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MODULE quadratures
IMPLICIT none
integer(kind=8) :: subintervals = 100
INTERFACE
FUNCTION integrate(ibeg, iend, myfun, p) result(val)
IMPLICIT none
! beginning of integration interval
real(kind=8), intent(in) :: ibeg
! ending of integration interval
real(kind=8), intent(in) :: iend
! function to be integrated
procedure(funint) :: myfun
! polynomial order
integer(kind=4), intent(in) :: p
! result of integration
real(kind=8) :: val
END FUNCTION integrate
END INTERFACE
INTERFACE
FUNCTION quadrature(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val
END FUNCTION quadrature
END INTERFACE
INTERFACE
FUNCTION funint(x) result(y)
IMPLICIT none
real(kind=8), intent(in) :: x
real(kind=8) :: y
END FUNCTION funint
END INTERFACE
CONTAINS
FUNCTION integrate_generic(ibeg, iend, fun, quad) result(val)
IMPLICIT none
real(kind=8), intent(in) :: ibeg
real(kind=8), intent(in) :: iend
procedure(funint) :: fun
procedure(quadrature) :: quad
real(kind=8) :: val, subinterval_width, qbeg, qend
real(kind=8), allocatable :: partval[:]
integer(kind=8) :: min_i, max_i, i, subintervals_per_thread
integer(kind=4) :: im
allocate(partval[*])
subintervals_per_thread = &
(subintervals + num_images() - 1) / num_images()
min_i = subintervals_per_thread * (this_image() - 1) + 1
max_i = min(subintervals, subintervals_per_thread * this_image())
subinterval_width = (iend - ibeg) / subintervals
! compute integral using quadrature pointed by quad
partval = 0
DO i = min_i, max_i
qend = ibeg + i * subinterval_width
qbeg = ibeg + (i - 1) * subinterval_width
partval = partval + quad(qbeg, qend, fun)
END DO
IF (this_image() == 1 .and. num_images() > 1) THEN
sync images([(im, im = 2, num_images())])
partval = partval + sum([(partval[im], im = 2, num_images())])
sync images([(im, im = 2, num_images())])
END IF
IF (this_image() /= 1) THEN
sync images(1)
sync images(1)
END IF
val = partval[1]
END FUNCTION integrate_generic
FUNCTION newton_cotes(ibeg, iend, fun, p) result(val)
IMPLICIT none
real(kind=8), intent(in) :: ibeg
real(kind=8), intent(in) :: iend
procedure(funint) :: fun
integer(kind=4), intent(in) :: p
real(kind=8) :: val
procedure(quadrature), pointer :: quad
SELECT CASE (p)
CASE (:-1)
STOP "negative interpolationg polynomial order passed"
CASE (0)
quad => rectangle
CASE (1)
quad => trapeze
CASE (2)
quad => simpson_1_third
CASE default
STOP "Newton-Cotes quadratures only implemented for order < 3"
END SELECT
val = integrate_generic(ibeg, iend, fun, quad)
END FUNCTION newton_cotes
FUNCTION rectangle(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val
val = (qend - qbeg) * fun((qend + qbeg) * 0.5)
END FUNCTION rectangle
FUNCTION trapeze(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val
val = (qend - qbeg) * 0.5 * (fun(qbeg) + fun(qend))
END FUNCTION trapeze
FUNCTION simpson_1_third(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val
val = (qend - qbeg) * (1/6.0) * &
(fun(qbeg) + 4 * fun ((qbeg + qend) * 0.5) + fun(qend))
END FUNCTION simpson_1_third
FUNCTION gauss(ibeg, iend, fun, p) result(val)
IMPLICIT none
real(kind=8), intent(in) :: ibeg
real(kind=8), intent(in) :: iend
procedure(funint) :: fun
integer(kind=4), intent(in) :: p
real(kind=8) :: val
procedure(quadrature), pointer :: quad
SELECT CASE (p)
CASE (:0)
STOP "non-positive Legendre polynomial order passed"
CASE (1)
quad => gauss_n1
CASE (2)
quad => gauss_n2
CASE (3)
quad => gauss_n3
CASE default
STOP "Gauss quadratures only implemented with roots of" &
// " Legendgre polynomial of order <= 3"
END SELECT
val = integrate_generic(ibeg, iend, fun, quad)
END FUNCTION gauss
FUNCTION gauss_generic(mid, halfwidth, fun, points, weights) &
result(val)
IMPLICIT none
real(kind=8), intent(in) :: mid, halfwidth, points(:), weights(:)
procedure(funint) :: fun
real(kind=8) :: val
integer(kind=4) :: i
val = halfwidth * sum(weights * &
[(fun(points(i) * halfwidth + mid), i = 1, size(points))])
END FUNCTION gauss_generic
FUNCTION gauss_n1(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val, weights(1) = [2], points(1) = [0]
val = gauss_generic((qbeg + qend) * 0.5, (qend - qbeg) * 0.5, &
fun, points, weights)
END FUNCTION gauss_n1
FUNCTION gauss_n2(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val, weights(2) = [1, 1], &
points(2) = [1 / sqrt(3.0), -1 / sqrt(3.0)]
val = gauss_generic((qbeg + qend) * 0.5, (qend - qbeg) * 0.5, &
fun, points, weights)
END FUNCTION gauss_n2
FUNCTION gauss_n3(qbeg, qend, fun) result(val)
IMPLICIT none
real(kind=8), intent(in) :: qbeg, qend
procedure(funint) :: fun
real(kind=8) :: val, weights(3) = [8 / 9.0, 5 / 9.0, 5 / 9.0],&
points(3) = [0.0, sqrt(3 / 5.0), -sqrt(3 / 5.0)]
val = gauss_generic((qbeg + qend) * 0.5, (qend - qbeg) * 0.5, &
fun, points, weights)
END FUNCTION gauss_n3
END MODULE quadratures
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