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

   1 [section:error_function Error Functions]
   2
   3 [h4 Synopsis]
   4
   5 ``
   6 #include <boost/math/special_functions/erf.hpp>
   7 ``
   8
   9    namespace boost{ namespace math{
  10
  11    template <class T>
  12    ``__sf_result`` erf(T z);
  13
  14    template <class T, class ``__Policy``>
  15    ``__sf_result`` erf(T z, const ``__Policy``&);
  16
  17    template <class T>
  18    ``__sf_result`` erfc(T z);
  19
  20    template <class T, class ``__Policy``>
  21    ``__sf_result`` erfc(T z, const ``__Policy``&);
  22
  23    }} // namespaces
  24
  25 The return type of these functions is computed using the __arg_promotion_rules:
  26 the return type is `double` if T is an integer type, and T otherwise.
  27
  28 [optional_policy]
  29
  30 [h4 Description]
  31
  32    template <class T>
  33    ``__sf_result`` erf(T z);
  34
  35    template <class T, class ``__Policy``>
  36    ``__sf_result`` erf(T z, const ``__Policy``&);
  37
  38 Returns the [@http://en.wikipedia.org/wiki/Error_function error function]
  39 [@http://functions.wolfram.com/GammaBetaErf/Erf/ erf] of z:
  40
  41 [equation erf1]
  42
  43 [graph erf]
  44
  45    template <class T>
  46    ``__sf_result`` erfc(T z);
  47
  48    template <class T, class ``__Policy``>
  49    ``__sf_result`` erfc(T z, const ``__Policy``&);
  50
  51 Returns the complement of the [@http://functions.wolfram.com/GammaBetaErf/Erfc/ error function] of z:
  52
  53 [equation erf2]
  54
  55 [graph erfc]
  56
  57 [h4 Accuracy]
  58
  59 The following table shows the peak errors (in units of epsilon)
  60 found on various platforms with various floating point types,
  61 along with comparisons to the __gsl, __glibc, __hpc and __cephes libraries.
  62 Unless otherwise specified any floating point type that is narrower
  63 than the one shown will have __zero_error.
  64
  65 [table_erf]
  66
  67 [table_erfc]
  68
  69 [h4 Testing]
  70
  71 The tests for these functions come in two parts:
  72 basic sanity checks use spot values calculated using
  73 [@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Erf Mathworld's online evaluator],
  74 while accuracy checks use high-precision test values calculated at 1000-bit precision with
  75 [@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
  76 Note that the generic and type-specific
  77 versions of these functions use differing implementations internally, so this
  78 gives us reasonably independent test data.  Using our test data to test other
  79 "known good" implementations also provides an additional sanity check.
  80
  81 [h4 Implementation]
  82
  83 All versions of these functions first use the usual reflection formulas
  84 to make their arguments positive:
  85
  86    erf(-z) = 1 - erf(z);
  87
  88    erfc(-z) = 2 - erfc(z);  // preferred when -z < -0.5
  89
  90    erfc(-z) = 1 + erf(z);   // preferred when -0.5 <= -z < 0
  91
  92 The generic versions of these functions are implemented in terms of
  93 the incomplete gamma function.
  94
  95 When the significand (mantissa) size is recognised
  96 (currently for 53, 64 and 113-bit reals, plus single-precision 24-bit handled via promotion to double)
  97 then a series of rational approximations [jm_rationals] are used.
  98
  99 For `z <= 0.5` then a rational approximation to erf is used, based on the
 100 observation that erf is an odd function and therefore erf is calculated using:
 101
 102    erf(z) = z * (C + R(z*z));
 103
 104 where the rational approximation R(z*z) is optimised for absolute error:
 105 as long as its absolute error is small enough compared to the constant C, then any
 106 round-off error incurred during the computation of R(z*z) will effectively
 107 disappear from the result.  As a result the error for erf and erfc in this
 108 region is very low: the last bit is incorrect in only a very small number of
 109 cases.
 110
 111 For `z > 0.5` we observe that over a small interval \[a, b) then:
 112
 113    erfc(z) * exp(z*z) * z ~ c
 114
 115 for some constant c.
 116
 117 Therefore for `z > 0.5` we calculate erfc using:
 118
 119    erfc(z) = exp(-z*z) * (C + R(z - B)) / z;
 120
 121 Again R(z - B) is optimised for absolute error, and the constant `C` is
 122 the average of `erfc(z) * exp(z*z) * z` taken at the endpoints of the range.
 123 Once again, as long as the absolute error in R(z - B) is small
 124 compared to `c` then `c + R(z - B)` will be correctly rounded, and the error
 125 in the result will depend only on the accuracy of the exp function.  In practice,
 126 in all but a very small number of cases, the error is confined to the last bit
 127 of the result.  The constant `B` is chosen so that the left hand end of the range
 128 of the rational approximation is 0.
 129
 130 For large `z` over a range \[a, +[infin]\] the above approximation is modified to:
 131
 132    erfc(z) = exp(-z*z) * (C + R(1 / z)) / z;
 133
 134 [endsect]
 135 [/ :error_function The Error Functions]
 136
 137 [/
 138   Copyright 2006 John Maddock and Paul A. Bristow.
 139   Distributed under the Boost Software License, Version 1.0.
 140   (See accompanying file LICENSE_1_0.txt or copy at
 141   http://www.boost.org/LICENSE_1_0.txt).
 142 ]