tgamma.qbk

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[section:tgamma Gamma]

[h4 Synopsis]

``
#include <boost/math/special_functions/gamma.hpp>
``

  namespace boost{ namespace math{
  
  template <class T>
  ``__sf_result`` tgamma(T z);
  
  template <class T, class ``__Policy``>
  ``__sf_result`` tgamma(T z, const ``__Policy``&);
  
  template <class T>
  ``__sf_result`` tgamma1pm1(T dz);
  
  template <class T, class ``__Policy``>
  ``__sf_result`` tgamma1pm1(T dz, const ``__Policy``&);
  
  }} // namespaces
  
[h4 Description]

  template <class T>
  ``__sf_result`` tgamma(T z);
  
  template <class T, class ``__Policy``>
  ``__sf_result`` tgamma(T z, const ``__Policy``&);
  
Returns the "true gamma" (hence name tgamma) of value z:

[equation gamm1]

[graph tgamma]

[optional_policy]

There are effectively two versions of the [@http://en.wikipedia.org/wiki/Gamma_function tgamma]
function internally: a fully
generic version that is slow, but reasonably accurate, and a much more
efficient approximation that is used where the number of digits in the significand
of T correspond to a certain __lanczos.  In practice any built in
floating point type you will encounter has an appropriate __lanczos
defined for it.  It is also possible, given enough machine time, to generate
further __lanczos's using the program libs/math/tools/lanczos_generator.cpp.

The return type of this function is computed using the __arg_pomotion_rules:
the result is `double` when T is an integer type, and T otherwise.

  template <class T>
  ``__sf_result`` tgamma1pm1(T dz);
  
  template <class T, class ``__Policy``>
  ``__sf_result`` tgamma1pm1(T dz, const ``__Policy``&);
  
Returns `tgamma(dz + 1) - 1`.  Internally the implementation does not make
use of the addition and subtraction implied by the definition, leading to
accurate results even for very small `dz`.  However, the implementation is
capped to either 35 digit accuracy, or to the precision of the __lanczos
associated with type T, whichever is more accurate.
  
The return type of this function is computed using the __arg_pomotion_rules:
the result is `double` when T is an integer type, and T otherwise.

[optional_policy]

[h4 Accuracy]

The following table shows the peak errors (in units of epsilon) 
found on various platforms with various floating point types, 
along with comparisons to the __gsl, __glibc, __hpc and __cephes libraries.
Unless otherwise specified any floating point type that is narrower
than the one shown will have __zero_error.

[table
[[Significand Size] [Platform and Compiler] [Factorials and Half factorials] [Values Near Zero] [Values Near 1 or 2] [Values Near a Negative Pole]]
[[53]            [Win32 Visual C++ 8]  
[Peak=1.9 Mean=0.7

(GSL=3.9)

(__cephes=3.0)] 
[Peak=2.0 Mean=1.1

(GSL=4.5)

(__cephes=1)]    
[Peak=2.0 Mean=1.1

(GSL=7.9)

(__cephes=1.0)] 
[Peak=2.6 Mean=1.3

(GSL=2.5)

(__cephes=2.7)] ]
[[64]            [Linux IA32 / GCC]  
[Peak=300 Mean=49.5

(__glibc Peak=395 Mean=89)]           
[Peak=3.0 Mean=1.4

(__glibc Peak=11 Mean=3.3)]    
[Peak=5.0 Mean=1.8

(__glibc Peak=0.92 Mean=0.2)]       
[Peak=157 Mean=65

(__glibc Peak=205 Mean=108)]   ]
[[64]            [Linux IA64 / GCC]  
[__glibc Peak 2.8 Mean=0.9

(__glibc Peak 0.7)]           
[Peak=4.8 Mean=1.5

(__glibc Peak 0)]    
[Peak=4.8 Mean=1.5

(__glibc Peak 0)]       

[Peak=5.0 Mean=1.7
(__glibc Peak 0)]           ]
[[113]           [HPUX IA64, aCC A.06.06]    
[Peak=2.5 Mean=1.1

(__hpc Peak 0)] 
[Peak=3.5 Mean=1.7

(__hpc Peak 0)]    
[Peak=3.5 Mean=1.6

(__hpc Peak 0)] 
[Peak=5.2 Mean=1.92

(__hpc Peak 0)]  ]
]

[h4 Testing]

The gamma is relatively easy to test: factorials and half-integer factorials
can be calculated exactly by other means and compared with the gamma function.
In addition, some accuracy tests in known tricky areas were computed at high precision
using the generic version of this function.

The function `tgamma1pm1` is tested against values calculated very naively
using the formula `tgamma(1+dz)-1` with a lanczos approximation accurate
to around 100 decimal digits.

[h4 Implementation]

The generic version of the `tgamma` function is implemented by combining the series and 
continued fraction representations for the incomplete gamma function:

[equation gamm2]

where /l/ is an arbitrary integration limit: choosing [^l = max(10, a)] 
seems to work fairly well.

For types of known precision the __lanczos is used, a traits class 
`boost::math::lanczos::lanczos_traits` maps type T to an appropriate
approximation.  

For z in the range -20 < z < 1 then recursion is used to shift to z > 1 via:

[equation gamm3]

For very small z, this helps to preserve the identity:

[equation gamm4]

For z < -20 the reflection formula:

[equation gamm5]

is used.  Particular care has to be taken to evaluate the `z * sin([pi][space] * z)` part: 
a special routine is used to reduce z prior to multiplying by [pi][space] to ensure that the
result in is the range [0, [pi]/2]. Without this an excessive amount of error occurs
in this region (which is hard enough already, as the rate of change near a negative pole
is /exceptionally/ high).

Finally if the argument is a small integer then table lookup of the factorial
is used.

The function `tgamma1pm1` is implemented using rational approximations [jm_rationals] in the
region `-0.5 < dz < 2`.  These are the same approximations (and internal routines)
that are used for __lgamma, and so aren't detailed further here.  The result of
the approximation is `log(tgamma(dz+1))` which can fed into __expm1 to give
the desired result.  Outside the range `-0.5 < dz < 2` then the naive formula
`tgamma1pm1(dz) = tgamma(dz+1)-1` can be used directly.

[endsect][/section:tgamma The Gamma Function]
[/ 
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]