Created
February 27, 2011 17:14
-
-
Save pozdneev/846338 to your computer and use it in GitHub Desktop.
Gaussian partitioned elimination without pivoting using straightforward formulas and BLAS routines
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
! Gaussian partitioned elimination without pivoting using straightforward | |
! formulas and BLAS routines | |
!!--------------------------------------------------------------------------- | |
! gfortran -ffree-form -Wall -Wextra -lblas -fopenmp | |
!!--------------------------------------------------------------------------- | |
! | |
! [1] J. Demmel "Numerical Linear Algebra" | |
! [2] N. J. Nigham "Gaussian Elimination" | |
! | |
! ELIMINATION | |
! for k = 1: n-1 | |
! for i = k+1: n | |
! a(i, k) /= a(k, k) | |
! for i = k+1: n | |
! for j = k+1: n | |
! a(i, j) -= a(i, k) * a(k, j) | |
! end | |
! end | |
! end | |
! | |
! BACKSUBSTITUTION - L U y = f => L x = f => x = L \ f => y = U \ x | |
! for i = 1: n | |
! x(i) = f(i) | |
! for j = 1: i-1 | |
! x(i) -= a(i, j) * x(j) | |
! end | |
! end | |
! | |
! for i = n: 1 | |
! y(i) = x(i) | |
! for j = n: i+1 | |
! y(i) -= a(i, j) * y(j) | |
! end | |
! y(i) /= a(i, i) | |
! end | |
!!--------------------------------------------------------------------------- | |
program ge | |
!!--------------------------------------------------------------------------- | |
use omp_lib, only: omp_get_wtime | |
implicit none | |
! Matrix of coefficients; the one is filled by random_number() | |
real, dimension(:, :), allocatable :: A | |
! "Analytical" solution; the one is filled by random_number() | |
real, dimension(:), allocatable :: u | |
! Right-hand side (RHS); the one is calculated as f = A * u | |
! Numerical solution (NS) of the equation A y = f | |
! RHS is overwritten by NS | |
real, dimension(:), allocatable :: y | |
! Size of block | |
integer, parameter :: b = 4 | |
! Size of matrix; must be a multiple of b | |
integer, parameter :: n = b * 3 | |
! Time marks | |
real(kind(0.d0)) :: elimination_start, elimination_finish | |
real(kind(0.d0)) :: backsubstition_start, backsubstition_finish | |
! Allocate memory | |
allocate(A(1: n, 1: n)) | |
allocate(u(1: n)) | |
allocate(y(1: n)) | |
! Algorithm uses straightforward formulas | |
call Generate_Data() | |
call Straightforward_Elimination() | |
call Straightforward_Backsubstition() | |
call Print_Norms() | |
call Print_Times() | |
! Algorithm uses BLAS | |
call Generate_Data() | |
call BLAS_Elimination() | |
call BLAS_Backsubstition() | |
call Print_Norms() | |
call Print_Times() | |
! Free memory | |
deallocate(A) | |
deallocate(u) | |
deallocate(y) | |
!!--------------------------------------------------------------------------- | |
contains | |
!!--------------------------------------------------------------------------- | |
subroutine Print_Norms() | |
write (*, *) "Norms:", maxval(abs(u)), maxval(abs(y - u)) | |
end subroutine Print_Norms | |
!!--------------------------------------------------------------------------- | |
subroutine Print_Times() | |
write (*, *) "Times:", & | |
elimination_finish - elimination_start, & | |
backsubstition_finish - backsubstition_start | |
end subroutine Print_Times | |
!!--------------------------------------------------------------------------- | |
! This version is a simplified modification of | |
! http://gcc.gnu.org/onlinedocs/gfortran/RANDOM_005fSEED.html | |
subroutine Init_Random_Seed() | |
integer :: i, n | |
integer, dimension(:), allocatable :: seed | |
call random_seed(size = n) | |
allocate(seed(n)) | |
seed = 37 * (/ (i - 1, i = 1, n) /) | |
call random_seed(put = seed) | |
deallocate(seed) | |
end subroutine Init_Random_Seed | |
!!--------------------------------------------------------------------------- | |
subroutine Generate_Data() | |
call Init_Random_Seed() | |
call random_number(A) | |
call random_number(u) | |
y = matmul(A, u) | |
end subroutine Generate_Data | |
!!--------------------------------------------------------------------------- | |
subroutine Straightforward_Elimination() | |
integer :: ib | |
integer :: i, j, k | |
integer :: p1b, p1e, p2b, p2e | |
elimination_start = omp_get_wtime() | |
do ib = 0, n/b - 1 | |
p1b = ib * b + 1 | |
p1e = (ib + 1) * b | |
p2b = p1e + 1 | |
p2e = n | |
! L_{11} U_{11} = A_{11} | |
do k = p1b, p1e | |
do i = k+1, p1e | |
a(i, k) = a(i, k) / a(k, k) | |
end do | |
do j = k+1, p1e | |
do i = k+1, p1e | |
a(i, j) = a(i, j) - a(i, k) * a(k, j) | |
end do | |
end do | |
end do | |
! L_{21} = A_{21} U_{11}^{-1} | |
do k = p1b, p1e | |
do i = p2b, p2e | |
do j = p1b, k-1 | |
a(i, k) = a(i, k) - a(i, j) * a(j, k) | |
end do | |
a(i, k) = a(i, k) / a(k, k) | |
end do | |
end do | |
! U_{12} = L_{11} \ A_{12} | |
do j = p2b, p2e | |
do k = p1b, p1e | |
do i = p1b, k-1 | |
a(k, j) = a(k, j) - a(k, i) * a(i, j) | |
end do | |
end do | |
end do | |
! A_{22} -= L_{21} U_{12} | |
do j = p2b, p2e | |
do k = p1b, p1e | |
do i = p2b, p2e | |
a(i, j) = a(i, j) - a(i, k) * a(k, j) | |
end do | |
end do | |
end do | |
end do | |
elimination_finish = omp_get_wtime() | |
end subroutine Straightforward_Elimination | |
!!--------------------------------------------------------------------------- | |
subroutine BLAS_Elimination() | |
integer :: ib | |
integer :: k | |
integer :: p1b, p1e, p2b, p2e | |
elimination_start = omp_get_wtime() | |
do ib = 0, n/b - 1 | |
p1b = ib * b + 1 | |
p1e = (ib + 1) * b | |
p2b = p1e + 1 | |
p2e = n | |
! L_{11} U_{11} = A_{11} | |
do k = p1b, p1e | |
! x = a*x | |
call sscal(p1e-k, 1.0 / a(k, k), a(k+1, k), 1) | |
! A := alpha*x*y'+ A | |
call sger(p1e-k, p1e-k, -1.0, & | |
a(k+1, k), 1, & | |
a(k, k+1), n, & | |
a(k+1, k+1), n) | |
end do | |
! L_{21} U_{11} = A_{21} => L_{21} = A_{21} U_{11}^{-1} | |
! X*op(A) = alpha*B | |
call strsm('R', 'U', 'N', 'N', p2e-p2b+1, b, 1.0, & | |
a(p1b, p1b), n, a(p2b, p1b), n) | |
! L_{11} U_{12} = A_{12} => U_{12} = L_{11} \ A_{12} | |
! op(A)*X = alpha*B | |
call strsm('L', 'L', 'N', 'U', b, p2e-p2b+1, 1.0, & | |
a(p1b, p1b), n, a(p1b, p2b), n) | |
! A_{22} -= L_{21} U_{12} | |
call sgemm('N', 'N', p2e-p2b+1, p2e-p2b+1, b, -1.0, & | |
a(p2b, p1b), n, a(p1b, p2b), n, 1.0, a(p2b, p2b), n) | |
end do | |
elimination_finish = omp_get_wtime() | |
end subroutine BLAS_Elimination | |
!!--------------------------------------------------------------------------- | |
subroutine Straightforward_Backsubstition() | |
integer :: i, j | |
backsubstition_start = omp_get_wtime() | |
! L x = f => x = L \ f | |
do i = 1, n | |
do j = 1, i-1 | |
y(i) = y(i) - a(i, j) * y(j) | |
end do | |
end do | |
! U y = x => y = U \ x | |
do i = n, 1, -1 | |
do j = i+1, n | |
y(i) = y(i) - a(i, j) * y(j) | |
end do | |
y(i) = y(i) / a(i, i) | |
end do | |
backsubstition_finish = omp_get_wtime() | |
end subroutine Straightforward_Backsubstition | |
!!--------------------------------------------------------------------------- | |
subroutine BLAS_Backsubstition() | |
backsubstition_start = omp_get_wtime() | |
! L x = f => x = L \ f | |
! op(A)*X = alpha*B | |
!call strsm('L', 'L', 'N', 'U', n, 1, 1.0, a, n, y, n) | |
! A * x = b | |
call strsv('L', 'N', 'U', n, a, n, y, 1) | |
! U y = x => y = U \ x | |
! op(A)*X = alpha*B | |
!call strsm('L', 'U', 'N', 'N', n, 1, 1.0, a, n, y, n) | |
! A * x = b | |
call strsv('U', 'N', 'N', n, a, n, y, 1) | |
backsubstition_finish = omp_get_wtime() | |
end subroutine BLAS_Backsubstition | |
!!--------------------------------------------------------------------------- | |
end program ge | |
!!--------------------------------------------------------------------------- |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment