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program testGamma
implicit None
integer,parameter::dp=selected_real_kind(10,30)
real ( kind = dp ) bound,scale,shape,p,q,score
shape=6
scale=20
score=1800
write(*,*)(score/scale),shape," --- ",1.0_dp-gammad(score/scale,shape)
contains
!======================================================================
! Gamma density function
!======================================================================
FUNCTION gammad(x, p) RESULT(fn_val)
! ALGORITHM AS239 APPL. STATIST. (1988) VOL. 37, NO. 3
! Computation of the Incomplete Gamma Integral
! Auxiliary functions required: ALOGAM = logarithm of the gamma
! function, and ALNORM = algorithm AS66
! ELF90-compatible version by Alan Miller
! Latest revision - 27 October 2000
! N.B. Argument IFAULT has been removed
IMPLICIT NONE
INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(12, 60)
REAL (dp), INTENT(IN) :: x, p
REAL (dp) :: fn_val
! Local variables
REAL (dp) :: pn1, pn2, pn3, pn4, pn5, pn6, arg, c, rn, a, b, an
REAL (dp), PARAMETER :: zero = 0.d0, one = 1.d0, two = 2.d0, &
oflo = 1.d+37, three = 3.d0, nine = 9.d0, &
tol = 1.d-20, xbig = 1.d+8, plimit = 1000.d0, &
elimit = -88.d0
INTEGER :: ifault
! EXTERNAL alogam, alnorm
fn_val = zero
! Check that we have valid values for X and P
IF (p <= zero .OR. x < zero) THEN
WRITE(*, *) 'AS239: Either p <= 0 or x < 0'
RETURN
END IF
IF (x == zero) RETURN
! Use a normal approximation if P > PLIMIT
IF (p > plimit) THEN
pn1 = three * SQRT(p) * ((x / p) ** (one / three) + one /(nine * p) - one)
fn_val = alnorm(pn1, .false.)
RETURN
END IF
! If X is extremely large compared to P then set fn_val = 1
IF (x > xbig) THEN
fn_val = one
RETURN
END IF
IF (x <= one .OR. x < p) THEN
! Use Pearson's series expansion.
! (Note that P is not large enough to force overflow in ALOGAM).
! No need to test IFAULT on exit since P > 0.
arg = p * LOG(x) - x - alogam(p + one, ifault)
c = one
fn_val = one
a = p
40 a = a + one
c = c * x / a
fn_val = fn_val + c
IF (c > tol) GO TO 40
arg = arg + LOG(fn_val)
fn_val = zero
IF (arg >= elimit) fn_val = EXP(arg)
ELSE
! Use a continued fraction expansion
arg = p * LOG(x) - x - alogam(p, ifault)
a = one - p
b = a + x + one
c = zero
pn1 = one
pn2 = x
pn3 = x + one
pn4 = x * b
fn_val = pn3 / pn4
60 a = a + one
b = b + two
c = c + one
an = a * c
pn5 = b * pn3 - an * pn1
pn6 = b * pn4 - an * pn2
IF (ABS(pn6) > zero) THEN
rn = pn5 / pn6
IF (ABS(fn_val - rn) <= MIN(tol, tol * rn)) GO TO 80
fn_val = rn
END IF
pn1 = pn3
pn2 = pn4
pn3 = pn5
pn4 = pn6
IF (ABS(pn5) >= oflo) THEN
! Re-scale terms in continued fraction if terms are large
pn1 = pn1 / oflo
pn2 = pn2 / oflo
pn3 = pn3 / oflo
pn4 = pn4 / oflo
END IF
GO TO 60
80 arg = arg + LOG(fn_val)
fn_val = one
IF (arg >= elimit) fn_val = one - EXP(arg)
END IF
write(*,*)EXP(arg),fn_val
RETURN
END FUNCTION gammad
function alogam ( x, ifault )
!*****************************************************************************80
!
!! ALOGAM computes the logarithm of the Gamma function.
!
! Modified:
!
! 28 March 1999
!
! Author:
!
! Original FORTRAN77 version by Malcolm Pike, David Hill.
! FORTRAN90 version by John Burkardt.
!
! Reference:
!
! Malcolm Pike, David Hill,
! Algorithm 291:
! Logarithm of Gamma Function,
! Communications of the ACM,
! Volume 9, Number 9, September 1966, page 684.
!
! Parameters:
!
! Input, real ( kind = 8 ) X, the argument of the Gamma function.
! X should be greater than 0.
!
! Output, integer ( kind = 4 ) IFAULT, error flag.
! 0, no error.
! 1, X <= 0.
!
! Output, real ( kind = 8 ) ALOGAM, the logarithm of the Gamma
! function of X.
!
implicit none
INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(12, 60)
real ( kind = dp ) alogam
real ( kind = dp ) f
integer :: ifault
real ( kind = dp ) x
real ( kind = dp ) y
real ( kind = dp ) z
if ( x <= 0.0D+00 ) then
ifault = 1
alogam = 0.0D+00
return
end if
ifault = 0
y = x
if ( x < 7.0D+00 ) then
f = 1.0D+00
z = y
do while ( z < 7.0D+00 )
f = f * z
z = z + 1.0D+00
end do
y = z
f = - log ( f )
else
f = 0.0D+00
end if
z = 1.0D+00 / y / y
alogam = f + ( y - 0.5D+00 ) * log ( y ) - y &
+ 0.918938533204673D+00 + &
((( &
- 0.000595238095238D+00 * z &
+ 0.000793650793651D+00 ) * z &
- 0.002777777777778D+00 ) * z &
+ 0.083333333333333D+00 ) / y
return
end function alogam
! Algorithm AS66 Applied Statistics (1973) vol.22, no.3
! Evaluates the tail area of the standardised normal curve
! from x to infinity if upper is .true. or
! from minus infinity to x if upper is .false.
! ELF90-compatible version by Alan Miller
! Latest revision - 29 November 2001
function alnorm ( x, upper )
!*****************************************************************************80
!
!! ALNORM computes the cumulative density of the standard normal distribution.
!
! Licensing:
!
! This code is distributed under the GNU LGPL license.
!
! Modified:
!
! 17 January 2008
!
! Author:
!
! Original FORTRAN77 version by David Hill
! MATLAB version by John Burkardt
!
! Reference:
!
! David Hill,
! Algorithm AS 66:
! The Normal Integral,
! Applied Statistics,
! Volume 22, Number 3, 1973, pages 424-427.
!
! Parameters:
!
! Input, real X, is one endpoint of the semi-infinite interval
! over which the integration takes place.
!
! Input, logical UPPER, determines whether the upper or lower
! interval is to be integrated:
! 1 => integrate from X to + Infinity;
! 0 => integrate from - Infinity to X.
!
! Output, real VALUE, the integral of the standard normal
! distribution over the desired interval.
!
implicit none
INTEGER, PARAMETER :: dp = SELECTED_REAL_KIND(12, 60)
real(kind=dp), parameter :: a1 = 5.75885480458
real(kind=dp), parameter :: a2 = 2.62433121679
real(kind=dp), parameter :: a3 = 5.92885724438
real(kind=dp), parameter :: b1 = -29.8213557807
real(kind=dp), parameter :: b2 = 48.6959930692
real(kind=dp), parameter :: c1 = -0.000000038052
real(kind=dp), parameter :: c2 = 0.000398064794
real(kind=dp), parameter :: c3 = -0.151679116635
real(kind=dp), parameter :: c4 = 4.8385912808
real(kind=dp), parameter :: c5 = 0.742380924027
real(kind=dp), parameter :: c6 = 3.99019417011
real(kind=dp), parameter :: con = 1.28
real(kind=dp), parameter :: d1 = 1.00000615302
real(kind=dp), parameter :: d2 = 1.98615381364
real(kind=dp), parameter :: d3 = 5.29330324926
real(kind=dp), parameter :: d4 = -15.1508972451
real(kind=dp), parameter :: d5 = 30.789933034
real(kind=dp), parameter :: ltone = 7.0
real(kind=dp), parameter :: p = 0.39894228044
real(kind=dp), parameter :: q = 0.39990348504
real(kind=dp), parameter :: r = 0.398942280385
real(kind=dp), parameter :: utzero = 18.66
real(kind=dp) ::x,y,z,alnorm
logical ::up,upper
up = upper
z = x
if (z .lt. 0.0D+00 )then
up = .not. up
z = - z
end if
if ( z .gt. ltone .and. ( ( .not. up ) .or. utzero .lt. z ) ) then
if ( up ) then
alnorm = 0.0D+00
else
alnorm = 1.0D+00
end if
return
end if
y = 0.5D+00 * z * z
if ( z .le. con ) then
alnorm = 0.5D+00 - z * (p-q*y/(y+a1+b1/(y+a2+b2/( y + a3 ))))
else
alnorm = r * exp ( - y )/(z+c1+d1/(z+c2+d2/(z+c3+d3/(z+c4+d4/(z+c5+d5/(z+c6))))))
end if
if ( .not. up ) then
alnorm = 1.0D+00 - alnorm
end if
return
end function alnorm
end program testGamma |
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