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  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.sf_gamma.digamma"></a><a class="link" href="digamma.html" title="Digamma">Digamma</a>
  28 </h3></div></div></div>
  29 <h5>
  30 <a name="math_toolkit.sf_gamma.digamma.h0"></a>
  31         <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.synopsis"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.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">digamma</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">digamma</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;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">digamma</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;20.&#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 <h5>
  46 <a name="math_toolkit.sf_gamma.digamma.h1"></a>
  47         <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.description"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.description">Description</a>
  48       </h5>
  49 <p>
  50         Returns the digamma or psi function of <span class="emphasis"><em>x</em></span>. Digamma is
  51         defined as the logarithmic derivative of the gamma function:
  52       </p>
  53 <div class="blockquote"><blockquote class="blockquote"><p>
  54           <span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span>
  55
  56         </p></blockquote></div>
  57 <div class="blockquote"><blockquote class="blockquote"><p>
  58           <span class="inlinemediaobject"><img src="../../../graphs/digamma.svg" align="middle"></span>
  59
  60         </p></blockquote></div>
  61 <p>
  62         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
  63         be used to control the behaviour of the function: how it handles errors,
  64         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
  65         documentation for more details</a>.
  66       </p>
  67 <p>
  68         The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
  69         type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type
  70         T otherwise.
  71       </p>
  72 <h5>
  73 <a name="math_toolkit.sf_gamma.digamma.h2"></a>
  74         <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.accuracy"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.accuracy">Accuracy</a>
  75       </h5>
  76 <p>
  77         The following table shows the peak errors (in units of epsilon) found on
  78         various platforms with various floating point types. Unless otherwise specified
  79         any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>.
  80       </p>
  81 <div class="table">
  82 <a name="math_toolkit.sf_gamma.digamma.table_digamma"></a><p class="title"><b>Table&#160;8.4.&#160;Error rates for digamma</b></p>
  83 <div class="table-contents"><table class="table" summary="Error rates for digamma">
  84 <colgroup>
  85 <col>
  86 <col>
  87 <col>
  88 <col>
  89 <col>
  90 </colgroup>
  91 <thead><tr>
  92 <th>
  93               </th>
  94 <th>
  95                 <p>
  96                   GNU C++ version 7.1.0<br> linux<br> double
  97                 </p>
  98               </th>
  99 <th>
 100                 <p>
 101                   GNU C++ version 7.1.0<br> linux<br> long double
 102                 </p>
 103               </th>
 104 <th>
 105                 <p>
 106                   Sun compiler version 0x5150<br> Sun Solaris<br> long double
 107                 </p>
 108               </th>
 109 <th>
 110                 <p>
 111                   Microsoft Visual C++ version 14.1<br> Win32<br> double
 112                 </p>
 113               </th>
 114 </tr></thead>
 115 <tbody>
 116 <tr>
 117 <td>
 118                 <p>
 119                   Digamma Function: Large Values
 120                 </p>
 121               </td>
 122 <td>
 123                 <p>
 124                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 125                   2.1:</em></span> Max = 1.84&#949; (Mean = 0.71&#949;))<br> (<span class="emphasis"><em>Rmath
 126                   3.2.3:</em></span> Max = 1.18&#949; (Mean = 0.331&#949;))
 127                 </p>
 128               </td>
 129 <td>
 130                 <p>
 131                   <span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
 132                 </p>
 133               </td>
 134 <td>
 135                 <p>
 136                   <span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
 137                 </p>
 138               </td>
 139 <td>
 140                 <p>
 141                   <span class="blue">Max = 0.98&#949; (Mean = 0.369&#949;)</span>
 142                 </p>
 143               </td>
 144 </tr>
 145 <tr>
 146 <td>
 147                 <p>
 148                   Digamma Function: Near the Positive Root
 149                 </p>
 150               </td>
 151 <td>
 152                 <p>
 153                   <span class="blue">Max = 0.891&#949; (Mean = 0.0995&#949;)</span><br>
 154                   <br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 135&#949; (Mean = 11.9&#949;))<br>
 155                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.02e+03&#949; (Mean = 256&#949;))
 156                 </p>
 157               </td>
 158 <td>
 159                 <p>
 160                   <span class="blue">Max = 1.37&#949; (Mean = 0.477&#949;)</span>
 161                 </p>
 162               </td>
 163 <td>
 164                 <p>
 165                   <span class="blue">Max = 1.31&#949; (Mean = 0.471&#949;)</span>
 166                 </p>
 167               </td>
 168 <td>
 169                 <p>
 170                   <span class="blue">Max = 0.997&#949; (Mean = 0.527&#949;)</span>
 171                 </p>
 172               </td>
 173 </tr>
 174 <tr>
 175 <td>
 176                 <p>
 177                   Digamma Function: Near Zero
 178                 </p>
 179               </td>
 180 <td>
 181                 <p>
 182                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 183                   2.1:</em></span> Max = 0.953&#949; (Mean = 0.348&#949;))<br> (<span class="emphasis"><em>Rmath
 184                   3.2.3:</em></span> Max = 1.17&#949; (Mean = 0.564&#949;))
 185                 </p>
 186               </td>
 187 <td>
 188                 <p>
 189                   <span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
 190                 </p>
 191               </td>
 192 <td>
 193                 <p>
 194                   <span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
 195                 </p>
 196               </td>
 197 <td>
 198                 <p>
 199                   <span class="blue">Max = 0.953&#949; (Mean = 0.337&#949;)</span>
 200                 </p>
 201               </td>
 202 </tr>
 203 <tr>
 204 <td>
 205                 <p>
 206                   Digamma Function: Negative Values
 207                 </p>
 208               </td>
 209 <td>
 210                 <p>
 211                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 212                   2.1:</em></span> Max = 4.56e+04&#949; (Mean = 3.91e+03&#949;))<br> (<span class="emphasis"><em>Rmath
 213                   3.2.3:</em></span> Max = 4.6e+04&#949; (Mean = 3.94e+03&#949;))
 214                 </p>
 215               </td>
 216 <td>
 217                 <p>
 218                   <span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
 219                 </p>
 220               </td>
 221 <td>
 222                 <p>
 223                   <span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
 224                 </p>
 225               </td>
 226 <td>
 227                 <p>
 228                   <span class="blue">Max = 214&#949; (Mean = 16.1&#949;)</span>
 229                 </p>
 230               </td>
 231 </tr>
 232 <tr>
 233 <td>
 234                 <p>
 235                   Digamma Function: Values near 0
 236                 </p>
 237               </td>
 238 <td>
 239                 <p>
 240                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 241                   2.1:</em></span> Max = 0.866&#949; (Mean = 0.387&#949;))<br> (<span class="emphasis"><em>Rmath
 242                   3.2.3:</em></span> Max = 3.58e+05&#949; (Mean = 1.6e+05&#949;))
 243                 </p>
 244               </td>
 245 <td>
 246                 <p>
 247                   <span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
 248                 </p>
 249               </td>
 250 <td>
 251                 <p>
 252                   <span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
 253                 </p>
 254               </td>
 255 <td>
 256                 <p>
 257                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
 258                 </p>
 259               </td>
 260 </tr>
 261 <tr>
 262 <td>
 263                 <p>
 264                   Digamma Function: Integer arguments
 265                 </p>
 266               </td>
 267 <td>
 268                 <p>
 269                   <span class="blue">Max = 0.992&#949; (Mean = 0.215&#949;)</span><br> <br>
 270                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.18&#949; (Mean = 0.607&#949;))<br>
 271                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.33&#949; (Mean = 0.982&#949;))
 272                 </p>
 273               </td>
 274 <td>
 275                 <p>
 276                   <span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
 277                 </p>
 278               </td>
 279 <td>
 280                 <p>
 281                   <span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
 282                 </p>
 283               </td>
 284 <td>
 285                 <p>
 286                   <span class="blue">Max = 0.992&#949; (Mean = 0.452&#949;)</span>
 287                 </p>
 288               </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                 <p>
 293                   Digamma Function: Half integer arguments
 294                 </p>
 295               </td>
 296 <td>
 297                 <p>
 298                   <span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
 299                   2.1:</em></span> Max = 1.09&#949; (Mean = 0.531&#949;))<br> (<span class="emphasis"><em>Rmath
 300                   3.2.3:</em></span> Max = 46.2&#949; (Mean = 7.24&#949;))
 301                 </p>
 302               </td>
 303 <td>
 304                 <p>
 305                   <span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
 306                 </p>
 307               </td>
 308 <td>
 309                 <p>
 310                   <span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
 311                 </p>
 312               </td>
 313 <td>
 314                 <p>
 315                   <span class="blue">Max = 0.78&#949; (Mean = 0.314&#949;)</span>
 316                 </p>
 317               </td>
 318 </tr>
 319 </tbody>
 320 </table></div>
 321 </div>
 322 <br class="table-break"><p>
 323         As shown above, error rates for positive arguments are generally very low.
 324         For negative arguments there are an infinite number of irrational roots:
 325         relative errors very close to these can be arbitrarily large, although absolute
 326         error will remain very low.
 327       </p>
 328 <p>
 329         The following error plot are based on an exhaustive search of the functions
 330         domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
 331         precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
 332         <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
 333       </p>
 334 <div class="blockquote"><blockquote class="blockquote"><p>
 335           <span class="inlinemediaobject"><img src="../../../graphs/digamma__double.svg" align="middle"></span>
 336
 337         </p></blockquote></div>
 338 <div class="blockquote"><blockquote class="blockquote"><p>
 339           <span class="inlinemediaobject"><img src="../../../graphs/digamma__80_bit_long_double.svg" align="middle"></span>
 340
 341         </p></blockquote></div>
 342 <div class="blockquote"><blockquote class="blockquote"><p>
 343           <span class="inlinemediaobject"><img src="../../../graphs/digamma____float128.svg" align="middle"></span>
 344
 345         </p></blockquote></div>
 346 <h5>
 347 <a name="math_toolkit.sf_gamma.digamma.h3"></a>
 348         <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.testing"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.testing">Testing</a>
 349       </h5>
 350 <p>
 351         There are two sets of tests: spot values are computed using the online calculator
 352         at functions.wolfram.com, while random test values are generated using the
 353         high-precision reference implementation (a differentiated <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
 354         approximation</a> see below).
 355       </p>
 356 <h5>
 357 <a name="math_toolkit.sf_gamma.digamma.h4"></a>
 358         <span class="phrase"><a name="math_toolkit.sf_gamma.digamma.implementation"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.implementation">Implementation</a>
 359       </h5>
 360 <p>
 361         The implementation is divided up into the following domains:
 362       </p>
 363 <p>
 364         For Negative arguments the reflection formula:
 365       </p>
 366 <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="number">1</span><span class="special">-</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">pi</span><span class="special">/</span><span class="identifier">tan</span><span class="special">(</span><span class="identifier">pi</span><span class="special">*</span><span class="identifier">x</span><span class="special">);</span>
 367 </pre>
 368 <p>
 369         is used to make <span class="emphasis"><em>x</em></span> positive.
 370       </p>
 371 <p>
 372         For arguments in the range [0,1] the recurrence relation:
 373       </p>
 374 <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span><span class="special">/</span><span class="identifier">x</span>
 375 </pre>
 376 <p>
 377         is used to shift the evaluation to [1,2].
 378       </p>
 379 <p>
 380         For arguments in the range [1,2] a rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 381         by JM</a> is used (see below).
 382       </p>
 383 <p>
 384         For arguments in the range [2,BIG] the recurrence relation:
 385       </p>
 386 <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">+</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">+</span> <span class="number">1</span><span class="special">/</span><span class="identifier">x</span><span class="special">;</span>
 387 </pre>
 388 <p>
 389         is used to shift the evaluation to the range [1,2].
 390       </p>
 391 <p>
 392         For arguments &gt; BIG the asymptotic expansion:
 393       </p>
 394 <div class="blockquote"><blockquote class="blockquote"><p>
 395           <span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span>
 396
 397         </p></blockquote></div>
 398 <p>
 399         can be used. However, this expansion is divergent after a few terms: exactly
 400         how many terms depends on the size of <span class="emphasis"><em>x</em></span>. Therefore the
 401         value of <span class="emphasis"><em>BIG</em></span> must be chosen so that the series can be
 402         truncated at a term that is too small to have any effect on the result when
 403         evaluated at <span class="emphasis"><em>BIG</em></span>. Choosing BIG=10 for up to 80-bit reals,
 404         and BIG=20 for 128-bit reals allows the series to truncated after a suitably
 405         small number of terms and evaluated as a polynomial in <code class="computeroutput"><span class="number">1</span><span class="special">/(</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>.
 406       </p>
 407 <p>
 408         The arbitrary precision version of this function uses recurrence relations
 409         until x &gt; BIG, and then evaluation via the asymptotic expansion above.
 410         As special cases integer and half integer arguments are handled via:
 411       </p>
 412 <div class="blockquote"><blockquote class="blockquote"><p>
 413           <span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span>
 414
 415         </p></blockquote></div>
 416 <div class="blockquote"><blockquote class="blockquote"><p>
 417           <span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span>
 418
 419         </p></blockquote></div>
 420 <p>
 421         The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 422         by JM</a> in the range [1,2] is derived as follows.
 423       </p>
 424 <p>
 425         First a high precision approximation to digamma was constructed using a 60-term
 426         differentiated <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>,
 427         the form used is:
 428       </p>
 429 <div class="blockquote"><blockquote class="blockquote"><p>
 430           <span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span>
 431
 432         </p></blockquote></div>
 433 <p>
 434         Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos
 435         sum, and P'(x) and Q'(x) are their first derivatives. The Lanzos part of
 436         this approximation has a theoretical precision of ~100 decimal digits. However,
 437         cancellation in the above sum will reduce that to around <code class="computeroutput"><span class="number">99</span><span class="special">-(</span><span class="number">1</span><span class="special">/</span><span class="identifier">y</span><span class="special">)</span></code> digits
 438         if <span class="emphasis"><em>y</em></span> is the result. This approximation was used to calculate
 439         the positive root of digamma, and was found to agree with the value used
 440         by Cody to 25 digits (See Math. Comp. 27, 123-127 (1973) by Cody, Strecok
 441         and Thacher) and with the value used by Morris to 35 digits (See TOMS Algorithm
 442         708).
 443       </p>
 444 <p>
 445         Likewise a few spot tests agreed with values calculated using functions.wolfram.com
 446         to &gt;40 digits. That's sufficiently precise to insure that the approximation
 447         below is accurate to double precision. Achieving 128-bit long double precision
 448         requires that the location of the root is known to ~70 digits, and it's not
 449         clear whether the value calculated by this method meets that requirement:
 450         the difficulty lies in independently verifying the value obtained.
 451       </p>
 452 <p>
 453         The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
 454         by JM</a> was optimised for absolute error using the form:
 455       </p>
 456 <pre class="programlisting"><span class="identifier">digamma</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">X0</span><span class="special">)(</span><span class="identifier">Y</span> <span class="special">+</span> <span class="identifier">R</span><span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">));</span>
 457 </pre>
 458 <p>
 459         Where X0 is the positive root of digamma, Y is a constant, and R(x - 1) is
 460         the rational approximation. Note that since X0 is irrational, we need twice
 461         as many digits in X0 as in x in order to avoid cancellation error during
 462         the subtraction (this assumes that <span class="emphasis"><em>x</em></span> is an exact value,
 463         if it's not then all bets are off). That means that even when x is the value
 464         of the root rounded to the nearest representable value, the result of digamma(x)
 465         <span class="emphasis"><em><span class="bold"><strong>will not be zero</strong></span></em></span>.
 466       </p>
 467 </div>
 468 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 469 <td align="left"></td>
 470 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 471       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 472       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 473       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 474       Daryle Walker and Xiaogang Zhang<p>
 475         Distributed under the Boost Software License, Version 1.0. (See accompanying
 476         file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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