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Manual Reference Pages  - gamma (3fortran)

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NAME

GAMMA(3) - [MATHEMATICS] Gamma function, which yields factorials for positive whole numbers

SYNOPSIS

result = gamma(x)

         elemental real(kind=**) function gamma( x)

          type(real,kind=**),intent(in) :: x

CHARACTERISTICS

o	X is a real value of any available KIND
o	returns a real value with the same kind as X.

DESCRIPTION

GAMMA(X) computes Gamma of X. For positive whole number values of N the Gamma function can be used to calculate factorials, as (N-1)! == GAMMA(REAL(N)). That is

    n! == gamma(real(n+1))

$$ \GAMMA(x) = \int_0**\infty t**{x-1}{\mathrm{e}}**{-T}\,{\mathrm{d}}t $$

OPTIONS

o	X : Shall be of type real and neither zero nor a negative integer.

RESULT

The return value is of type real of the same kind as x. The result has a value equal to a processor-dependent approximation to the gamma function of X.

EXAMPLES

Sample program:

    program demo_gamma
    use, intrinsic :: iso_fortran_env, only : wp=>real64, int64
    implicit none
    real :: x, xa(4)
    integer :: i, j

       ! basic usage
       x = gamma(1.0)
       write(*,*)’gamma(1.0)=’,x

       ! elemental
       xa=gamma([1.0,2.0,3.0,4.0])
       write(*,*)xa
       write(*,*)


       ! gamma() is related to the factorial function
       do i = 1, 171
          ! check value is not too big for default integer type
          if (factorial(i)  <=  huge(0)) then
             write(*,*) i, nint(factorial(i)), ’integer’
          elseif (factorial(i)  <=  huge(0_int64)) then
             write(*,*) i, nint(factorial(i),kind=int64),’integer(kind=int64)’
          else
             write(*,*) i, factorial(i) , ’user factorial function’
             write(*,*) i, product([(real(j, kind=wp), j=1, i)]), ’product’
             write(*,*) i, gamma(real(i + 1, kind=wp)), ’gamma directly’
          endif
       enddo


    contains
    function factorial(i) result(f)
    !  GAMMA(X) computes Gamma of X. For positive whole number values of N the
    !  Gamma function can be used to calculate factorials, as (N-1)! ==
    !  GAMMA(REAL(N)). That is
    !
    !      n! == gamma(real(n+1))
    !
    integer, intent(in) :: i
    real(kind=wp) :: f
       if (i  <=  0) then
          write(*,’(*(g0))’) ’<ERROR> gamma(3) function value ’, i, ’ <= 0’
          stop ’<STOP> bad value in gamma function’
       endif
       f = anint(gamma(real(i + 1,kind=wp)))
    end function factorial

    end program demo_gamma

Results:

     >  gamma(1.0)=   1.00000000
     >    1.00000000       1.00000000       2.00000000       6.00000000
     >
     >            1           1 integer
     >            2           2 integer
     >            3           6 integer
     >            4          24 integer
     >            5         120 integer
     >            6         720 integer
     >            7        5040 integer
     >            8       40320 integer
     >            9      362880 integer
     >           10     3628800 integer
     >           11    39916800 integer
     >           12   479001600 integer
     >           13           6227020800 integer(kind=int64)
     >           14          87178291200 integer(kind=int64)
     >           15        1307674368000 integer(kind=int64)
     >           16       20922789888000 integer(kind=int64)
     >           17      355687428096000 integer(kind=int64)
     >           18     6402373705728001 integer(kind=int64)
     >           19   121645100408832000 integer(kind=int64)
     >           20  2432902008176640000 integer(kind=int64)
     >           21   5.1090942171709440E+019 user factorial function
     >           21   5.1090942171709440E+019 product
     >           21   5.1090942171709440E+019 gamma directly
     >            :
     >            :
     >            :
     >          170   7.2574156153079990E+306 user factorial function
     >          170   7.2574156153079940E+306 product
     >          170   7.2574156153079990E+306 gamma directly
     >          171                  Infinity user factorial function
     >          171                  Infinity product
     >          171                  Infinity gamma directly

STANDARD

Fortran 2008

SEE ALSO

Logarithm of the Gamma function: LOG_GAMMA(3)

RESOURCES

Wikipedia: Gamma_function

fortran-lang intrinsic descriptions

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Nemo Release 3.1

gamma (3fortran)

April 28, 2024

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