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

   1 [section:expint Exponential Integrals]
   2
   3 [section:expint_n Exponential Integral En]
   4
   5 [h4 Synopsis]
   6
   7 ``
   8 #include <boost/math/special_functions/expint.hpp>
   9 ``
  10
  11    namespace boost{ namespace math{
  12
  13    template <class T>
  14    ``__sf_result`` expint(unsigned n, T z);
  15
  16    template <class T, class ``__Policy``>
  17    ``__sf_result`` expint(unsigned n, T z, const ``__Policy``&);
  18
  19    }} // namespaces
  20
  21 The return type of these functions is computed using the __arg_promotion_rules:
  22 the return type is `double` if T is an integer type, and T otherwise.
  23
  24 [optional_policy]
  25
  26 [h4 Description]
  27
  28    template <class T>
  29    ``__sf_result`` expint(unsigned n, T z);
  30
  31    template <class T, class ``__Policy``>
  32    ``__sf_result`` expint(unsigned n, T z, const ``__Policy``&);
  33
  34 Returns the [@http://mathworld.wolfram.com/En-Function.html exponential integral En]
  35 of z:
  36
  37 [equation expint_n_1]
  38
  39 [graph expint2]
  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 other libraries.
  46 Unless otherwise specified any floating point type that is narrower
  47 than the one shown will have __zero_error.
  48
  49 [table_expint_En_]
  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=ExpIntegralE 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 The generic version of this function uses the continued fraction:
  66
  67 [equation expint_n_3]
  68
  69 for large /x/ and the infinite series:
  70
  71 [equation expint_n_2]
  72
  73 for small /x/.
  74
  75 Where the precision of /x/ is known at compile time and is 113 bits or fewer
  76 in precision, then rational approximations [jm_rationals] are used for the
  77 `n == 1` case.
  78
  79 For `x < 1` the approximating form is a minimax approximation:
  80
  81 [equation expint_n_4]
  82
  83 and for `x > 1` a Chebyshev interpolated approximation of the form:
  84
  85 [equation expint_n_5]
  86
  87 is used.
  88
  89
  90 [endsect]
  91
  92 [section:expint_i Exponential Integral Ei]
  93
  94 [h4 Synopsis]
  95
  96 ``
  97 #include <boost/math/special_functions/expint.hpp>
  98 ``
  99
 100    namespace boost{ namespace math{
 101
 102    template <class T>
 103    ``__sf_result`` expint(T z);
 104
 105    template <class T, class ``__Policy``>
 106    ``__sf_result`` expint(T z, const ``__Policy``&);
 107
 108    }} // namespaces
 109
 110 The return type of these functions is computed using the __arg_promotion_rules:
 111 the return type is `double` if T is an integer type, and T otherwise.
 112
 113 [optional_policy]
 114
 115 [h4 Description]
 116
 117    template <class T>
 118    ``__sf_result`` expint(T z);
 119
 120    template <class T, class ``__Policy``>
 121    ``__sf_result`` expint(T z, const ``__Policy``&);
 122
 123 Returns the [@http://mathworld.wolfram.com/ExponentialIntegral.html exponential integral]
 124 of z:
 125
 126 [equation expint_i_1]
 127
 128 [graph expint_i]
 129
 130 [h4 Accuracy]
 131
 132 The following table shows the peak errors (in units of epsilon)
 133 found on various platforms with various floating point types,
 134 along with comparisons to Cody's SPECFUN implementation and the __gsl library.
 135 Unless otherwise specified any floating point type that is narrower
 136 than the one shown will have __zero_error.
 137
 138 [table_expint_Ei_]
 139
 140 It should be noted that all three libraries tested above
 141 offer sub-epsilon precision over most of their range.
 142
 143 GSL has the greatest difficulty near the positive root of En, while
 144 Cody's SPECFUN along with this implementation increase their
 145 error rates very slightly over the range \[4,6\].
 146
 147 [h4 Testing]
 148
 149 The tests for these functions come in two parts:
 150 basic sanity checks use spot values calculated using
 151 [@http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ExpIntegralEi Mathworld's online evaluator],
 152 while accuracy checks use high-precision test values calculated at 1000-bit precision with
 153 [@http://shoup.net/ntl/doc/RR.txt NTL::RR] and this implementation.
 154 Note that the generic and type-specific
 155 versions of these functions use differing implementations internally, so this
 156 gives us reasonably independent test data.  Using our test data to test other
 157 "known good" implementations also provides an additional sanity check.
 158
 159 [h4 Implementation]
 160
 161 For x < 0 this function just calls __expint_n(1, -x): which in turn is implemented
 162 in terms of rational approximations when the type of x has 113 or fewer bits of
 163 precision.
 164
 165 For x > 0 the generic version is implemented using the infinte series:
 166
 167 [equation expint_i_2]
 168
 169 However, when the precision of the argument type is known at compile time
 170 and is 113 bits or less, then rational approximations [jm_rationals] are used.
 171
 172 For 0 < z < 6 a root-preserving approximation of the form:
 173
 174 [equation expint_i_3]
 175
 176 is used, where z[sub 0] is the positive root of the function, and
 177 R(z/3 - 1) is a minimax rational approximation rescaled so that
 178 it is evaluated over \[-1,1\].  Note that while the rational approximation
 179 over \[0,6\] converges rapidly to the minimax solution it is rather
 180 ill-conditioned in practice.  Cody and Thacher
 181 [footnote W. J. Cody and H. C. Thacher, Jr.,
 182 Rational Chebyshev approximations for the exponential integral E[sub 1](x),
 183 Math. Comp. 22 (1968), 641-649,
 184 and W. J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the
 185 exponential integral Ei(x), Math. Comp. 23 (1969), 289-303.]
 186 experienced the same issue and
 187 converted the polynomials into Chebeshev form to ensure stable
 188 computation.  By experiment we found that the polynomials are just as stable
 189 in polynomial as Chebyshev form, /provided/ they are computed
 190 over the interval \[-1,1\].
 191
 192 Over the a series of intervals [a,b] and [b,INF] the rational approximation
 193 takes the form:
 194
 195 [equation expint_i_4]
 196
 197 where /c/ is a constant, and R(t) is a minimax solution optimised for low
 198 absolute error compared to /c/.  Variable /t/ is `1/z` when the range in infinite
 199 and `2z/(b-a) - (2a/(b-a) + 1)` otherwise: this has the effect of scaling z to the
 200 interval \[-1,1\].  As before rational approximations over arbitrary intervals
 201 were found to be ill-conditioned: Cody and Thacher solved this issue by
 202 converting the polynomials to their J-Fraction equivalent.  However, as long
 203 as the interval of evaluation was \[-1,1\] and the number of terms carefully chosen,
 204 it was found that the polynomials /could/ be evaluated to suitable precision:
 205 error rates are typically 2 to 3 epsilon which is comparible to the error
 206 rate that Cody and Thacher achieved using J-Fractions, but marginally more
 207 efficient given that fewer divisions are involved.
 208
 209 [endsect]
 210 [endsect]
 211
 212 [/
 213   Copyright 2006 John Maddock and Paul A. Bristow.
 214   Distributed under the Boost Software License, Version 1.0.
 215   (See accompanying file LICENSE_1_0.txt or copy at
 216   http://www.boost.org/LICENSE_1_0.txt).
 217 ]