在Fortran 95中编程为从数学计算Gamma函数值的函数无法产生正确的值。
我正在尝试在Fortran 95中实现一个递归函数,该递归函数使用Lanczos approximation计算Gamma函数的值(是的,我知道在2003标准及更高版本中对此具有内在函数)。我已经非常严格地遵循了标准公式,所以不确定什么地方出了问题。 Gamma函数的正确值对于我正在进行的其他一些数值计算至关重要,这涉及通过递归关系对Jacobi多项式进行数值计算。
program testGam
implicit none
integer, parameter :: dp = selected_real_kind(15,307)
real(dp), parameter :: pi = 3.14159265358979324
real(dp), dimension(10) :: xGam, Gam
integer :: n
xGam = (/ -3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 /)
do n = 1,10
Gam(n) = GammaFun(xGam(n))
end do
do n = 1,10
write(*,*) xGam(n), Gam(n)
end do
contains
recursive function GammaFun(x) result(G)
real(dp), intent(in) :: x
real(dp) :: G
real(dp), dimension(0:8), parameter :: q = &
(/ 0.99999999999980993, 676.5203681218851, -1259.1392167224028, &
771.32342877765313, -176.61502916214059, 12.507343278686905, &
-0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 /)
real(dp) :: t, w, xx
integer :: n
xx = x
if ( xx < 0.5_dp ) then
G = pi / ( sin(pi*xx)*GammaFun(1.0_dp - xx) )
else
xx = xx - 1.0_dp
t = q(0)
do n = 1,9
t = t + q(n) / (xx + real(n, dp))
end do
w = xx + 7.5_dp
G = sqrt(2.0_dp*pi)*(w**(xx + 0.5_dp))*exp(-w)*t
end if
end function GammaFun
end program testGam
尽管此代码应在整个实线上为Gamma函数生成正确的值,但无论输入如何,似乎只能生成接近122的恒定值。我怀疑我没有看到一些奇怪的浮点算术问题。
答案 0 :(得分:3)
您的代码有两个明显的问题
do n = 1,9 t = t + q(n) / (xx + real(n, dp)) end do
至少在我看来,使我相信这些是对程序的正确修复,它们可以正确编译,链接和运行。见下文:
ian@eris:~/work/stackoverflow$ cat g.f90
program testGam
implicit none
integer, parameter :: dp = selected_real_kind(15,307)
real(dp), parameter :: pi = 3.14159265358979324_dp
real(dp), dimension(10) :: xGam, Gam
integer :: n
xGam = (/ -3.5_dp, -2.5_dp, -1.5_dp, -0.5_dp, 0.5_dp, 1.5_dp, 2.5_dp, 3.5_dp, 4.5_dp, 5.5_dp /)
do n = 1,10
Gam(n) = GammaFun(xGam(n))
end do
do n = 1,10
write(*,*) xGam(n), Gam(n), gamma( xGam( n ) ), Abs( Gam( n ) - gamma( xGam( n ) ) )
end do
contains
recursive function GammaFun(x) result(G)
real(dp), intent(in) :: x
real(dp) :: G
real(dp), dimension(0:8), parameter :: q = &
(/ 0.99999999999980993_dp, 676.5203681218851_dp, -1259.1392167224028_dp, &
771.32342877765313_dp, -176.61502916214059_dp, 12.507343278686905_dp, &
-0.13857109526572012_dp, 9.9843695780195716e-6_dp, 1.5056327351493116e-7_dp /)
real(dp) :: t, w, xx
integer :: n
xx = x
if ( xx < 0.5_dp ) then
G = pi / ( sin(pi*xx)*GammaFun(1.0_dp - xx) )
else
xx = xx - 1.0_dp
t = q(0)
do n = 1,8
t = t + q(n) / (xx + real(n, dp))
end do
w = xx + 7.5_dp
G = sqrt(2.0_dp*pi)*(w**(xx + 0.5_dp))*exp(-w)*t
end if
end function GammaFun
end program testGam
ian@eris:~/work/stackoverflow$ gfortran -O -std=f2008 -Wall -Wextra -fcheck=all g.f90
ian@eris:~/work/stackoverflow$ ./a.out
-3.5000000000000000 0.27008820585226917 0.27008820585226906 1.1102230246251565E-016
-2.5000000000000000 -0.94530872048294168 -0.94530872048294179 1.1102230246251565E-016
-1.5000000000000000 2.3632718012073521 2.3632718012073548 2.6645352591003757E-015
-0.50000000000000000 -3.5449077018110295 -3.5449077018110318 2.2204460492503131E-015
0.50000000000000000 1.7724538509055159 1.7724538509055161 2.2204460492503131E-016
1.5000000000000000 0.88622692545275861 0.88622692545275805 5.5511151231257827E-016
2.5000000000000000 1.3293403881791384 1.3293403881791370 1.3322676295501878E-015
3.5000000000000000 3.3233509704478430 3.3233509704478426 4.4408920985006262E-016
4.5000000000000000 11.631728396567446 11.631728396567450 3.5527136788005009E-015
5.5000000000000000 52.342777784553583 52.342777784553519 6.3948846218409017E-014
ian@eris:~/work/stackoverflow$