887                 </p>
 888               </th>
 889 <th>
 890                 <p>
 891                   GNU C++ version 5.1.0<br> linux<br> long double
 892                 </p>
 893               </th>
 894 <th>
 895                 <p>
 896                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 897                 </p>
 898               </th>
 899 </tr></thead>
 900 <tbody>
 901 <tr>
 902 <td>
 903                 <p>
 904                   Y0: Mathworld Data
 905                 </p>
 906               </td>
 907 <td>
 908                 <p>
 909                   <span class="blue">Max = 4.61&#949; (Mean = 2.29&#949;)</span>
 910                 </p>
 911               </td>
 912 <td>
 913                 <p>
 914                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 915                   1.16:</em></span> Max = 60.9&#949; (Mean = 20.4&#949;))<br> (<span class="emphasis"><em>Rmath
 916                   3.0.2:</em></span> Max = 167&#949; (Mean = 56.5&#949;))
 917                 </p>
 918               </td>
 919 <td>
 920                 <p>
 921                   <span class="blue">Max = 5.59&#949; (Mean = 2.54&#949;)</span><br> <br>
 922                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.05e+05&#949; (Mean =
 923                   6.87e+04&#949;))
 924                 </p>
 925               </td>
 926 <td>
 927                 <p>
 928                   <span class="blue">Max = 5.53&#949; (Mean = 2.4&#949;)</span>
 929                 </p>
 930               </td>
 931 </tr>
 932 <tr>
 933 <td>
 934                 <p>
 935                   Y1: Mathworld Data
 936                 </p>
 937               </td>
 938 <td>
 939                 <p>
 940                   <span class="blue">Max = 4.75&#949; (Mean = 1.72&#949;)</span>
 941                 </p>
 942               </td>
 943 <td>
 944                 <p>
 945                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 946                   1.16:</em></span> Max = 23.4&#949; (Mean = 8.1&#949;))<br> (<span class="emphasis"><em>Rmath
 947                   3.0.2:</em></span> Max = 193&#949; (Mean = 64.4&#949;))
 948                 </p>
 949               </td>
 950 <td>
 951                 <p>
 952                   <span class="blue">Max = 12.7&#949; (Mean = 4.34&#949;)</span><br> <br>
 953                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 9.71e+03&#949; (Mean =
 954                   4.08e+03&#949;))
 955                 </p>
 956               </td>
 957 <td>
 958                 <p>
 959                   <span class="blue">Max = 6.33&#949; (Mean = 2.29&#949;)</span>
 960                 </p>
 961               </td>
 962 </tr>
 963 <tr>
 964 <td>
 965                 <p>
 966                   Yn: Mathworld Data
 967                 </p>
 968               </td>
 969 <td>
 970                 <p>
 971                   <span class="blue">Max = 35&#949; (Mean = 11.8&#949;)</span>
 972                 </p>
 973               </td>
 974 <td>
 975                 <p>
 976                   <span class="blue">Max = 0.993&#949; (Mean = 0.314&#949;)</span><br> <br>
 977                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 2.41e+05&#949; (Mean = 7.62e+04&#949;)
 978                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yn_Mathworld_Data">And
 979                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 980                   Max = 1.24e+04&#949; (Mean = 4e+03&#949;))
 981                 </p>
 982               </td>
 983 <td>
 984                 <p>
 985                   <span class="blue">Max = 55.2&#949; (Mean = 17.7&#949;)</span><br> <br>
 986                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
 987                   = 2.2e+20&#949; (Mean = 6.97e+19&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yn_Mathworld_Data">And
 988                   other failures.</a>)</span>
 989                 </p>
 990               </td>
 991 <td>
 992                 <p>
 993                   <span class="blue">Max = 55.2&#949; (Mean = 17.8&#949;)</span>
 994                 </p>
 995               </td>
 996 </tr>
 997 <tr>
 998 <td>
 999                 <p>
1000                   Yv: Mathworld Data
1001                 </p>
1002               </td>
1003 <td>
1004                 <p>
1005                   <span class="blue">Max = 7.89&#949; (Mean = 3.27&#949;)</span>
1006                 </p>
1007               </td>
1008 <td>
1009                 <p>
1010                   <span class="blue">Max = 10&#949; (Mean = 3.02&#949;)</span><br> <br>
1011                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.07e+05&#949; (Mean = 3.22e+04&#949;)
1012                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yv_Mathworld_Data">And
1013                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
1014                   Max = 1.05e+03&#949; (Mean = 326&#949;))
1015                 </p>
1016               </td>
1017 <td>
1018                 <p>
1019                   <span class="blue">Max = 10.7&#949; (Mean = 4.92&#949;)</span><br> <br>
1020                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
1021                   = 3.49e+15&#949; (Mean = 1.05e+15&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yv_Mathworld_Data">And
1022                   other failures.</a>)</span>
1023                 </p>
1024               </td>
1025 <td>
1026                 <p>
1027                   <span class="blue">Max = 10.7&#949; (Mean = 5.1&#949;)</span>
1028                 </p>
1029               </td>
1030 </tr>
1031 <tr>
1032 <td>
1033                 <p>
1034                   Yv: Mathworld Data (large values)
1035                 </p>
1036               </td>
1037 <td>
1038                 <p>
1039                   <span class="blue">Max = 0.682&#949; (Mean = 0.35&#949;)</span>
1040                 </p>
1041               </td>
1042 <td>
1043                 <p>
1044                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1045                   1.16:</em></span> Max = 60.8&#949; (Mean = 23&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_GSL_1_16_Yv_Mathworld_Data_large_values_">And
1046                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
1047                   <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_neumann_Rmath_3_0_2_Yv_Mathworld_Data_large_values_">And
1048                   other failures.</a>)</span>
1049                 </p>
1050               </td>
1051 <td>
1052                 <p>
1053                   <span class="blue">Max = 1.57&#949; (Mean = 1.17&#949;)</span><br> <br>
1054                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 43.2&#949; (Mean = 16.3&#949;)
1055                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yv_Mathworld_Data_large_values_">And
1056                   other failures.</a>)
1057                 </p>
1058               </td>
1059 <td>
1060                 <p>
1061                   <span class="blue">Max = 1.57&#949; (Mean = 1.24&#949;)</span>
1062                 </p>
1063               </td>
1064 </tr>
1065 <tr>
1066 <td>
1067                 <p>
1068                   Y0 and Y1: Random Data
1069                 </p>
1070               </td>
1071 <td>
1072                 <p>
1073                   <span class="blue">Max = 4.17&#949; (Mean = 1.24&#949;)</span>
1074                 </p>
1075               </td>
1076 <td>
1077                 <p>
1078                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1079                   1.16:</em></span> Max = 34.4&#949; (Mean = 8.9&#949;))<br> (<span class="emphasis"><em>Rmath
1080                   3.0.2:</em></span> Max = 83&#949; (Mean = 14.2&#949;))
1081                 </p>
1082               </td>
1083 <td>
1084                 <p>
1085                   <span class="blue">Max = 11.8&#949; (Mean = 3.28&#949;)</span><br> <br>
1086                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.59e+03&#949; (Mean =
1087                   500&#949;))
1088                 </p>
1089               </td>
1090 <td>
1091                 <p>
1092                   <span class="blue">Max = 10.8&#949; (Mean = 3.04&#949;)</span>
1093                 </p>
1094               </td>
1095 </tr>
1096 <tr>
1097 <td>
1098                 <p>
1099                   Yn: Random Data
1100                 </p>
1101               </td>
1102 <td>
1103                 <p>
1104                   <span class="blue">Max = 117&#949; (Mean = 10.2&#949;)</span>
1105                 </p>
1106               </td>
1107 <td>
1108                 <p>
1109                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1110                   1.16:</em></span> Max = 500&#949; (Mean = 47.8&#949;))<br> (<span class="emphasis"><em>Rmath
1111                   3.0.2:</em></span> Max = 691&#949; (Mean = 67.9&#949;))
1112                 </p>
1113               </td>
1114 <td>
1115                 <p>
1116                   <span class="blue">Max = 338&#949; (Mean = 28.2&#949;)</span><br> <br>
1117                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 4.01e+03&#949; (Mean =
1118                   348&#949;))
1119                 </p>
1120               </td>
1121 <td>
1122                 <p>
1123                   <span class="blue">Max = 338&#949; (Mean = 27.5&#949;)</span>
1124                 </p>
1125               </td>
1126 </tr>
1127 <tr>
1128 <td>
1129                 <p>
1130                   Yv: Random Data
1131                 </p>
1132               </td>
1133 <td>
1134                 <p>
1135                   <span class="blue">Max = 1.23e+003&#949; (Mean = 69.9&#949;)</span>
1136                 </p>
1137               </td>
1138 <td>
1139                 <p>
1140                   <span class="blue">Max = 1.53&#949; (Mean = 0.102&#949;)</span><br> <br>
1141                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.41e+06&#949; (Mean = 7.67e+04&#949;))<br>
1142                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.79e+05&#949; (Mean = 9.64e+03&#949;))
1143                 </p>
1144               </td>
1145 <td>
1146                 <p>
1147                   <span class="blue">Max = 2.08e+03&#949; (Mean = 149&#949;)</span><br>
1148                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> <span class="red">Max
1149                   = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_neumann__tr1_cmath__Yv_Random_Data">And
1150                   other failures.</a>)</span>
1151                 </p>
1152               </td>
1153 <td>
1154                 <p>
1155                   <span class="blue">Max = 2.08e+03&#949; (Mean = 149&#949;)</span>
1156                 </p>
1157               </td>
1158 </tr>
1159 </tbody>
1160 </table></div>
1161 </div>
1162 <br class="table-break"><p>
1163         Note that for large <span class="emphasis"><em>x</em></span> these functions are largely dependent
1164         on the accuracy of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span></code> and
1165         <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cos</span></code> functions.
1166       </p>
1167 <p>
1168         Comparison to GSL and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
1169         is interesting: both <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a>
1170         and this library optimise the integer order case - leading to identical results
1171         - simply using the general case is for the most part slightly more accurate
1172         though, as noted by the better accuracy of GSL in the integer argument cases.
1173         This implementation tends to perform much better when the arguments become
1174         large, <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> in particular
1175         produces some remarkably inaccurate results with some of the test data (no
1176         significant figures correct), and even GSL performs badly with some inputs
1177         to J<sub>v</sub>. Note that by way of double-checking these results, the worst performing
1178         <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> and GSL cases were
1179         recomputed using <a href="http://functions.wolfram.com" target="_top">functions.wolfram.com</a>,
1180         and the result checked against our test data: no errors in the test data
1181         were found.
1182       </p>
1183 <h5>
1184 <a name="math_toolkit.bessel.bessel_first.h4"></a>
1185         <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>
1186       </h5>
1187 <p>
1188         The implementation is mostly about filtering off various special cases:
1189       </p>
1190 <p>
1191         When <span class="emphasis"><em>x</em></span> is negative, then the order <span class="emphasis"><em>v</em></span>
1192         must be an integer or the result is a domain error. If the order is an integer
1193         then the function is odd for odd orders and even for even orders, so we reflect
1194         to <span class="emphasis"><em>x &gt; 0</em></span>.
1195       </p>
1196 <p>
1197         When the order <span class="emphasis"><em>v</em></span> is negative then the reflection formulae
1198         can be used to move to <span class="emphasis"><em>v &gt; 0</em></span>:
1199       </p>
1200 <p>
1201         <span class="inlinemediaobject"><img src="../../../equations/bessel9.svg"></span>
1202       </p>
1203 <p>
1204         <span class="inlinemediaobject"><img src="../../../equations/bessel10.svg"></span>
1205       </p>
1206 <p>
1207         Note that if the order is an integer, then these formulae reduce to:
1208       </p>
1209 <p>
1210         J<sub>-n</sub> = (-1)<sup>n</sup>J<sub>n</sub>
1211       </p>
1212 <p>
1213         Y<sub>-n</sub> = (-1)<sup>n</sup>Y<sub>n</sub>
1214       </p>
1215 <p>
1216         However, in general, a negative order implies that we will need to compute
1217         both J and Y.
1218       </p>
1219 <p>
1220         When <span class="emphasis"><em>x</em></span> is large compared to the order <span class="emphasis"><em>v</em></span>
1221         then the asymptotic expansions for large <span class="emphasis"><em>x</em></span> in M. Abramowitz
1222         and I.A. Stegun, <span class="emphasis"><em>Handbook of Mathematical Functions</em></span>
1223         9.2.19 are used (these were found to be more reliable than those in A&amp;S
1224         9.2.5).
1225       </p>
1226 <p>
1227         When the order <span class="emphasis"><em>v</em></span> is an integer the method first relates
1228         the result to J<sub>0</sub>, J<sub>1</sub>, Y<sub>0</sub> &#160; and Y<sub>1</sub> &#160; using either forwards or backwards recurrence
1229         (Miller's algorithm) depending upon which is stable. The values for J<sub>0</sub>, J<sub>1</sub>,
1230         Y<sub>0</sub> &#160; and Y<sub>1</sub> &#160; are calculated using the rational minimax approximations on root-bracketing
1231         intervals for small <span class="emphasis"><em>|x|</em></span> and Hankel asymptotic expansion
1232         for large <span class="emphasis"><em>|x|</em></span>. The coefficients are from:
1233       </p>
1234 <p>
1235         W.J. Cody, <span class="emphasis"><em>ALGORITHM 715: SPECFUN - A Portable FORTRAN Package
1236         of Special Function Routines and Test Drivers</em></span>, ACM Transactions
1237         on Mathematical Software, vol 19, 22 (1993).
1238       </p>
1239 <p>
1240         and
1241       </p>
1242 <p>
1243         J.F. Hart et al, <span class="emphasis"><em>Computer Approximations</em></span>, John Wiley
1244         &amp; Sons, New York, 1968.
1245       </p>
1246 <p>
1247         These approximations are accurate to around 19 decimal digits: therefore
1248         these methods are not used when type T has more than 64 binary digits.
1249       </p>
1250 <p>
1251         When <span class="emphasis"><em>x</em></span> is smaller than machine epsilon then the following
1252         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>,
1253         <a href="http://functions.wolfram.com/03.03.06.0038.01" target="_top">http://functions.wolfram.com/03.03.06.0038.01</a>,
1254         <a href="http://functions.wolfram.com/03.03.06.0039.01" target="_top">http://functions.wolfram.com/03.03.06.0039.01</a>
1255         and <a href="http://functions.wolfram.com/03.03.06.0040.01" target="_top">http://functions.wolfram.com/03.03.06.0040.01</a>):
1256       </p>
1257 <p>
1258         <span class="inlinemediaobject"><img src="../../../equations/bessel_y0_small_z.svg"></span>
1259       </p>
1260 <p>
1261         <span class="inlinemediaobject"><img src="../../../equations/bessel_y1_small_z.svg"></span>
1262       </p>
1263 <p>
1264         <span class="inlinemediaobject"><img src="../../../equations/bessel_y2_small_z.svg"></span>
1265       </p>
1266 <p>
1267         <span class="inlinemediaobject"><img src="../../../equations/bessel_yn_small_z.svg"></span>
1268       </p>
1269 <p>
1270         When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span> and
1271         <span class="emphasis"><em>v</em></span> is not an integer, then the following series approximation
1272         can be used for Y<sub>v</sub>(x), this is also an area where other approximations are
1273         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>):
1274       </p>
1275 <p>
1276         <span class="inlinemediaobject"><img src="../../../equations/bessel_yv_small_z.svg"></span>
1277       </p>
1278 <p>
1279         When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
1280         J<sub>v</sub>x &#160; is best computed directly from the series:
1281       </p>
1282 <p>
1283         <span class="inlinemediaobject"><img src="../../../equations/bessel2.svg"></span>
1284       </p>
1285 <p>
1286         In the general case we compute J<sub>v</sub> &#160; and Y<sub>v</sub> &#160; simultaneously.
1287       </p>
1288 <p>
1289         To get the initial values, let &#956; &#160; = &#957; - floor(&#957; + 1/2), then &#956; &#160; is the fractional part
1290         of &#957; &#160; such that |&#956;| &lt;= 1/2 (we need this for convergence later). The idea
1291         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).
1292       </p>
1293 <p>
1294         The algorithm is called Steed's method, which needs two continued fractions
1295         as well as the Wronskian:
1296       </p>
1297 <p>
1298         <span class="inlinemediaobject"><img src="../../../equations/bessel8.svg"></span>
1299       </p>
1300 <p>
1301         <span class="inlinemediaobject"><img src="../../../equations/bessel11.svg"></span>
1302       </p>
1303 <p>
1304         <span class="inlinemediaobject"><img src="../../../equations/bessel12.svg"></span>
1305       </p>
1306 <p>
1307         See: F.S. Acton, <span class="emphasis"><em>Numerical Methods that Work</em></span>, The Mathematical
1308         Association of America, Washington, 1997.
1309       </p>
1310 <p>
1311         The continued fractions are computed using the modified Lentz's method (W.J.
1312         Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
1313         using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
1314         Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need
1315         different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>.
1316       </p>
1317 <p>
1318         <span class="emphasis"><em>x &gt; v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations
1319         to converge, CF2 converges rapidly
1320       </p>
1321 <p>
1322         <span class="emphasis"><em>x &lt;= v</em></span>, CF1 converges rapidly, CF2 fails to converge
1323         when <span class="emphasis"><em>x</em></span> <code class="literal">-&gt;</code> 0
1324       </p>
1325 <p>
1326         When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> &gt; 2), both
1327         continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
1328         J<sub>&#956;</sub>, J<sub>&#956;+1</sub>, Y<sub>&#956;</sub>, Y<sub>&#956;+1</sub> can be calculated by
1329       </p>
1330 <p>
1331         <span class="inlinemediaobject"><img src="../../../equations/bessel13.svg"></span>
1332       </p>
1333 <p>
1334         where
1335       </p>
1336 <p>
1337         <span class="inlinemediaobject"><img src="../../../equations/bessel14.svg"></span>
1338       </p>
1339 <p>
1340         J<sub>&#957;</sub> and Y<sub>&#956;</sub> are then calculated using backward (Miller's algorithm) and forward
1341         recurrence respectively.
1342       </p>
1343 <p>
1344         When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> &lt;= 2), CF2
1345         convergence may fail (but CF1 works very well). The solution here is Temme's
1346         series:
1347       </p>
1348 <p>
1349         <span class="inlinemediaobject"><img src="../../../equations/bessel15.svg"></span>
1350       </p>
1351 <p>
1352         where
1353       </p>
1354 <p>
1355         <span class="inlinemediaobject"><img src="../../../equations/bessel16.svg"></span>
1356       </p>
1357 <p>
1358         g<sub>k</sub> &#160; and h<sub>k</sub> &#160;
1359 are also computed by recursions (involving gamma functions), but
1360         the formulas are a little complicated, readers are refered to N.M. Temme,
1361         <span class="emphasis"><em>On the numerical evaluation of the ordinary Bessel function of
1362         the second kind</em></span>, Journal of Computational Physics, vol 21, 343
1363         (1976). Note Temme's series converge only for |&#956;| &lt;= 1/2.
1364       </p>
1365 <p>
1366         As the previous case, Y<sub>&#957;</sub> &#160; is calculated from the forward recurrence, so is Y<sub>&#957;+1</sub>.
1367         With these two values and f<sub>&#957;</sub>, the Wronskian yields J<sub>&#957;</sub>(x) directly without backward
1368         recurrence.
1369       </p>
1370 </div>
1371 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1372 <td align="left"></td>
1373 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
1374       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1375       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
1376       Benjamin Sobotta, Thijs van den Berg, 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>)
1379       </p>
1380 </div></td>
1381 </tr></table>
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