ceph/src/boost/libs/math/doc/html/math_toolkit/zetas/zeta.html

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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.zetas.zeta"></a><a class="link" href="zeta.html" title="Riemann Zeta Function">Riemann Zeta Function</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.zetas.zeta.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.zetas.zeta.synopsis"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.synopsis">Synopsis</a>
  32       </h5>
  33 <pre class="programlisting"><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">zeta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34 </pre>
  35 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</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">T</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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</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="special">}}</span> <span class="comment">// namespaces</span>
  44 </pre>
  45 <p>
  46         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
  47         type calculation rules</em></span></a>: the return type is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, and T otherwise.
  48       </p>
  49 <p>
  50         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
  51         be used to control the behaviour of the function: how it handles errors,
  52         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
  53         documentation for more details</a>.
  54       </p>
  55 <h5>
  56 <a name="math_toolkit.zetas.zeta.h1"></a>
  57         <span class="phrase"><a name="math_toolkit.zetas.zeta.description"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.description">Description</a>
  58       </h5>
  59 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  60 <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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
  61
  62 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</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>
  63 <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">zeta</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</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>
  64 </pre>
  65 <p>
  66         Returns the <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html" target="_top">zeta
  67         function</a> of z:
  68       </p>
  69 <p>
  70         <span class="inlinemediaobject"><img src="../../../equations/zeta1.svg"></span>
  71       </p>
  72 <p>
  73         <span class="inlinemediaobject"><img src="../../../graphs/zeta1.svg" align="middle"></span>
  74       </p>
  75 <p>
  76         <span class="inlinemediaobject"><img src="../../../graphs/zeta2.svg" align="middle"></span>
  77       </p>
  78 <h5>
  79 <a name="math_toolkit.zetas.zeta.h2"></a>
  80         <span class="phrase"><a name="math_toolkit.zetas.zeta.accuracy"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.accuracy">Accuracy</a>
  81       </h5>
  82 <p>
  83         The following table shows the peak errors (in units of epsilon) found on
  84         various platforms with various floating point types, along with comparisons
  85         to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a> and
  86         <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries. Unless
  87         otherwise specified any floating point type that is narrower than the one
  88         shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
  89         zero error</a>.
  90       </p>
  91 <div class="table">
  92 <a name="math_toolkit.zetas.zeta.table_zeta"></a><p class="title"><b>Table&#160;6.73.&#160;Error rates for zeta</b></p>
  93 <div class="table-contents"><table class="table" summary="Error rates for zeta">
  94 <colgroup>
  95 <col>
  96 <col>
  97 <col>
  98 <col>
  99 <col>
 100 </colgroup>
 101 <thead><tr>
 102 <th>
 103               </th>
 104 <th>
 105                 <p>
 106                   Microsoft Visual C++ version 12.0<br> Win32<br> double
 107                 </p>
 108               </th>
 109 <th>
 110                 <p>
 111                   GNU C++ version 5.1.0<br> linux<br> double
 112                 </p>
 113               </th>
 114 <th>
 115                 <p>
 116                   GNU C++ version 5.1.0<br> linux<br> long double
 117                 </p>
 118               </th>
 119 <th>
 120                 <p>
 121                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 122                 </p>
 123               </th>
 124 </tr></thead>
 125 <tbody>
 126 <tr>
 127 <td>
 128                 <p>
 129                   Zeta: Random values greater than 1
 130                 </p>
 131               </td>
 132 <td>
 133                 <p>
 134                   <span class="blue">Max = 0.836&#949; (Mean = 0.093&#949;)</span>
 135                 </p>
 136               </td>
 137 <td>
 138                 <p>
 139                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 140                   1.16:</em></span> Max = 8.69&#949; (Mean = 1.03&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 141                   <span class="red">Max = 4.49e+33&#949; (Mean = 6.85e+32&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Random_values_greater_than_1">And
 142                   other failures.</a>)</span>
 143                 </p>
 144               </td>
 145 <td>
 146                 <p>
 147                   <span class="blue">Max = 0.846&#949; (Mean = 0.0833&#949;)</span><br>
 148                   <br> (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 5.45&#949; (Mean
 149                   = 1&#949;))
 150                 </p>
 151               </td>
 152 <td>
 153                 <p>
 154                   <span class="blue">Max = 0.846&#949; (Mean = 0.0743&#949;)</span>
 155                 </p>
 156               </td>
 157 </tr>
 158 <tr>
 159 <td>
 160                 <p>
 161                   Zeta: Random values less than 1
 162                 </p>
 163               </td>
 164 <td>
 165                 <p>
 166                   <span class="blue">Max = 7.03&#949; (Mean = 2.98&#949;)</span>
 167                 </p>
 168               </td>
 169 <td>
 170                 <p>
 171                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 172                   1.16:</em></span> Max = 137&#949; (Mean = 13.8&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 173                   <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_zeta_Cephes_Zeta_Random_values_less_than_1">And
 174                   other failures.</a>)</span>
 175                 </p>
 176               </td>
 177 <td>
 178                 <p>
 179                   <span class="blue">Max = 7.03&#949; (Mean = 2.71&#949;)</span><br> <br>
 180                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 538&#949; (Mean = 59.3&#949;))
 181                 </p>
 182               </td>
 183 <td>
 184                 <p>
 185                   <span class="blue">Max = 70.1&#949; (Mean = 17.1&#949;)</span>
 186                 </p>
 187               </td>
 188 </tr>
 189 <tr>
 190 <td>
 191                 <p>
 192                   Zeta: Values close to and greater than 1
 193                 </p>
 194               </td>
 195 <td>
 196                 <p>
 197                   <span class="blue">Max = 0.994&#949; (Mean = 0.421&#949;)</span>
 198                 </p>
 199               </td>
 200 <td>
 201                 <p>
 202                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 203                   1.16:</em></span> Max = 7.73&#949; (Mean = 4.07&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 204                   <span class="red">Max = 6.77e+15&#949; (Mean = 1.52e+15&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Values_close_to_and_greater_than_1">And
 205                   other failures.</a>)</span>
 206                 </p>
 207               </td>
 208 <td>
 209                 <p>
 210                   <span class="blue">Max = 0.995&#949; (Mean = 0.5&#949;)</span><br> <br>
 211                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.9e+06&#949; (Mean = 5.11e+05&#949;))
 212                 </p>
 213               </td>
 214 <td>
 215                 <p>
 216                   <span class="blue">Max = 0.995&#949; (Mean = 0.5&#949;)</span>
 217                 </p>
 218               </td>
 219 </tr>
 220 <tr>
 221 <td>
 222                 <p>
 223                   Zeta: Values close to and less than 1
 224                 </p>
 225               </td>
 226 <td>
 227                 <p>
 228                   <span class="blue">Max = 0.991&#949; (Mean = 0.375&#949;)</span>
 229                 </p>
 230               </td>
 231 <td>
 232                 <p>
 233                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 234                   1.16:</em></span> Max = 0.991&#949; (Mean = 0.28&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 235                   <span class="red">Max = 8.66e+15&#949; (Mean = 1.9e+15&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_zeta_Cephes_Zeta_Values_close_to_and_less_than_1">And
 236                   other failures.</a>)</span>
 237                 </p>
 238               </td>
 239 <td>
 240                 <p>
 241                   <span class="blue">Max = 0.998&#949; (Mean = 0.508&#949;)</span><br> <br>
 242                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 8.53e+06&#949; (Mean =
 243                   1.87e+06&#949;))
 244                 </p>
 245               </td>
 246 <td>
 247                 <p>
 248                   <span class="blue">Max = 0.998&#949; (Mean = 0.568&#949;)</span>
 249                 </p>
 250               </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                 <p>
 255                   Zeta: Integer arguments
 256                 </p>
 257               </td>
 258 <td>
 259                 <p>
 260                   <span class="blue">Max = 6.5&#949; (Mean = 2.17&#949;)</span>
 261                 </p>
 262               </td>
 263 <td>
 264                 <p>
 265                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 266                   1.16:</em></span> Max = 3.75&#949; (Mean = 1.1&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
 267                   <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_zeta_Cephes_Zeta_Integer_arguments">And
 268                   other failures.</a>)</span>
 269                 </p>
 270               </td>
 271 <td>
 272                 <p>
 273                   <span class="blue">Max = 9&#949; (Mean = 3.06&#949;)</span><br> <br>
 274                   (<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 70.3&#949; (Mean = 17.4&#949;))
 275                 </p>
 276               </td>
 277 <td>
 278                 <p>
 279                   <span class="blue">Max = 21&#949; (Mean = 7.13&#949;)</span>
 280                 </p>
 281               </td>
 282 </tr>
 283 </tbody>
 284 </table></div>
 285 </div>
 286 <br class="table-break"><h5>
 287 <a name="math_toolkit.zetas.zeta.h3"></a>
 288         <span class="phrase"><a name="math_toolkit.zetas.zeta.testing"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.testing">Testing</a>
 289       </h5>
 290 <p>
 291         The tests for these functions come in two parts: basic sanity checks use
 292         spot values calculated using <a href="http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta" target="_top">Mathworld's
 293         online evaluator</a>, while accuracy checks use high-precision test values
 294         calculated at 1000-bit precision with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL::RR</a>
 295         and this implementation. Note that the generic and type-specific versions
 296         of these functions use differing implementations internally, so this gives
 297         us reasonably independent test data. Using our test data to test other "known
 298         good" implementations also provides an additional sanity check.
 299       </p>
 300 <h5>
 301 <a name="math_toolkit.zetas.zeta.h4"></a>
 302         <span class="phrase"><a name="math_toolkit.zetas.zeta.implementation"></a></span><a class="link" href="zeta.html#math_toolkit.zetas.zeta.implementation">Implementation</a>
 303       </h5>
 304 <p>
 305         All versions of these functions first use the usual reflection formulas to
 306         make their arguments positive:
 307       </p>
 308 <p>
 309         <span class="inlinemediaobject"><img src="../../../equations/zeta3.svg"></span>
 310       </p>
 311 <p>
 312         The generic versions of these functions are implemented using the series:
 313       </p>
 314 <p>
 315         <span class="inlinemediaobject"><img src="../../../equations/zeta6.svg"></span>
 316       </p>
 317 <p>
 318         When the significand (mantissa) size is recognised (currently for 53, 64
 319         and 113-bit reals, plus single-precision 24-bit handled via promotion to
 320         double) then a series of rational approximations <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 321         by JM</a> are used.
 322       </p>
 323 <p>
 324         For 0 &lt; z &lt; 1 the approximating form is:
 325       </p>
 326 <p>
 327         <span class="inlinemediaobject"><img src="../../../equations/zeta4.svg"></span>
 328       </p>
 329 <p>
 330         For a rational approximation R(1-z) and a constant C.
 331       </p>
 332 <p>
 333         For 1 &lt; z &lt; 4 the approximating form is:
 334       </p>
 335 <p>
 336         <span class="inlinemediaobject"><img src="../../../equations/zeta5.svg"></span>
 337       </p>
 338 <p>
 339         For a rational approximation R(n-z) and a constant C and integer n.
 340       </p>
 341 <p>
 342         For z &gt; 4 the approximating form is:
 343       </p>
 344 <p>
 345         &#950;(z) = 1 + e<sup>R(z - n)</sup>
 346       </p>
 347 <p>
 348         For a rational approximation R(z-n) and integer n, note that the accuracy
 349         required for R(z-n) is not full machine precision, but an absolute error
 350         of: &#949;/R(0). This saves us quite a few digits when dealing with large z, especially
 351         when &#949; is small.
 352       </p>
 353 <p>
 354         Finally, there are some special cases for integer arguments, there are closed
 355         forms for negative or even integers:
 356       </p>
 357 <p>
 358         <span class="inlinemediaobject"><img src="../../../equations/zeta7.svg"></span>
 359       </p>
 360 <p>
 361         <span class="inlinemediaobject"><img src="../../../equations/zeta8.svg"></span>
 362       </p>
 363 <p>
 364         <span class="inlinemediaobject"><img src="../../../equations/zeta9.svg"></span>
 365       </p>
 366 <p>
 367         and for positive odd integers we simply cache pre-computed values as these
 368         are of great benefit to some infinite series calculations.
 369       </p>
 370 </div>
 371 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 372 <td align="left"></td>
 373 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 374       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 375       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 376       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 377         Distributed under the Boost Software License, Version 1.0. (See accompanying
 378         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>)
 379       </p>
 380 </div></td>
 381 </tr></table>
 382 <hr>
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