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module proc
implicit none
real :: x,y,z,a
contains
!! Definition des fonctions
function func(a)
implicit none
real, intent(in) :: a
real :: func, aire1
call romberg(func1,0.0,1.0,0.001,aire1)
func = aire1
end function func
function func1(z)
implicit none
real, intent(in):: z
real :: func1, aire2
call romberg(func2,0.0,z,0.001,aire2)
func1 = aire2
end function func1
function func2(y)
implicit none
real, intent(in):: y
real :: func2, aire3
call romberg(func3,0.0,y,0.001,aire3)
func2 = aire3
end function func2
function func3(x)
implicit none
real, intent(in):: x
real :: func3
!! Test function of 3 variables
func3 = a*(a**2*x+0.44*y+2.0*z)
end function func3
subroutine romberg(fn,a,b,eps,aire)
!! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - !!
!! !!
!! Integration par la methode de Romberg !!
!! Dominique Lefebvre fevrier 2007 !!
!! Inspire du code de A.Garcia dans Numerical Methods for Physics !!
!! Modifiee le 30/08/09 suite a une erreur d'estimation de la precision !!
!! signalee par Maelle Nodet (Universite de Grenoble) !!
!! a = borne inferieure d'integration !!
!! b = borne superieure d'integration !!
!! eps = precision souhaitee !!
!! aire = surface retournee !!
!! !!
!! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - !!
implicit none
real , intent(in) :: a, b, eps
real , intent(out):: aire
real :: h,xi, delta, sum
real :: c, fotom, t(136)
integer :: out, pieces, i, nx(16)
integer :: nt, ii, n, nn, l, ntra, k, m, j, l2
logical :: fini
!! Interface ajoutee par saidou Diallo
interface
function fn(x)
implicit none
real, intent(in) :: x
real :: fn
end function fn
end interface
!! calcul du premier terme
h = b-a
pieces = 1
nx(1) = 1
delta = h / pieces
c = (fn(a) + fn(b)) / 2.0
sum = c
t(1) = delta * c
n = 1
nn = 2
fini =.false.
do while (.not. fini)
n = n + 1
fotom = 4.0
nx(n) = nn
pieces = pieces * 2
l = pieces - 1
l2 = (l+1) / 2
delta = h / pieces
!! evaluation par la methode des trapezes
do ii = 1, l2
i = ii * 2 - 1
xi = a + delta * i
sum = sum + fn(xi)
enddo
t(nn) = sum * delta
ntra = nx(n-1)
k = n-1
!! calcul de la nieme colonne du tableau
do m = 1, k
j = nn + m
nt = nx(n-1) + m - 1
t(j) = (fotom * t(j-1) - t(nt)) / (fotom - 1)
fotom = fotom * 4
enddo
!! estimation de la precision
if (n < 5) then
nn = j+1
else if (t(nn+1) == 0.0) then
nn = j+1
else
if (abs(t(ntra+1)-t(nn+1)) <= abs(t(nn+1)*eps)) then
fini = .true.
elseif (abs(t(nn-1) - t(j)) <= abs(t(j)*eps)) then
fini = .true.
else
nn = j+1
endif
endif
enddo
!! on retourne la valeur finale de l'aire
aire = t(j)
return
end subroutine romberg
end module proc
!! Main program
program test
use proc
implicit none
a = 1.0
print*, func(a)
end program test |
Integrations successives d'une fonction à plusieurs variables
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