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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.bessel.bessel_derivatives"></a><a class="link" href="bessel_derivatives.html" title="Derivatives of the Bessel Functions">Derivatives of
  28       the Bessel Functions</a>
  29 </h3></div></div></div>
  30 <h5>
  31 <a name="math_toolkit.bessel.bessel_derivatives.h0"></a>
  32         <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.synopsis"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.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_prime</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_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  39
  40 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  41 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_j_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  42
  43 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
  44 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  48
  49 <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>
  50 <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_prime</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>
  51
  52 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  53 <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_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  54
  55 <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>
  56 <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_prime</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>
  57
  58 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  59 <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_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  60
  61 <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>
  62 <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">sph_bessel_prime</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>
  63
  64 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  65 <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">sph_bessel_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  66
  67 <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>
  68 <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">sph_neumann_prime</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>
  69
  70 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  71 <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">sph_neumann_prime</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  72 </pre>
  73 <h5>
  74 <a name="math_toolkit.bessel.bessel_derivatives.h1"></a>
  75         <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.description"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.description">Description</a>
  76       </h5>
  77 <p>
  78         These functions return the first derivative with respect to <span class="emphasis"><em>x</em></span>
  79         of the corresponding Bessel function.
  80       </p>
  81 <p>
  82         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
  83         type calculation rules</em></span></a> when T1 and T2 are different types.
  84         The functions are also optimised for the relatively common case that T1 is
  85         an integer.
  86       </p>
  87 <p>
  88         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  89         be used to control the behaviour of the function: how it handles errors,
  90         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
  91         documentation for more details</a>.
  92       </p>
  93 <p>
  94         The functions return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  95         whenever the result is undefined or complex.
  96       </p>
  97 <h5>
  98 <a name="math_toolkit.bessel.bessel_derivatives.h2"></a>
  99         <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.testing"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.testing">Testing</a>
 100       </h5>
 101 <p>
 102         There are two sets of test values: spot values calculated using <a href="http://www.wolframalpha.com/" target="_top">wolframalpha.com</a>,
 103         and a much larger set of tests computed using a relation to the underlying
 104         Bessel functions that the implementation does not use.
 105       </p>
 106 <h5>
 107 <a name="math_toolkit.bessel.bessel_derivatives.h3"></a>
 108         <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.accuracy"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.accuracy">Accuracy</a>
 109       </h5>
 110 <p>
 111         The accuracy of these functions is broadly similar to the underlying Bessel
 112         functions.
 113       </p>
 114 <div class="table">
 115 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime_integer_orders_"></a><p class="title"><b>Table&#160;8.50.&#160;Error rates for cyl_bessel_i_prime (integer orders)</b></p>
 116 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i_prime (integer orders)">
 117 <colgroup>
 118 <col>
 119 <col>
 120 <col>
 121 <col>
 122 <col>
 123 </colgroup>
 124 <thead><tr>
 125 <th>
 126               </th>
 127 <th>
 128                 <p>
 129                   GNU C++ version 7.1.0<br> linux<br> double
 130                 </p>
 131               </th>
 132 <th>
 133                 <p>
 134                   GNU C++ version 7.1.0<br> linux<br> long double
 135                 </p>
 136               </th>
 137 <th>
 138                 <p>
 139                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 140                 </p>
 141               </th>
 142 <th>
 143                 <p>
 144                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 145                 </p>
 146               </th>
 147 </tr></thead>
 148 <tbody>
 149 <tr>
 150 <td>
 151                 <p>
 152                   Bessel I'0: Mathworld Data (Integer Version)
 153                 </p>
 154               </td>
 155 <td>
 156                 <p>
 157                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 158                 </p>
 159               </td>
 160 <td>
 161                 <p>
 162                   <span class="blue">Max = 0.82&#949; (Mean = 0.259&#949;)</span>
 163                 </p>
 164               </td>
 165 <td>
 166                 <p>
 167                   <span class="blue">Max = 0.82&#949; (Mean = 0.259&#949;)</span>
 168                 </p>
 169               </td>
 170 <td>
 171                 <p>
 172                   <span class="blue">Max = 0.82&#949; (Mean = 0.354&#949;)</span>
 173                 </p>
 174               </td>
 175 </tr>
 176 <tr>
 177 <td>
 178                 <p>
 179                   Bessel I'1: Mathworld Data (Integer Version)
 180                 </p>
 181               </td>
 182 <td>
 183                 <p>
 184                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   <span class="blue">Max = 1.97&#949; (Mean = 0.757&#949;)</span>
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 1.97&#949; (Mean = 0.757&#949;)</span>
 195                 </p>
 196               </td>
 197 <td>
 198                 <p>
 199                   <span class="blue">Max = 1.36&#949; (Mean = 0.782&#949;)</span>
 200                 </p>
 201               </td>
 202 </tr>
 203 <tr>
 204 <td>
 205                 <p>
 206                   Bessel I'n: Mathworld Data (Integer Version)
 207                 </p>
 208               </td>
 209 <td>
 210                 <p>
 211                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 212                 </p>
 213               </td>
 214 <td>
 215                 <p>
 216                   <span class="blue">Max = 2.31&#949; (Mean = 1.41&#949;)</span>
 217                 </p>
 218               </td>
 219 <td>
 220                 <p>
 221                   <span class="blue">Max = 701&#949; (Mean = 212&#949;)</span>
 222                 </p>
 223               </td>
 224 <td>
 225                 <p>
 226                   <span class="blue">Max = 3.61&#949; (Mean = 1.22&#949;)</span>
 227                 </p>
 228               </td>
 229 </tr>
 230 </tbody>
 231 </table></div>
 232 </div>
 233 <br class="table-break"><div class="table">
 234 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_i_prime"></a><p class="title"><b>Table&#160;8.51.&#160;Error rates for cyl_bessel_i_prime</b></p>
 235 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i_prime">
 236 <colgroup>
 237 <col>
 238 <col>
 239 <col>
 240 <col>
 241 <col>
 242 </colgroup>
 243 <thead><tr>
 244 <th>
 245               </th>
 246 <th>
 247                 <p>
 248                   GNU C++ version 7.1.0<br> linux<br> double
 249                 </p>
 250               </th>
 251 <th>
 252                 <p>
 253                   GNU C++ version 7.1.0<br> linux<br> long double
 254                 </p>
 255               </th>
 256 <th>
 257                 <p>
 258                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 259                 </p>
 260               </th>
 261 <th>
 262                 <p>
 263                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 264                 </p>
 265               </th>
 266 </tr></thead>
 267 <tbody>
 268 <tr>
 269 <td>
 270                 <p>
 271                   Bessel I'0: Mathworld Data
 272                 </p>
 273               </td>
 274 <td>
 275                 <p>
 276                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 277                 </p>
 278               </td>
 279 <td>
 280                 <p>
 281                   <span class="blue">Max = 0.82&#949; (Mean = 0.259&#949;)</span>
 282                 </p>
 283               </td>
 284 <td>
 285                 <p>
 286                   <span class="blue">Max = 0.82&#949; (Mean = 0.259&#949;)</span>
 287                 </p>
 288               </td>
 289 <td>
 290                 <p>
 291                   <span class="blue">Max = 0.82&#949; (Mean = 0.354&#949;)</span>
 292                 </p>
 293               </td>
 294 </tr>
 295 <tr>
 296 <td>
 297                 <p>
 298                   Bessel I'1: Mathworld Data
 299                 </p>
 300               </td>
 301 <td>
 302                 <p>
 303                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 304                 </p>
 305               </td>
 306 <td>
 307                 <p>
 308                   <span class="blue">Max = 1.97&#949; (Mean = 0.757&#949;)</span>
 309                 </p>
 310               </td>
 311 <td>
 312                 <p>
 313                   <span class="blue">Max = 1.97&#949; (Mean = 0.757&#949;)</span>
 314                 </p>
 315               </td>
 316 <td>
 317                 <p>
 318                   <span class="blue">Max = 1.36&#949; (Mean = 0.782&#949;)</span>
 319                 </p>
 320               </td>
 321 </tr>
 322 <tr>
 323 <td>
 324                 <p>
 325                   Bessel I'n: Mathworld Data
 326                 </p>
 327               </td>
 328 <td>
 329                 <p>
 330                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 331                 </p>
 332               </td>
 333 <td>
 334                 <p>
 335                   <span class="blue">Max = 2.31&#949; (Mean = 1.41&#949;)</span>
 336                 </p>
 337               </td>
 338 <td>
 339                 <p>
 340                   <span class="blue">Max = 701&#949; (Mean = 212&#949;)</span>
 341                 </p>
 342               </td>
 343 <td>
 344                 <p>
 345                   <span class="blue">Max = 3.61&#949; (Mean = 1.22&#949;)</span>
 346                 </p>
 347               </td>
 348 </tr>
 349 <tr>
 350 <td>
 351                 <p>
 352                   Bessel I'v: Mathworld Data
 353                 </p>
 354               </td>
 355 <td>
 356                 <p>
 357                   <span class="blue">Max = 1.62&#949; (Mean = 0.512&#949;)</span>
 358                 </p>
 359               </td>
 360 <td>
 361                 <p>
 362                   <span class="blue">Max = 2.89e+03&#949; (Mean = 914&#949;)</span>
 363                 </p>
 364               </td>
 365 <td>
 366                 <p>
 367                   <span class="blue">Max = 2.89e+03&#949; (Mean = 914&#949;)</span>
 368                 </p>
 369               </td>
 370 <td>
 371                 <p>
 372                   <span class="blue">Max = 3.76e+03&#949; (Mean = 1.19e+03&#949;)</span>
 373                 </p>
 374               </td>
 375 </tr>
 376 <tr>
 377 <td>
 378                 <p>
 379                   Bessel I'n: Random Data
 380                 </p>
 381               </td>
 382 <td>
 383                 <p>
 384                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 385                 </p>
 386               </td>
 387 <td>
 388                 <p>
 389                   <span class="blue">Max = 3.95&#949; (Mean = 1.06&#949;)</span>
 390                 </p>
 391               </td>
 392 <td>
 393                 <p>
 394                   <span class="blue">Max = 195&#949; (Mean = 37.1&#949;)</span>
 395                 </p>
 396               </td>
 397 <td>
 398                 <p>
 399                   <span class="blue">Max = 9.85&#949; (Mean = 1.82&#949;)</span>
 400                 </p>
 401               </td>
 402 </tr>
 403 <tr>
 404 <td>
 405                 <p>
 406                   Bessel I'v: Random Data
 407                 </p>
 408               </td>
 409 <td>
 410                 <p>
 411                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 412                 </p>
 413               </td>
 414 <td>
 415                 <p>
 416                   <span class="blue">Max = 14.1&#949; (Mean = 2.93&#949;)</span>
 417                 </p>
 418               </td>
 419 <td>
 420                 <p>
 421                   <span class="blue">Max = 336&#949; (Mean = 68.5&#949;)</span>
 422                 </p>
 423               </td>
 424 <td>
 425                 <p>
 426                   <span class="blue">Max = 14&#949; (Mean = 2.5&#949;)</span>
 427                 </p>
 428               </td>
 429 </tr>
 430 <tr>
 431 <td>
 432                 <p>
 433                   Bessel I'v: Mathworld Data (large values)
 434                 </p>
 435               </td>
 436 <td>
 437                 <p>
 438                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 439                 </p>
 440               </td>
 441 <td>
 442                 <p>
 443                   <span class="blue">Max = 42.6&#949; (Mean = 20.2&#949;)</span>
 444                 </p>
 445               </td>
 446 <td>
 447                 <p>
 448                   <span class="blue">Max = 42.6&#949; (Mean = 20.2&#949;)</span>
 449                 </p>
 450               </td>
 451 <td>
 452                 <p>
 453                   <span class="blue">Max = 59.5&#949; (Mean = 26.6&#949;)</span>
 454                 </p>
 455               </td>
 456 </tr>
 457 </tbody>
 458 </table></div>
 459 </div>
 460 <br class="table-break"><div class="table">
 461 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime_integer_orders_"></a><p class="title"><b>Table&#160;8.52.&#160;Error rates for cyl_bessel_j_prime (integer orders)</b></p>
 462 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j_prime (integer orders)">
 463 <colgroup>
 464 <col>
 465 <col>
 466 <col>
 467 <col>
 468 <col>
 469 </colgroup>
 470 <thead><tr>
 471 <th>
 472               </th>
 473 <th>
 474                 <p>
 475                   GNU C++ version 7.1.0<br> linux<br> double
 476                 </p>
 477               </th>
 478 <th>
 479                 <p>
 480                   GNU C++ version 7.1.0<br> linux<br> long double
 481                 </p>
 482               </th>
 483 <th>
 484                 <p>
 485                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 486                 </p>
 487               </th>
 488 <th>
 489                 <p>
 490                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 491                 </p>
 492               </th>
 493 </tr></thead>
 494 <tbody>
 495 <tr>
 496 <td>
 497                 <p>
 498                   Bessel J0': Mathworld Data (Integer Version)
 499                 </p>
 500               </td>
 501 <td>
 502                 <p>
 503                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 504                 </p>
 505               </td>
 506 <td>
 507                 <p>
 508                   <span class="blue">Max = 18.9&#949; (Mean = 6.82&#949;)</span>
 509                 </p>
 510               </td>
 511 <td>
 512                 <p>
 513                   <span class="blue">Max = 18.9&#949; (Mean = 6.72&#949;)</span>
 514                 </p>
 515               </td>
 516 <td>
 517                 <p>
 518                   <span class="blue">Max = 6.62&#949; (Mean = 2.55&#949;)</span>
 519                 </p>
 520               </td>
 521 </tr>
 522 <tr>
 523 <td>
 524                 <p>
 525                   Bessel J0': Mathworld Data (Tricky cases) (Integer Version)
 526                 </p>
 527               </td>
 528 <td>
 529                 <p>
 530                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 531                 </p>
 532               </td>
 533 <td>
 534                 <p>
 535                   <span class="blue">Max = 7.44&#949; (Mean = 3.34&#949;)</span>
 536                 </p>
 537               </td>
 538 <td>
 539                 <p>
 540                   <span class="blue">Max = 7.44&#949; (Mean = 3.31&#949;)</span>
 541                 </p>
 542               </td>
 543 <td>
 544                 <p>
 545                   <span class="blue">Max = 3.67&#949; (Mean = 1.74&#949;)</span>
 546                 </p>
 547               </td>
 548 </tr>
 549 <tr>
 550 <td>
 551                 <p>
 552                   Bessel J1': Mathworld Data (Integer Version)
 553                 </p>
 554               </td>
 555 <td>
 556                 <p>
 557                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 558                 </p>
 559               </td>
 560 <td>
 561                 <p>
 562                   <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
 563                 </p>
 564               </td>
 565 <td>
 566                 <p>
 567                   <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
 568                 </p>
 569               </td>
 570 <td>
 571                 <p>
 572                   <span class="blue">Max = 0.999&#949; (Mean = 0.627&#949;)</span>
 573                 </p>
 574               </td>
 575 </tr>
 576 <tr>
 577 <td>
 578                 <p>
 579                   Bessel J1': Mathworld Data (tricky cases) (Integer Version)
 580                 </p>
 581               </td>
 582 <td>
 583                 <p>
 584                   <span class="blue">Max = 287&#949; (Mean = 129&#949;)</span>
 585                 </p>
 586               </td>
 587 <td>
 588                 <p>
 589                   <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
 590                 </p>
 591               </td>
 592 <td>
 593                 <p>
 594                   <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
 595                 </p>
 596               </td>
 597 <td>
 598                 <p>
 599                   <span class="blue">Max = 288&#949; (Mean = 129&#949;)</span>
 600                 </p>
 601               </td>
 602 </tr>
 603 <tr>
 604 <td>
 605                 <p>
 606                   Bessel JN': Mathworld Data (Integer Version)
 607                 </p>
 608               </td>
 609 <td>
 610                 <p>
 611                   <span class="blue">Max = 0.527&#949; (Mean = 0.128&#949;)</span>
 612                 </p>
 613               </td>
 614 <td>
 615                 <p>
 616                   <span class="blue">Max = 1.29e+03&#949; (Mean = 312&#949;)</span>
 617                 </p>
 618               </td>
 619 <td>
 620                 <p>
 621                   <span class="blue">Max = 1.29e+03&#949; (Mean = 355&#949;)</span>
 622                 </p>
 623               </td>
 624 <td>
 625                 <p>
 626                   <span class="blue">Max = 14&#949; (Mean = 6.13&#949;)</span>
 627                 </p>
 628               </td>
 629 </tr>
 630 </tbody>
 631 </table></div>
 632 </div>
 633 <br class="table-break"><div class="table">
 634 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_j_prime"></a><p class="title"><b>Table&#160;8.53.&#160;Error rates for cyl_bessel_j_prime</b></p>
 635 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_j_prime">
 636 <colgroup>
 637 <col>
 638 <col>
 639 <col>
 640 <col>
 641 <col>
 642 </colgroup>
 643 <thead><tr>
 644 <th>
 645               </th>
 646 <th>
 647                 <p>
 648                   GNU C++ version 7.1.0<br> linux<br> double
 649                 </p>
 650               </th>
 651 <th>
 652                 <p>
 653                   GNU C++ version 7.1.0<br> linux<br> long double
 654                 </p>
 655               </th>
 656 <th>
 657                 <p>
 658                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 659                 </p>
 660               </th>
 661 <th>
 662                 <p>
 663                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 664                 </p>
 665               </th>
 666 </tr></thead>
 667 <tbody>
 668 <tr>
 669 <td>
 670                 <p>
 671                   Bessel J0': Mathworld Data
 672                 </p>
 673               </td>
 674 <td>
 675                 <p>
 676                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 677                 </p>
 678               </td>
 679 <td>
 680                 <p>
 681                   <span class="blue">Max = 18.9&#949; (Mean = 6.82&#949;)</span>
 682                 </p>
 683               </td>
 684 <td>
 685                 <p>
 686                   <span class="blue">Max = 18.9&#949; (Mean = 6.72&#949;)</span>
 687                 </p>
 688               </td>
 689 <td>
 690                 <p>
 691                   <span class="blue">Max = 6.62&#949; (Mean = 2.55&#949;)</span>
 692                 </p>
 693               </td>
 694 </tr>
 695 <tr>
 696 <td>
 697                 <p>
 698                   Bessel J0': Mathworld Data (Tricky cases)
 699                 </p>
 700               </td>
 701 <td>
 702                 <p>
 703                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 704                 </p>
 705               </td>
 706 <td>
 707                 <p>
 708                   <span class="blue">Max = 7.44&#949; (Mean = 3.34&#949;)</span>
 709                 </p>
 710               </td>
 711 <td>
 712                 <p>
 713                   <span class="blue">Max = 7.44&#949; (Mean = 3.31&#949;)</span>
 714                 </p>
 715               </td>
 716 <td>
 717                 <p>
 718                   <span class="blue">Max = 3.67&#949; (Mean = 1.74&#949;)</span>
 719                 </p>
 720               </td>
 721 </tr>
 722 <tr>
 723 <td>
 724                 <p>
 725                   Bessel J1': Mathworld Data
 726                 </p>
 727               </td>
 728 <td>
 729                 <p>
 730                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 731                 </p>
 732               </td>
 733 <td>
 734                 <p>
 735                   <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
 736                 </p>
 737               </td>
 738 <td>
 739                 <p>
 740                   <span class="blue">Max = 7.9&#949; (Mean = 3.37&#949;)</span>
 741                 </p>
 742               </td>
 743 <td>
 744                 <p>
 745                   <span class="blue">Max = 0.999&#949; (Mean = 0.627&#949;)</span>
 746                 </p>
 747               </td>
 748 </tr>
 749 <tr>
 750 <td>
 751                 <p>
 752                   Bessel J1': Mathworld Data (tricky cases)
 753                 </p>
 754               </td>
 755 <td>
 756                 <p>
 757                   <span class="blue">Max = 287&#949; (Mean = 129&#949;)</span>
 758                 </p>
 759               </td>
 760 <td>
 761                 <p>
 762                   <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
 763                 </p>
 764               </td>
 765 <td>
 766                 <p>
 767                   <span class="blue">Max = 5.88e+05&#949; (Mean = 2.63e+05&#949;)</span>
 768                 </p>
 769               </td>
 770 <td>
 771                 <p>
 772                   <span class="blue">Max = 288&#949; (Mean = 129&#949;)</span>
 773                 </p>
 774               </td>
 775 </tr>
 776 <tr>
 777 <td>
 778                 <p>
 779                   Bessel JN': Mathworld Data
 780                 </p>
 781               </td>
 782 <td>
 783                 <p>
 784                   <span class="blue">Max = 0.527&#949; (Mean = 0.128&#949;)</span>
 785                 </p>
 786               </td>
 787 <td>
 788                 <p>
 789                   <span class="blue">Max = 1.29e+03&#949; (Mean = 312&#949;)</span>
 790                 </p>
 791               </td>
 792 <td>
 793                 <p>
 794                   <span class="blue">Max = 1.29e+03&#949; (Mean = 355&#949;)</span>
 795                 </p>
 796               </td>
 797 <td>
 798                 <p>
 799                   <span class="blue">Max = 14&#949; (Mean = 6.13&#949;)</span>
 800                 </p>
 801               </td>
 802 </tr>
 803 <tr>
 804 <td>
 805                 <p>
 806                   Bessel J': Mathworld Data
 807                 </p>
 808               </td>
 809 <td>
 810                 <p>
 811                   <span class="blue">Max = 21.5&#949; (Mean = 4.7&#949;)</span>
 812                 </p>
 813               </td>
 814 <td>
 815                 <p>
 816                   <span class="blue">Max = 42.5&#949; (Mean = 9.31&#949;)</span>
 817                 </p>
 818               </td>
 819 <td>
 820                 <p>
 821                   <span class="blue">Max = 42.5&#949; (Mean = 9.32&#949;)</span>
 822                 </p>
 823               </td>
 824 <td>
 825                 <p>
 826                   <span class="blue">Max = 23.7&#949; (Mean = 8&#949;)</span>
 827                 </p>
 828               </td>
 829 </tr>
 830 <tr>
 831 <td>
 832                 <p>
 833                   Bessel J': Mathworld Data (large values)
 834                 </p>
 835               </td>
 836 <td>
 837                 <p>
 838                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 839                 </p>
 840               </td>
 841 <td>
 842                 <p>
 843                   <span class="blue">Max = 989&#949; (Mean = 495&#949;)</span>
 844                 </p>
 845               </td>
 846 <td>
 847                 <p>
 848                   <span class="blue">Max = 989&#949; (Mean = 495&#949;)</span>
 849                 </p>
 850               </td>
 851 <td>
 852                 <p>
 853                   <span class="blue">Max = 2.9&#949; (Mean = 1.61&#949;)</span>
 854                 </p>
 855               </td>
 856 </tr>
 857 <tr>
 858 <td>
 859                 <p>
 860                   Bessel JN': Random Data
 861                 </p>
 862               </td>
 863 <td>
 864                 <p>
 865                   <span class="blue">Max = 0.593&#949; (Mean = 0.0396&#949;)</span>
 866                 </p>
 867               </td>
 868 <td>
 869                 <p>
 870                   <span class="blue">Max = 11.3&#949; (Mean = 1.85&#949;)</span>
 871                 </p>
 872               </td>
 873 <td>
 874                 <p>
 875                   <span class="blue">Max = 79.4&#949; (Mean = 16.2&#949;)</span>
 876                 </p>
 877               </td>
 878 <td>
 879                 <p>
 880                   <span class="blue">Max = 6.34&#949; (Mean = 0.999&#949;)</span>
 881                 </p>
 882               </td>
 883 </tr>
 884 <tr>
 885 <td>
 886                 <p>
 887                   Bessel J': Random Data
 888                 </p>
 889               </td>
 890 <td>
 891                 <p>
 892                   <span class="blue">Max = 0.885&#949; (Mean = 0.033&#949;)</span>
 893                 </p>
 894               </td>
 895 <td>
 896                 <p>
 897                   <span class="blue">Max = 139&#949; (Mean = 6.47&#949;)</span>
 898                 </p>
 899               </td>
 900 <td>
 901                 <p>
 902                   <span class="blue">Max = 279&#949; (Mean = 27.2&#949;)</span>
 903                 </p>
 904               </td>
 905 <td>
 906                 <p>
 907                   <span class="blue">Max = 176&#949; (Mean = 9.75&#949;)</span>
 908                 </p>
 909               </td>
 910 </tr>
 911 <tr>
 912 <td>
 913                 <p>
 914                   Bessel J': Random Data (Tricky large values)
 915                 </p>
 916               </td>
 917 <td>
 918                 <p>
 919                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 920                 </p>
 921               </td>
 922 <td>
 923                 <p>
 924                   <span class="blue">Max = 474&#949; (Mean = 62.2&#949;)</span>
 925                 </p>
 926               </td>
 927 <td>
 928                 <p>
 929                   <span class="blue">Max = 474&#949; (Mean = 64.5&#949;)</span>
 930                 </p>
 931               </td>
 932 <td>
 933                 <p>
 934                   <span class="blue">Max = 379&#949; (Mean = 45.4&#949;)</span>
 935                 </p>
 936               </td>
 937 </tr>
 938 </tbody>
 939 </table></div>
 940 </div>
 941 <br class="table-break"><div class="table">
 942 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime_integer_orders_"></a><p class="title"><b>Table&#160;8.54.&#160;Error rates for cyl_bessel_k_prime (integer orders)</b></p>
 943 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k_prime (integer orders)">
 944 <colgroup>
 945 <col>
 946 <col>
 947 <col>
 948 <col>
 949 <col>
 950 </colgroup>
 951 <thead><tr>
 952 <th>
 953               </th>
 954 <th>
 955                 <p>
 956                   GNU C++ version 7.1.0<br> linux<br> double
 957                 </p>
 958               </th>
 959 <th>
 960                 <p>
 961                   GNU C++ version 7.1.0<br> linux<br> long double
 962                 </p>
 963               </th>
 964 <th>
 965                 <p>
 966                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 967                 </p>
 968               </th>
 969 <th>
 970                 <p>
 971                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 972                 </p>
 973               </th>
 974 </tr></thead>
 975 <tbody>
 976 <tr>
 977 <td>
 978                 <p>
 979                   Bessel K'0: Mathworld Data (Integer Version)
 980                 </p>
 981               </td>
 982 <td>
 983                 <p>
 984                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 985                 </p>
 986               </td>
 987 <td>
 988                 <p>
 989                   <span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
 990                 </p>
 991               </td>
 992 <td>
 993                 <p>
 994                   <span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
 995                 </p>
 996               </td>
 997 <td>
 998                 <p>
 999                   <span class="blue">Max = 0.786&#949; (Mean = 0.39&#949;)</span>
1000                 </p>
1001               </td>
1002 </tr>
1003 <tr>
1004 <td>
1005                 <p>
1006                   Bessel K'1: Mathworld Data (Integer Version)
1007                 </p>
1008               </td>
1009 <td>
1010                 <p>
1011                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1012                 </p>
1013               </td>
1014 <td>
1015                 <p>
1016                   <span class="blue">Max = 0.736&#949; (Mean = 0.389&#949;)</span>
1017                 </p>
1018               </td>
1019 <td>
1020                 <p>
1021                   <span class="blue">Max = 0.736&#949; (Mean = 0.389&#949;)</span>
1022                 </p>
1023               </td>
1024 <td>
1025                 <p>
1026                   <span class="blue">Max = 0.761&#949; (Mean = 0.444&#949;)</span>
1027                 </p>
1028               </td>
1029 </tr>
1030 <tr>
1031 <td>
1032                 <p>
1033                   Bessel K'n: Mathworld Data (Integer Version)
1034                 </p>
1035               </td>
1036 <td>
1037                 <p>
1038                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1039                 </p>
1040               </td>
1041 <td>
1042                 <p>
1043                   <span class="blue">Max = 2.16&#949; (Mean = 1.08&#949;)</span>
1044                 </p>
1045               </td>
1046 <td>
1047                 <p>
1048                   <span class="blue">Max = 2.16&#949; (Mean = 1.08&#949;)</span>
1049                 </p>
1050               </td>
1051 <td>
1052                 <p>
1053                   <span class="blue">Max = 4.17&#949; (Mean = 1.75&#949;)</span>
1054                 </p>
1055               </td>
1056 </tr>
1057 </tbody>
1058 </table></div>
1059 </div>
1060 <br class="table-break"><div class="table">
1061 <a name="math_toolkit.bessel.bessel_derivatives.table_cyl_bessel_k_prime"></a><p class="title"><b>Table&#160;8.55.&#160;Error rates for cyl_bessel_k_prime</b></p>
1062 <div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k_prime">
1063 <colgroup>
1064 <col>
1065 <col>
1066 <col>
1067 <col>
1068 <col>
1069 </colgroup>
1070 <thead><tr>
1071 <th>
1072               </th>
1073 <th>
1074                 <p>
1075                   GNU C++ version 7.1.0<br> linux<br> double
1076                 </p>
1077               </th>
1078 <th>
1079                 <p>
1080                   GNU C++ version 7.1.0<br> linux<br> long double
1081                 </p>
1082               </th>
1083 <th>
1084                 <p>
1085                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
1086                 </p>
1087               </th>
1088 <th>
1089                 <p>
1090                   Microsoft Visual C++ version 14.1<br> Win32<br> double
1091                 </p>
1092               </th>
1093 </tr></thead>
1094 <tbody>
1095 <tr>
1096 <td>
1097                 <p>
1098                   Bessel K'0: Mathworld Data
1099                 </p>
1100               </td>
1101 <td>
1102                 <p>
1103                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1104                 </p>
1105               </td>
1106 <td>
1107                 <p>
1108                   <span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
1109                 </p>
1110               </td>
1111 <td>
1112                 <p>
1113                   <span class="blue">Max = 0.786&#949; (Mean = 0.329&#949;)</span>
1114                 </p>
1115               </td>
1116 <td>
1117                 <p>
1118                   <span class="blue">Max = 0.786&#949; (Mean = 0.39&#949;)</span>
1119                 </p>
1120               </td>
1121 </tr>
1122 <tr>
1123 <td>
1124                 <p>
1125                   Bessel K'1: Mathworld Data
1126                 </p>
1127               </td>
1128 <td>
1129                 <p>
1130                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1131                 </p>
1132               </td>
1133 <td>
1134                 <p>
1135                   <span class="blue">Max = 0.736&#949; (Mean = 0.389&#949;)</span>
1136                 </p>
1137               </td>
1138 <td>
1139                 <p>
1140                   <span class="blue">Max = 0.736&#949; (Mean = 0.389&#949;)</span>
1141                 </p>
1142               </td>
1143 <td>
1144                 <p>
1145                   <span class="blue">Max = 0.761&#949; (Mean = 0.444&#949;)</span>
1146                 </p>
1147               </td>
1148 </tr>
1149 <tr>
1150 <td>
1151                 <p>
1152                   Bessel K'n: Mathworld Data
1153                 </p>
1154               </td>
1155 <td>
1156                 <p>
1157                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1158                 </p>
1159               </td>
1160 <td>
1161                 <p>
1162                   <span class="blue">Max = 2.16&#949; (Mean = 1.08&#949;)</span>
1163                 </p>
1164               </td>
1165 <td>
1166                 <p>
1167                   <span class="blue">Max = 2.16&#949; (Mean = 1.08&#949;)</span>
1168                 </p>
1169               </td>
1170 <td>
1171                 <p>
1172                   <span class="blue">Max = 4.17&#949; (Mean = 1.75&#949;)</span>
1173                 </p>
1174               </td>
1175 </tr>
1176 <tr>
1177 <td>
1178                 <p>
1179                   Bessel K'v: Mathworld Data
1180                 </p>
1181               </td>
1182 <td>
1183                 <p>
1184                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1185                 </p>
1186               </td>
1187 <td>
1188                 <p>
1189                   <span class="blue">Max = 3.94&#949; (Mean = 2.44&#949;)</span>
1190                 </p>
1191               </td>
1192 <td>
1193                 <p>
1194                   <span class="blue">Max = 3.94&#949; (Mean = 2.34&#949;)</span>
1195                 </p>
1196               </td>
1197 <td>
1198                 <p>
1199                   <span class="blue">Max = 3.94&#949; (Mean = 1.47&#949;)</span>
1200                 </p>
1201               </td>
1202 </tr>
1203 <tr>
1204 <td>
1205                 <p>
1206                   Bessel K'v: Mathworld Data (large values)
1207                 </p>
1208               </td>
1209 <td>
1210                 <p>
1211                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1212                 </p>
1213               </td>
1214 <td>
1215                 <p>
1216                   <span class="blue">Max = 59.2&#949; (Mean = 42.9&#949;)</span>
1217                 </p>
1218               </td>
1219 <td>
1220                 <p>
1221                   <span class="blue">Max = 58.7&#949; (Mean = 42.6&#949;)</span>
1222                 </p>
1223               </td>
1224 <td>
1225                 <p>
1226                   <span class="blue">Max = 18.6&#949; (Mean = 11.8&#949;)</span>
1227                 </p>
1228               </td>
1229 </tr>
1230 <tr>
1231 <td>
1232                 <p>
1233                   Bessel K'n: Random Data
1234                 </p>
1235               </td>
1236 <td>
1237                 <p>
1238                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1239                 </p>
1240               </td>
1241 <td>
1242                 <p>
1243                   <span class="blue">Max = 4.45&#949; (Mean = 1.19&#949;)</span>
1244                 </p>
1245               </td>
1246 <td>
1247                 <p>
1248                   <span class="blue">Max = 4.45&#949; (Mean = 1.19&#949;)</span>
1249                 </p>
1250               </td>
1251 <td>
1252                 <p>
1253                   <span class="blue">Max = 9.67&#949; (Mean = 1.73&#949;)</span>
1254                 </p>
1255               </td>
1256 </tr>
1257 <tr>
1258 <td>
1259                 <p>
1260                   Bessel K'v: Random Data
1261                 </p>
1262               </td>
1263 <td>
1264                 <p>
1265                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
1266                 </p>
1267               </td>
1268 <td>
1269                 <p>
1270                   <span class="blue">Max = 7.95&#949; (Mean = 1.53&#949;)</span>
1271                 </p>
1272               </td>
1273 <td>
1274                 <p>
1275                   <span class="blue">Max = 7.95&#949; (Mean = 1.52&#949;)</span>
1276                 </p>
1277               </td>
1278 <td>
1279                 <p>
1280                   <span class="blue">Max = 8.32&#949; (Mean = 1.65&#949;)</span>
1281                 </p>
1282               </td>
1283 </tr>
1284 </tbody>
1285 </table></div>
1286 </div>
1287 <br class="table-break"><div class="table">
1288 <a name="math_toolkit.bessel.bessel_derivatives.table_sph_bessel_prime"></a><p class="title"><b>Table&#160;8.56.&#160;Error rates for sph_bessel_prime</b></p>
1289 <div class="table-contents"><table class="table" summary="Error rates for sph_bessel_prime">
1290 <colgroup>
1291 <col>
1292 <col>
1293 <col>
1294 <col>
1295 <col>
1296 </colgroup>
1297 <thead><tr>
1298 <th>
1299               </th>
1300 <th>
1301                 <p>
1302                   GNU C++ version 7.1.0<br> linux<br> double
1303                 </p>
1304               </th>
1305 <th>
1306                 <p>
1307                   GNU C++ version 7.1.0<br> linux<br> long double
1308                 </p>
1309               </th>
1310 <th>
1311                 <p>
1312                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
1313                 </p>
1314               </th>
1315 <th>
1316                 <p>
1317                   Microsoft Visual C++ version 14.1<br> Win32<br> double
1318                 </p>
1319               </th>
1320 </tr></thead>
1321 <tbody><tr>
1322 <td>
1323                 <p>
1324                   Bessel j': Random Data
1325                 </p>
1326               </td>
1327 <td>
1328                 <p>
1329                   <span class="blue">Max = 0.753&#949; (Mean = 0.0343&#949;)</span>
1330                 </p>
1331               </td>
1332 <td>
1333                 <p>
1334                   <span class="blue">Max = 167&#949; (Mean = 12&#949;)</span>
1335                 </p>
1336               </td>
1337 <td>
1338                 <p>
1339                   <span class="blue">Max = 167&#949; (Mean = 33.2&#949;)</span>
1340                 </p>
1341               </td>
1342 <td>
1343                 <p>
1344                   <span class="blue">Max = 307&#949; (Mean = 25.2&#949;)</span>
1345                 </p>
1346               </td>
1347 </tr></tbody>
1348 </table></div>
1349 </div>
1350 <br class="table-break"><div class="table">
1351 <a name="math_toolkit.bessel.bessel_derivatives.table_sph_neumann_prime"></a><p class="title"><b>Table&#160;8.57.&#160;Error rates for sph_neumann_prime</b></p>
1352 <div class="table-contents"><table class="table" summary="Error rates for sph_neumann_prime">
1353 <colgroup>
1354 <col>
1355 <col>
1356 <col>
1357 <col>
1358 <col>
1359 </colgroup>
1360 <thead><tr>
1361 <th>
1362               </th>
1363 <th>
1364                 <p>
1365                   GNU C++ version 7.1.0<br> linux<br> double
1366                 </p>
1367               </th>
1368 <th>
1369                 <p>
1370                   GNU C++ version 7.1.0<br> linux<br> long double
1371                 </p>
1372               </th>
1373 <th>
1374                 <p>
1375                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
1376                 </p>
1377               </th>
1378 <th>
1379                 <p>
1380                   Microsoft Visual C++ version 14.1<br> Win32<br> double
1381                 </p>
1382               </th>
1383 </tr></thead>
1384 <tbody><tr>
1385 <td>
1386                 <p>
1387                   y': Random Data
1388                 </p>
1389               </td>
1390 <td>
1391                 <p>
1392                   <span class="blue">Max = 0.988&#949; (Mean = 0.0869&#949;)</span>
1393                 </p>
1394               </td>
1395 <td>
1396                 <p>
1397                   <span class="blue">Max = 158&#949; (Mean = 18.8&#949;)</span>
1398                 </p>
1399               </td>
1400 <td>
1401                 <p>
1402                   <span class="blue">Max = 158&#949; (Mean = 20.2&#949;)</span>
1403                 </p>
1404               </td>
1405 <td>
1406                 <p>
1407                   <span class="blue">Max = 296&#949; (Mean = 25.6&#949;)</span>
1408                 </p>
1409               </td>
1410 </tr></tbody>
1411 </table></div>
1412 </div>
1413 <br class="table-break"><h5>
1414 <a name="math_toolkit.bessel.bessel_derivatives.h4"></a>
1415         <span class="phrase"><a name="math_toolkit.bessel.bessel_derivatives.implementation"></a></span><a class="link" href="bessel_derivatives.html#math_toolkit.bessel.bessel_derivatives.implementation">Implementation</a>
1416       </h5>
1417 <p>
1418         In the general case, the derivatives are calculated using the relations:
1419       </p>
1420 <div class="blockquote"><blockquote class="blockquote"><p>
1421           <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives1.svg"></span>
1422
1423         </p></blockquote></div>
1424 <p>
1425         There are also a number of special cases, for large x we have:
1426       </p>
1427 <div class="blockquote"><blockquote class="blockquote"><p>
1428           <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives4.svg"></span>
1429
1430         </p></blockquote></div>
1431 <p>
1432         And for small x:
1433       </p>
1434 <div class="blockquote"><blockquote class="blockquote"><p>
1435           <span class="inlinemediaobject"><img src="../../../equations/bessel_derivatives5.svg"></span>
1436
1437         </p></blockquote></div>
1438 </div>
1439 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
1440 <td align="left"></td>
1441 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
1442       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
1443       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
1444       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
1445       Daryle Walker and Xiaogang Zhang<p>
1446         Distributed under the Boost Software License, Version 1.0. (See accompanying
1447         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>)
1448       </p>
1449 </div></td>
1450 </tr></table>
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