ceph/src/boost/libs/math/doc/sf/zeta.qbk

   1 [section:zeta Riemann Zeta Function]
   2
   3 [h4 Synopsis]
   4
   5 ``
   6 #include <boost/math/special_functions/zeta.hpp>
   7 ``
   8
   9    namespace boost{ namespace math{
  10
  11    template <class T>
  12    ``__sf_result`` zeta(T z);
  13
  14    template <class T, class ``__Policy``>
  15    ``__sf_result`` zeta(T z, const ``__Policy``&);
  16
  17    }} // namespaces
  18
  19 The return type of these functions is computed using the __arg_promotion_rules:
  20 the return type is `double` if T is an integer type, and T otherwise.
  21
  22 [optional_policy]
  23
  24 [h4 Description]
  25
  26    template <class T>
  27    ``__sf_result`` zeta(T z);
  28
  29    template <class T, class ``__Policy``>
  30    ``__sf_result`` zeta(T z, const ``__Policy``&);
  31
  32 Returns the [@http://mathworld.wolfram.com/RiemannZetaFunction.html zeta function]
  33 of z:
  34
  35 [equation zeta1]
  36
  37 [graph zeta1]
  38
  39 [graph zeta2]
  40
  41 [h4 Accuracy]
  42
  43 The following table shows the peak errors (in units of epsilon)
  44 found on various platforms with various floating point types,
  45 along with comparisons to the __gsl and __cephes libraries.
  46 Unless otherwise specified any floating point type that is narrower
  47 than the one shown will have __zero_error.
  48
  49 [table_zeta]
  50
  51 [h4 Testing]
  52
  53 The tests for these functions come in two parts:
  54 basic sanity checks use spot values calculated using
  55 [@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Zeta Mathworld's online evaluator],
  56 while accuracy checks use high-precision test values calculated at 1000-bit precision with
  57 [@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
  58 Note that the generic and type-specific
  59 versions of these functions use differing implementations internally, so this
  60 gives us reasonably independent test data.  Using our test data to test other
  61 "known good" implementations also provides an additional sanity check.
  62
  63 [h4 Implementation]
  64
  65 All versions of these functions first use the usual reflection formulas
  66 to make their arguments positive:
  67
  68 [equation zeta3]
  69
  70 The generic versions of these functions are implemented using the series:
  71
  72 [equation zeta6]
  73
  74 When the significand (mantissa) size is recognised
  75 (currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
  76 then a series of rational approximations [jm_rationals] are used.
  77
  78 For 0 < z < 1 the approximating form is:
  79
  80 [equation zeta4]
  81
  82 For a rational approximation R(1-z) and a constant C.
  83
  84 For 1 < z < 4 the approximating form is:
  85
  86 [equation zeta5]
  87
  88 For a rational approximation R(n-z) and a constant C and integer n.
  89
  90 For z > 4 the approximating form is:
  91
  92 [zeta](z) = 1 + e[super R(z - n)]
  93
  94 For a rational approximation R(z-n) and integer n, note that the accuracy
  95 required for R(z-n) is not full machine precision, but an absolute error
  96 of: [epsilon]/R(0).  This saves us quite a few digits when dealing with large
  97 z, especially when [epsilon] is small.
  98
  99 Finally, there are some special cases for integer arguments, there are
 100 closed forms for negative or even integers:
 101
 102 [equation zeta7]
 103
 104 [equation zeta8]
 105
 106 [equation zeta9]
 107
 108 and for positive odd integers we simply cache pre-computed values as these are of great
 109 benefit to some infinite series calculations.
 110
 111 [endsect]
 112 [/ :error_function The Error Functions]
 113
 114 [/
 115   Copyright 2006 John Maddock and Paul A. Bristow.
 116   Distributed under the Boost Software License, Version 1.0.
 117   (See accompanying file LICENSE_1_0.txt or copy at
 118   http://www.boost.org/LICENSE_1_0.txt).
 119 ]