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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.bessel.mbessel"></a><a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">Modified Bessel Functions
  28       of the First and Second Kinds</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.bessel.mbessel.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.bessel.mbessel.synopsis"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.synopsis">Synopsis</a>
  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_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  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_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#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_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  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_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#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.mbessel.h1"></a>
  51         <span class="phrase"><a name="math_toolkit.bessel.mbessel.description"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.description">Description</a>
  52       </h5>
  53 <p>
  54         The functions <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_i</a>
  55         and <a class="link" href="mbessel.html" title="Modified Bessel Functions of the First and Second Kinds">cyl_bessel_k</a> return
  56         the result of the modified Bessel functions of the first and second kind
  57         respectively:
  58       </p>
  59 <p>
  60         cyl_bessel_i(v, x) = I<sub>v</sub>(x)
  61       </p>
  62 <p>
  63         cyl_bessel_k(v, x) = K<sub>v</sub>(x)
  64       </p>
  65 <p>
  66         where:
  67       </p>
  68 <p>
  69         <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
  70       </p>
  71 <p>
  72         <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
  73       </p>
  74 <p>
  75         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
  76         type calculation rules</em></span></a> when T1 and T2 are different types.
  77         The functions are also optimised for the relatively common case that T1 is
  78         an integer.
  79       </p>
  80 <p>
  81         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
  82         be used to control the behaviour of the function: how it handles errors,
  83         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
  84         documentation for more details</a>.
  85       </p>
  86 <p>
  87         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  88         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>
  89         this occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;</span>
  90         <span class="number">0</span></code> and v is not an integer, or when
  91         <code class="computeroutput"><span class="identifier">x</span> <span class="special">==</span>
  92         <span class="number">0</span></code> and <code class="computeroutput"><span class="identifier">v</span>
  93         <span class="special">!=</span> <span class="number">0</span></code>.
  94         For <a class="link" href="bessel_first.html" title="Bessel Functions of the First and Second Kinds">cyl_neumann</a> this
  95         occurs when <code class="computeroutput"><span class="identifier">x</span> <span class="special">&lt;=</span>
  96         <span class="number">0</span></code>.
  97       </p>
  98 <p>
  99         The following graph illustrates the exponential behaviour of I<sub>v</sub>.
 100       </p>
 101 <p>
 102         <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_i.svg" align="middle"></span>
 103       </p>
 104 <p>
 105         The following graph illustrates the exponential decay of K<sub>v</sub>.
 106       </p>
 107 <p>
 108         <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_k.svg" align="middle"></span>
 109       </p>
 110 <h5>
 111 <a name="math_toolkit.bessel.mbessel.h2"></a>
 112         <span class="phrase"><a name="math_toolkit.bessel.mbessel.testing"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.testing">Testing</a>
 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.mbessel.h3"></a>
 121         <span class="phrase"><a name="math_toolkit.bessel.mbessel.accuracy"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.accuracy">Accuracy</a>
 122       </h5>
 123 <p>
 124         The following tables show how the accuracy of these functions varies on various
 125         platforms, along with comparison to other libraries. Note that only results
 126         for the widest floating-point type on the system are given, as narrower types
 127         have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero
 128         error</a>. All values are relative errors in units of epsilon. Note that
 129         our test suite includes some fairly extreme inputs which results in most
 130         of the worst problem cases in other libraries:
 131       </p>
 132 <div class="table">
 133 <a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table&#160;6.44.&#160;Error rates for cyl_bessel_i (integer orders)</b></p>
 134 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i (integer orders)">
 135 <colgroup>
 136 <col>
 137 <col>
 138 <col>
 139 <col>
 140 <col>
 141 </colgroup>
 142 <thead><tr>
 143 <th>
 144               </th>
 145 <th>
 146                 <p>
 147                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 148                 </p>
 149               </th>
 150 <th>
 151                 <p>
 152                   GNU C++ version 5.1.0<br> linux<br> long double
 153                 </p>
 154               </th>
 155 <th>
 156                 <p>
 157                   GNU C++ version 5.1.0<br> linux<br> double
 158                 </p>
 159               </th>
 160 <th>
 161                 <p>
 162                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 163                 </p>
 164               </th>
 165 </tr></thead>
 166 <tbody>
 167 <tr>
 168 <td>
 169                 <p>
 170                   Bessel I0: Mathworld Data (Integer Version)
 171                 </p>
 172               </td>
 173 <td>
 174                 <p>
 175                   <span class="blue">Max = 0.877&#949; (Mean = 0.549&#949;)</span>
 176                 </p>
 177               </td>
 178 <td>
 179                 <p>
 180                   <span class="blue">Max = 4.57&#949; (Mean = 2.1&#949;)</span><br> <br>
 181                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 8.49&#949; (Mean = 3.46&#949;)
 182                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_I0_Mathworld_Data_Integer_Version_">And
 183                   other failures.</a>)
 184                 </p>
 185               </td>
 186 <td>
 187                 <p>
 188                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 189                   1.16:</em></span> Max = 0.79&#949; (Mean = 0.482&#949;))<br> (<span class="emphasis"><em>Rmath
 190                   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_i_integer_orders__Rmath_3_0_2_Bessel_I0_Mathworld_Data_Integer_Version_">And
 191                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 192                   <span class="red">Max = 2.55e+43&#949; (Mean = 8.06e+42&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_I0_Mathworld_Data_Integer_Version_">And
 193                   other failures.</a>)</span>
 194                 </p>
 195               </td>
 196 <td>
 197                 <p>
 198                   <span class="blue">Max = 4.54&#949; (Mean = 2.11&#949;)</span>
 199                 </p>
 200               </td>
 201 </tr>
 202 <tr>
 203 <td>
 204                 <p>
 205                   Bessel I1: Mathworld Data (Integer Version)
 206                 </p>
 207               </td>
 208 <td>
 209                 <p>
 210                   <span class="blue">Max = 0.885&#949; (Mean = 0.55&#949;)</span>
 211                 </p>
 212               </td>
 213 <td>
 214                 <p>
 215                   <span class="blue">Max = 7.83&#949; (Mean = 2.79&#949;)</span><br> <br>
 216                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 5&#949; (Mean = 2.15&#949;)
 217                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_I1_Mathworld_Data_Integer_Version_">And
 218                   other failures.</a>)
 219                 </p>
 220               </td>
 221 <td>
 222                 <p>
 223                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 224                   1.16:</em></span> Max = 0.82&#949; (Mean = 0.456&#949;))<br> (<span class="emphasis"><em>Rmath
 225                   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_i_integer_orders__Rmath_3_0_2_Bessel_I1_Mathworld_Data_Integer_Version_">And
 226                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 227                   <span class="red">Max = 1.28e+43&#949; (Mean = 4.05e+42&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_I1_Mathworld_Data_Integer_Version_">And
 228                   other failures.</a>)</span>
 229                 </p>
 230               </td>
 231 <td>
 232                 <p>
 233                   <span class="blue">Max = 6.52&#949; (Mean = 2.25&#949;)</span>
 234                 </p>
 235               </td>
 236 </tr>
 237 <tr>
 238 <td>
 239                 <p>
 240                   Bessel In: Mathworld Data (Integer Version)
 241                 </p>
 242               </td>
 243 <td>
 244                 <p>
 245                   <span class="blue">Max = 3.46&#949; (Mean = 1.32&#949;)</span>
 246                 </p>
 247               </td>
 248 <td>
 249                 <p>
 250                   <span class="blue">Max = 1.8&#949; (Mean = 1.33&#949;)</span><br> <br>
 251                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 430&#949; (Mean = 163&#949;)
 252                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i_integer_orders___tr1_cmath__Bessel_In_Mathworld_Data_Integer_Version_">And
 253                   other failures.</a>)
 254                 </p>
 255               </td>
 256 <td>
 257                 <p>
 258                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 259                   1.16:</em></span> Max = 5.15&#949; (Mean = 2.13&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__GSL_1_16_Bessel_In_Mathworld_Data_Integer_Version_">And
 260                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 261                   <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_i_integer_orders__Rmath_3_0_2_Bessel_In_Mathworld_Data_Integer_Version_">And
 262                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 263                   <span class="red">Max = 3.67e+177&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_integer_orders__Cephes_Bessel_In_Mathworld_Data_Integer_Version_">And
 264                   other failures.</a>)</span>
 265                 </p>
 266               </td>
 267 <td>
 268                 <p>
 269                   <span class="blue">Max = 463&#949; (Mean = 140&#949;)</span>
 270                 </p>
 271               </td>
 272 </tr>
 273 </tbody>
 274 </table></div>
 275 </div>
 276 <br class="table-break"><div class="table">
 277 <a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table&#160;6.45.&#160;Error rates for cyl_bessel_i</b></p>
 278 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i">
 279 <colgroup>
 280 <col>
 281 <col>
 282 <col>
 283 <col>
 284 <col>
 285 </colgroup>
 286 <thead><tr>
 287 <th>
 288               </th>
 289 <th>
 290                 <p>
 291                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 292                 </p>
 293               </th>
 294 <th>
 295                 <p>
 296                   GNU C++ version 5.1.0<br> linux<br> long double
 297                 </p>
 298               </th>
 299 <th>
 300                 <p>
 301                   GNU C++ version 5.1.0<br> linux<br> double
 302                 </p>
 303               </th>
 304 <th>
 305                 <p>
 306                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 307                 </p>
 308               </th>
 309 </tr></thead>
 310 <tbody>
 311 <tr>
 312 <td>
 313                 <p>
 314                   Bessel I0: Mathworld Data
 315                 </p>
 316               </td>
 317 <td>
 318                 <p>
 319                   <span class="blue">Max = 0.877&#949; (Mean = 0.549&#949;)</span>
 320                 </p>
 321               </td>
 322 <td>
 323                 <p>
 324                   <span class="blue">Max = 4.57&#949; (Mean = 2.1&#949;)</span><br> <br>
 325                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 8.49&#949; (Mean = 3.46&#949;)
 326                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_I0_Mathworld_Data">And
 327                   other failures.</a>)
 328                 </p>
 329               </td>
 330 <td>
 331                 <p>
 332                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 333                   1.16:</em></span> Max = 270&#949; (Mean = 91.6&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_I0_Mathworld_Data">And
 334                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 335                   <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_i_Rmath_3_0_2_Bessel_I0_Mathworld_Data">And
 336                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 337                   <span class="red">Max = 2.55e+43&#949; (Mean = 8.06e+42&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_I0_Mathworld_Data">And
 338                   other failures.</a>)</span>
 339                 </p>
 340               </td>
 341 <td>
 342                 <p>
 343                   <span class="blue">Max = 4.54&#949; (Mean = 2.11&#949;)</span>
 344                 </p>
 345               </td>
 346 </tr>
 347 <tr>
 348 <td>
 349                 <p>
 350                   Bessel I1: Mathworld Data
 351                 </p>
 352               </td>
 353 <td>
 354                 <p>
 355                   <span class="blue">Max = 0.885&#949; (Mean = 0.55&#949;)</span>
 356                 </p>
 357               </td>
 358 <td>
 359                 <p>
 360                   <span class="blue">Max = 7.83&#949; (Mean = 2.79&#949;)</span><br> <br>
 361                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 5&#949; (Mean = 2.15&#949;)
 362                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_I1_Mathworld_Data">And
 363                   other failures.</a>)
 364                 </p>
 365               </td>
 366 <td>
 367                 <p>
 368                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 369                   1.16:</em></span> Max = 128&#949; (Mean = 41&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_I1_Mathworld_Data">And
 370                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 371                   <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_i_Rmath_3_0_2_Bessel_I1_Mathworld_Data">And
 372                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 373                   <span class="red">Max = 1.28e+43&#949; (Mean = 4.05e+42&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_I1_Mathworld_Data">And
 374                   other failures.</a>)</span>
 375                 </p>
 376               </td>
 377 <td>
 378                 <p>
 379                   <span class="blue">Max = 6.52&#949; (Mean = 2.25&#949;)</span>
 380                 </p>
 381               </td>
 382 </tr>
 383 <tr>
 384 <td>
 385                 <p>
 386                   Bessel In: Mathworld Data
 387                 </p>
 388               </td>
 389 <td>
 390                 <p>
 391                   <span class="blue">Max = 3.46&#949; (Mean = 1.32&#949;)</span>
 392                 </p>
 393               </td>
 394 <td>
 395                 <p>
 396                   <span class="blue">Max = 1.8&#949; (Mean = 1.33&#949;)</span><br> <br>
 397                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 430&#949; (Mean = 163&#949;)
 398                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_In_Mathworld_Data">And
 399                   other failures.</a>)
 400                 </p>
 401               </td>
 402 <td>
 403                 <p>
 404                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 405                   1.16:</em></span> Max = 2.31&#949; (Mean = 0.838&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_In_Mathworld_Data">And
 406                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 407                   <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_i_Rmath_3_0_2_Bessel_In_Mathworld_Data">And
 408                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 409                   <span class="red">Max = 3.67e+177&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_In_Mathworld_Data">And
 410                   other failures.</a>)</span>
 411                 </p>
 412               </td>
 413 <td>
 414                 <p>
 415                   <span class="blue">Max = 463&#949; (Mean = 140&#949;)</span>
 416                 </p>
 417               </td>
 418 </tr>
 419 <tr>
 420 <td>
 421                 <p>
 422                   Bessel Iv: Mathworld Data
 423                 </p>
 424               </td>
 425 <td>
 426                 <p>
 427                   <span class="blue">Max = 2.97&#949; (Mean = 1.33&#949;)</span>
 428                 </p>
 429               </td>
 430 <td>
 431                 <p>
 432                   <span class="blue">Max = 4.12&#949; (Mean = 1.85&#949;)</span><br> <br>
 433                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 616&#949; (Mean = 221&#949;)
 434                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_Mathworld_Data">And
 435                   other failures.</a>)
 436                 </p>
 437               </td>
 438 <td>
 439                 <p>
 440                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 441                   1.16:</em></span> Max = 5.95&#949; (Mean = 2.08&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Mathworld_Data">And
 442                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 443                   Max = 1e+04&#949; (Mean = 3.18e+03&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 444                   <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_i_Cephes_Bessel_Iv_Mathworld_Data">And
 445                   other failures.</a>)</span>
 446                 </p>
 447               </td>
 448 <td>
 449                 <p>
 450                   <span class="blue">Max = 4.12&#949; (Mean = 1.95&#949;)</span>
 451                 </p>
 452               </td>
 453 </tr>
 454 <tr>
 455 <td>
 456                 <p>
 457                   Bessel In: Random Data
 458                 </p>
 459               </td>
 460 <td>
 461                 <p>
 462                   <span class="blue">Max = 9.67&#949; (Mean = 1.89&#949;)</span>
 463                 </p>
 464               </td>
 465 <td>
 466                 <p>
 467                   <span class="blue">Max = 6.79&#949; (Mean = 1.15&#949;)</span><br> <br>
 468                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 645&#949; (Mean = 132&#949;))
 469                 </p>
 470               </td>
 471 <td>
 472                 <p>
 473                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 474                   1.16:</em></span> Max = 261&#949; (Mean = 53.2&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_In_Random_Data">And
 475                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 476                   Max = 7.37&#949; (Mean = 2.4&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span> Max
 477                   = 4.22e+06&#949; (Mean = 2.26e+05&#949;))
 478                 </p>
 479               </td>
 480 <td>
 481                 <p>
 482                   <span class="blue">Max = 176&#949; (Mean = 39.2&#949;)</span>
 483                 </p>
 484               </td>
 485 </tr>
 486 <tr>
 487 <td>
 488                 <p>
 489                   Bessel Iv: Random Data
 490                 </p>
 491               </td>
 492 <td>
 493                 <p>
 494                   <span class="blue">Max = 7.46&#949; (Mean = 1.54&#949;)</span>
 495                 </p>
 496               </td>
 497 <td>
 498                 <p>
 499                   <span class="blue">Max = 8.35&#949; (Mean = 1.49&#949;)</span><br> <br>
 500                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.05e+03&#949; (Mean =
 501                   224&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_Random_Data">And
 502                   other failures.</a>)
 503                 </p>
 504               </td>
 505 <td>
 506                 <p>
 507                   <span class="blue">Max = 0.661&#949; (Mean = 0.0441&#949;)</span><br>
 508                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 6.18e+03&#949; (Mean = 1.55e+03&#949;)
 509                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Random_Data">And
 510                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 511                   <span class="red">Max = 4.28e+08&#949; (Mean = 2.85e+07&#949;))</span><br>
 512                   (<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 6e+30&#949; (Mean
 513                   = 4e+29&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Cephes_Bessel_Iv_Random_Data">And
 514                   other failures.</a>)</span>
 515                 </p>
 516               </td>
 517 <td>
 518                 <p>
 519                   <span class="blue">Max = 283&#949; (Mean = 88.4&#949;)</span>
 520                 </p>
 521               </td>
 522 </tr>
 523 <tr>
 524 <td>
 525                 <p>
 526                   Bessel Iv: Mathworld Data (large values)
 527                 </p>
 528               </td>
 529 <td>
 530                 <p>
 531                   <span class="blue">Max = 3.67&#949; (Mean = 1.64&#949;)</span>
 532                 </p>
 533               </td>
 534 <td>
 535                 <p>
 536                   <span class="blue">Max = 14.7&#949; (Mean = 6.57&#949;)</span><br> <br>
 537                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 118&#949; (Mean = 57.2&#949;)
 538                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_i__tr1_cmath__Bessel_Iv_Mathworld_Data_large_values_">And
 539                   other failures.</a>)
 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 = 37&#949; (Mean = 18&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_GSL_1_16_Bessel_Iv_Mathworld_Data_large_values_">And
 546                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 547                   <span class="red">Max = 3.77e+168&#949; (Mean = 2.39e+168&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_i_Rmath_3_0_2_Bessel_Iv_Mathworld_Data_large_values_">And
 548                   other failures.</a>)</span><br> (<span class="emphasis"><em>Cephes:</em></span>
 549                   Max = 73.7&#949; (Mean = 58.5&#949;))
 550                 </p>
 551               </td>
 552 <td>
 553                 <p>
 554                   <span class="blue">Max = 14.7&#949; (Mean = 6.59&#949;)</span>
 555                 </p>
 556               </td>
 557 </tr>
 558 </tbody>
 559 </table></div>
 560 </div>
 561 <br class="table-break"><div class="table">
 562 <a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table&#160;6.46.&#160;Error rates for cyl_bessel_k (integer orders)</b></p>
 563 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k (integer orders)">
 564 <colgroup>
 565 <col>
 566 <col>
 567 <col>
 568 <col>
 569 <col>
 570 </colgroup>
 571 <thead><tr>
 572 <th>
 573               </th>
 574 <th>
 575                 <p>
 576                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 577                 </p>
 578               </th>
 579 <th>
 580                 <p>
 581                   GNU C++ version 5.1.0<br> linux<br> long double
 582                 </p>
 583               </th>
 584 <th>
 585                 <p>
 586                   GNU C++ version 5.1.0<br> linux<br> double
 587                 </p>
 588               </th>
 589 <th>
 590                 <p>
 591                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 592                 </p>
 593               </th>
 594 </tr></thead>
 595 <tbody>
 596 <tr>
 597 <td>
 598                 <p>
 599                   Bessel K0: Mathworld Data (Integer Version)
 600                 </p>
 601               </td>
 602 <td>
 603                 <p>
 604                   <span class="blue">Max = 1.55&#949; (Mean = 0.837&#949;)</span>
 605                 </p>
 606               </td>
 607 <td>
 608                 <p>
 609                   <span class="blue">Max = 4.16&#949; (Mean = 1.46&#949;)</span><br> <br>
 610                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 9.33&#949; (Mean = 3.25&#949;))
 611                 </p>
 612               </td>
 613 <td>
 614                 <p>
 615                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 616                   1.16:</em></span> Max = 1.2&#949; (Mean = 0.733&#949;))<br> (<span class="emphasis"><em>Rmath
 617                   3.0.2:</em></span> Max = 0.833&#949; (Mean = 0.601&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 618                   Max = 1.1e+06&#949; (Mean = 3.68e+05&#949;))
 619                 </p>
 620               </td>
 621 <td>
 622                 <p>
 623                   <span class="blue">Max = 4.16&#949; (Mean = 1.49&#949;)</span>
 624                 </p>
 625               </td>
 626 </tr>
 627 <tr>
 628 <td>
 629                 <p>
 630                   Bessel K1: Mathworld Data (Integer Version)
 631                 </p>
 632               </td>
 633 <td>
 634                 <p>
 635                   <span class="blue">Max = 1&#949; (Mean = 0.573&#949;)</span>
 636                 </p>
 637               </td>
 638 <td>
 639                 <p>
 640                   <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span><br> <br>
 641                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 8.94&#949; (Mean = 3.19&#949;))
 642                 </p>
 643               </td>
 644 <td>
 645                 <p>
 646                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 647                   1.16:</em></span> Max = 0.626&#949; (Mean = 0.333&#949;))<br> (<span class="emphasis"><em>Rmath
 648                   3.0.2:</em></span> Max = 0.894&#949; (Mean = 0.516&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 649                   Max = 5.38e+05&#949; (Mean = 1.79e+05&#949;))
 650                 </p>
 651               </td>
 652 <td>
 653                 <p>
 654                   <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
 655                 </p>
 656               </td>
 657 </tr>
 658 <tr>
 659 <td>
 660                 <p>
 661                   Bessel Kn: Mathworld Data (Integer Version)
 662                 </p>
 663               </td>
 664 <td>
 665                 <p>
 666                   <span class="blue">Max = 3.63&#949; (Mean = 1.46&#949;)</span>
 667                 </p>
 668               </td>
 669 <td>
 670                 <p>
 671                   <span class="blue">Max = 4.48&#949; (Mean = 2.14&#949;)</span><br> <br>
 672                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 12.9&#949; (Mean = 4.91&#949;)
 673                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k_integer_orders___tr1_cmath__Bessel_Kn_Mathworld_Data_Integer_Version_">And
 674                   other failures.</a>)
 675                 </p>
 676               </td>
 677 <td>
 678                 <p>
 679                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 680                   1.16:</em></span> Max = 168&#949; (Mean = 59.5&#949;))<br> (<span class="emphasis"><em>Rmath
 681                   3.0.2:</em></span> Max = 8.48&#949; (Mean = 2.98&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 682                   <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_k_integer_orders__Cephes_Bessel_Kn_Mathworld_Data_Integer_Version_">And
 683                   other failures.</a>)</span>
 684                 </p>
 685               </td>
 686 <td>
 687                 <p>
 688                   <span class="blue">Max = 4.48&#949; (Mean = 1.98&#949;)</span>
 689                 </p>
 690               </td>
 691 </tr>
 692 </tbody>
 693 </table></div>
 694 </div>
 695 <br class="table-break"><div class="table">
 696 <a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table&#160;6.47.&#160;Error rates for cyl_bessel_k</b></p>
 697 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k">
 698 <colgroup>
 699 <col>
 700 <col>
 701 <col>
 702 <col>
 703 <col>
 704 </colgroup>
 705 <thead><tr>
 706 <th>
 707               </th>
 708 <th>
 709                 <p>
 710                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 711                 </p>
 712               </th>
 713 <th>
 714                 <p>
 715                   GNU C++ version 5.1.0<br> linux<br> long double
 716                 </p>
 717               </th>
 718 <th>
 719                 <p>
 720                   GNU C++ version 5.1.0<br> linux<br> double
 721                 </p>
 722               </th>
 723 <th>
 724                 <p>
 725                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 726                 </p>
 727               </th>
 728 </tr></thead>
 729 <tbody>
 730 <tr>
 731 <td>
 732                 <p>
 733                   Bessel K0: Mathworld Data
 734                 </p>
 735               </td>
 736 <td>
 737                 <p>
 738                   <span class="blue">Max = 1.55&#949; (Mean = 0.837&#949;)</span>
 739                 </p>
 740               </td>
 741 <td>
 742                 <p>
 743                   <span class="blue">Max = 4.16&#949; (Mean = 1.46&#949;)</span><br> <br>
 744                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 9.33&#949; (Mean = 3.25&#949;))
 745                 </p>
 746               </td>
 747 <td>
 748                 <p>
 749                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 750                   1.16:</em></span> Max = 6.04&#949; (Mean = 2.16&#949;))<br> (<span class="emphasis"><em>Rmath
 751                   3.0.2:</em></span> Max = 0.833&#949; (Mean = 0.601&#949;))
 752                 </p>
 753               </td>
 754 <td>
 755                 <p>
 756                   <span class="blue">Max = 4.16&#949; (Mean = 1.49&#949;)</span>
 757                 </p>
 758               </td>
 759 </tr>
 760 <tr>
 761 <td>
 762                 <p>
 763                   Bessel K1: Mathworld Data
 764                 </p>
 765               </td>
 766 <td>
 767                 <p>
 768                   <span class="blue">Max = 1&#949; (Mean = 0.573&#949;)</span>
 769                 </p>
 770               </td>
 771 <td>
 772                 <p>
 773                   <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span><br> <br>
 774                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 8.94&#949; (Mean = 3.19&#949;))
 775                 </p>
 776               </td>
 777 <td>
 778                 <p>
 779                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 780                   1.16:</em></span> Max = 6.26&#949; (Mean = 2.21&#949;))<br> (<span class="emphasis"><em>Rmath
 781                   3.0.2:</em></span> Max = 0.894&#949; (Mean = 0.516&#949;))
 782                 </p>
 783               </td>
 784 <td>
 785                 <p>
 786                   <span class="blue">Max = 1.8&#949; (Mean = 1.02&#949;)</span>
 787                 </p>
 788               </td>
 789 </tr>
 790 <tr>
 791 <td>
 792                 <p>
 793                   Bessel Kn: Mathworld Data
 794                 </p>
 795               </td>
 796 <td>
 797                 <p>
 798                   <span class="blue">Max = 3.63&#949; (Mean = 1.46&#949;)</span>
 799                 </p>
 800               </td>
 801 <td>
 802                 <p>
 803                   <span class="blue">Max = 4.48&#949; (Mean = 2.14&#949;)</span><br> <br>
 804                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 12.9&#949; (Mean = 4.91&#949;)
 805                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_cmath__Bessel_Kn_Mathworld_Data">And
 806                   other failures.</a>)
 807                 </p>
 808               </td>
 809 <td>
 810                 <p>
 811                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 812                   1.16:</em></span> Max = 3.36&#949; (Mean = 1.43&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kn_Mathworld_Data">And
 813                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 814                   Max = 8.48&#949; (Mean = 2.98&#949;))
 815                 </p>
 816               </td>
 817 <td>
 818                 <p>
 819                   <span class="blue">Max = 4.48&#949; (Mean = 1.98&#949;)</span>
 820                 </p>
 821               </td>
 822 </tr>
 823 <tr>
 824 <td>
 825                 <p>
 826                   Bessel Kv: Mathworld Data
 827                 </p>
 828               </td>
 829 <td>
 830                 <p>
 831                   <span class="blue">Max = 4.78&#949; (Mean = 2.2&#949;)</span>
 832                 </p>
 833               </td>
 834 <td>
 835                 <p>
 836                   <span class="blue">Max = 3.58&#949; (Mean = 2.44&#949;)</span><br> <br>
 837                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 13&#949; (Mean = 4.81&#949;)
 838                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_cmath__Bessel_Kv_Mathworld_Data">And
 839                   other failures.</a>)
 840                 </p>
 841               </td>
 842 <td>
 843                 <p>
 844                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 845                   1.16:</em></span> Max = 5.47&#949; (Mean = 2.04&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Mathworld_Data">And
 846                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 847                   Max = 3.15&#949; (Mean = 1.35&#949;))
 848                 </p>
 849               </td>
 850 <td>
 851                 <p>
 852                   <span class="blue">Max = 3.58&#949; (Mean = 2.29&#949;)</span>
 853                 </p>
 854               </td>
 855 </tr>
 856 <tr>
 857 <td>
 858                 <p>
 859                   Bessel Kv: Mathworld Data (large values)
 860                 </p>
 861               </td>
 862 <td>
 863                 <p>
 864                   <span class="blue">Max = 59.8&#949; (Mean = 26.9&#949;)</span>
 865                 </p>
 866               </td>
 867 <td>
 868                 <p>
 869                   <span class="blue">Max = 42.3&#949; (Mean = 21&#949;)</span><br> <br>
 870                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 42.3&#949; (Mean = 19.8&#949;)
 871                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_cmath__Bessel_Kv_Mathworld_Data_large_values_">And
 872                   other failures.</a>)
 873                 </p>
 874               </td>
 875 <td>
 876                 <p>
 877                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 878                   1.16:</em></span> Max = 308&#949; (Mean = 142&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Mathworld_Data_large_values_">And
 879                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 880                   Max = 84.6&#949; (Mean = 37.8&#949;))
 881                 </p>
 882               </td>
 883 <td>
 884                 <p>
 885                   <span class="blue">Max = 43.1&#949; (Mean = 21.3&#949;)</span>
 886                 </p>
 887               </td>
 888 </tr>
 889 <tr>
 890 <td>
 891                 <p>
 892                   Bessel Kn: Random Data
 893                 </p>
 894               </td>
 895 <td>
 896                 <p>
 897                   <span class="blue">Max = 7.47&#949; (Mean = 1.4&#949;)</span>
 898                 </p>
 899               </td>
 900 <td>
 901                 <p>
 902                   <span class="blue">Max = 4.55&#949; (Mean = 1.09&#949;)</span><br> <br>
 903                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 13.9&#949; (Mean = 2.91&#949;))
 904                 </p>
 905               </td>
 906 <td>
 907                 <p>
 908                   <span class="blue">Max = 0.764&#949; (Mean = 0.0348&#949;)</span><br>
 909                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 8.71&#949; (Mean = 1.76&#949;)
 910                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kn_Random_Data">And
 911                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 912                   Max = 7.47&#949; (Mean = 1.34&#949;))
 913                 </p>
 914               </td>
 915 <td>
 916                 <p>
 917                   <span class="blue">Max = 4.55&#949; (Mean = 1.21&#949;)</span>
 918                 </p>
 919               </td>
 920 </tr>
 921 <tr>
 922 <td>
 923                 <p>
 924                   Bessel Kv: Random Data
 925                 </p>
 926               </td>
 927 <td>
 928                 <p>
 929                   <span class="blue">Max = 8.33&#949; (Mean = 1.62&#949;)</span>
 930                 </p>
 931               </td>
 932 <td>
 933                 <p>
 934                   <span class="blue">Max = 7.88&#949; (Mean = 1.48&#949;)</span><br> <br>
 935                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 13.6&#949; (Mean = 2.68&#949;)
 936                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_long_double_cyl_bessel_k__tr1_cmath__Bessel_Kv_Random_Data">And
 937                   other failures.</a>)
 938                 </p>
 939               </td>
 940 <td>
 941                 <p>
 942                   <span class="blue">Max = 0.507&#949; (Mean = 0.0313&#949;)</span><br>
 943                   <br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 9.71&#949; (Mean = 1.47&#949;)
 944                   <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_cyl_bessel_k_GSL_1_16_Bessel_Kv_Random_Data">And
 945                   other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span>
 946                   Max = 7.37&#949; (Mean = 1.49&#949;))
 947                 </p>
 948               </td>
 949 <td>
 950                 <p>
 951                   <span class="blue">Max = 7.88&#949; (Mean = 1.49&#949;)</span>
 952                 </p>
 953               </td>
 954 </tr>
 955 </tbody>
 956 </table></div>
 957 </div>
 958 <br class="table-break"><h5>
 959 <a name="math_toolkit.bessel.mbessel.h4"></a>
 960         <span class="phrase"><a name="math_toolkit.bessel.mbessel.implementation"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.implementation">Implementation</a>
 961       </h5>
 962 <p>
 963         The following are handled as special cases first:
 964       </p>
 965 <p>
 966         When computing I<sub>v</sub> &#160; for <span class="emphasis"><em>x &lt; 0</em></span>, then &#957; &#160; must be an integer
 967         or a domain error occurs. If &#957; &#160; is an integer, then the function is odd if &#957; &#160; is
 968         odd and even if &#957; &#160; is even, and we can reflect to <span class="emphasis"><em>x &gt; 0</em></span>.
 969       </p>
 970 <p>
 971         For I<sub>v</sub> &#160; with v equal to 0, 1 or 0.5 are handled as special cases.
 972       </p>
 973 <p>
 974         The 0 and 1 cases use minimax rational approximations on finite and infinite
 975         intervals. The coefficients are from:
 976       </p>
 977 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 978 <li class="listitem">
 979             J.M. Blair and C.A. Edwards, <span class="emphasis"><em>Stable rational minimax approximations
 980             to the modified Bessel functions I_0(x) and I_1(x)</em></span>, Atomic
 981             Energy of Canada Limited Report 4928, Chalk River, 1974.
 982           </li>
 983 <li class="listitem">
 984             S. Moshier, <span class="emphasis"><em>Methods and Programs for Mathematical Functions</em></span>,
 985             Ellis Horwood Ltd, Chichester, 1989.
 986           </li>
 987 </ul></div>
 988 <p>
 989         While the 0.5 case is a simple trigonometric function:
 990       </p>
 991 <p>
 992         I<sub>0.5</sub>(x) = sqrt(2 / &#960;x) * sinh(x)
 993       </p>
 994 <p>
 995         For K<sub>v</sub> &#160; with <span class="emphasis"><em>v</em></span> an integer, the result is calculated using
 996         the recurrence relation:
 997       </p>
 998 <p>
 999         <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
1000       </p>
1001 <p>
1002         starting from K<sub>0</sub> &#160; and K<sub>1</sub> &#160; which are calculated using rational the approximations
1003         above. These rational approximations are accurate to around 19 digits, and
1004         are therefore only used when T has no more than 64 binary digits of precision.
1005       </p>
1006 <p>
1007         When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
1008         I<sub>v</sub>x &#160; is best computed directly from the series:
1009       </p>
1010 <p>
1011         <span class="inlinemediaobject"><img src="../../../equations/mbessel17.svg"></span>
1012       </p>
1013 <p>
1014         In the general case, we first normalize &#957; &#160; to [<code class="literal">0, [inf]</code>)
1015         with the help of the reflection formulae:
1016       </p>
1017 <p>
1018         <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
1019       </p>
1020 <p>
1021         <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
1022       </p>
1023 <p>
1024         Let &#956; &#160; = &#957; - floor(&#957; + 1/2), then &#956; &#160; is the fractional part of &#957; &#160; such that |&#956;| &lt;= 1/2
1025         (we need this for convergence later). The idea is to calculate K<sub>&#956;</sub>(x) and K<sub>&#956;+1</sub>(x),
1026         and use them to obtain I<sub>&#957;</sub>(x) and K<sub>&#957;</sub>(x).
1027       </p>
1028 <p>
1029         The algorithm is proposed by Temme in N.M. Temme, <span class="emphasis"><em>On the numerical
1030         evaluation of the modified bessel function of the third kind</em></span>,
1031         Journal of Computational Physics, vol 19, 324 (1975), which needs two continued
1032         fractions as well as the Wronskian:
1033       </p>
1034 <p>
1035         <span class="inlinemediaobject"><img src="../../../equations/mbessel11.svg"></span>
1036       </p>
1037 <p>
1038         <span class="inlinemediaobject"><img src="../../../equations/mbessel12.svg"></span>
1039       </p>
1040 <p>
1041         <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
1042       </p>
1043 <p>
1044         The continued fractions are computed using the modified Lentz's method (W.J.
1045         Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
1046         using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
1047         Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need
1048         different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>.
1049       </p>
1050 <p>
1051         <span class="emphasis"><em>x &gt; v</em></span>, CF1 needs O(<span class="emphasis"><em>x</em></span>) iterations
1052         to converge, CF2 converges rapidly.
1053       </p>
1054 <p>
1055         <span class="emphasis"><em>x &lt;= v</em></span>, CF1 converges rapidly, CF2 fails to converge
1056         when <span class="emphasis"><em>x</em></span> <code class="literal">-&gt;</code> 0.
1057       </p>
1058 <p>
1059         When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> &gt; 2), both
1060         continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
1061         K<sub>&#956;</sub> &#160; and K<sub>&#956;+1</sub> &#160;
1062 can be calculated by
1063       </p>
1064 <p>
1065         <span class="inlinemediaobject"><img src="../../../equations/mbessel13.svg"></span>
1066       </p>
1067 <p>
1068         where
1069       </p>
1070 <p>
1071         <span class="inlinemediaobject"><img src="../../../equations/mbessel14.svg"></span>
1072       </p>
1073 <p>
1074         <span class="emphasis"><em>S</em></span> is also a series that is summed along with CF2, see
1075         I.J. Thompson and A.R. Barnett, <span class="emphasis"><em>Modified Bessel functions I_v and
1076         K_v of real order and complex argument to selected accuracy</em></span>, Computer
1077         Physics Communications, vol 47, 245 (1987).
1078       </p>
1079 <p>
1080         When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> &lt;= 2), CF2
1081         convergence may fail (but CF1 works very well). The solution here is Temme's
1082         series:
1083       </p>
1084 <p>
1085         <span class="inlinemediaobject"><img src="../../../equations/mbessel15.svg"></span>
1086       </p>
1087 <p>
1088         where
1089       </p>
1090 <p>
1091         <span class="inlinemediaobject"><img src="../../../equations/mbessel16.svg"></span>
1092       </p>
1093 <p>
1094         f<sub>k</sub> &#160; and h<sub>k</sub> &#160;
1095 are also computed by recursions (involving gamma functions), but
1096         the formulas are a little complicated, readers are referred to N.M. Temme,
1097         <span class="emphasis"><em>On the numerical evaluation of the modified Bessel function of
1098         the third kind</em></span>, Journal of Computational Physics, vol 19, 324
1099         (1975). Note: Temme's series converge only for |&#956;| &lt;= 1/2.
1100       </p>
1101 <p>
1102         K<sub>&#957;</sub>(x) is then calculated from the forward recurrence, as is K<sub>&#957;+1</sub>(x). With these
1103         two values and f<sub>&#957;</sub>, the Wronskian yields I<sub>&#957;</sub>(x) directly.
1104       </p>
1105 </div>
1106 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1107 <td align="left"></td>
1108 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
1109       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
1110       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
1111       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
1112         Distributed under the Boost Software License, Version 1.0. (See accompanying
1113         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>)
1114       </p>
1115 </div></td>
1116 </tr></table>
1117 <hr>
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