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cosolver/FM/FMsamples/LinSolveZM.f95
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LinSolveZM.f95
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SUBROUTINE ZM_LIN_SOLVE(A,X,B,N,DET)
USE FMVALS
USE FMZM
IMPLICIT NONE
! Gauss elimination to solve the linear system A X = B, where:
! A is the matrix of the system, containing the N x N coefficient matrix.
! B is the N x 1 right-hand-side vector.
! X is the returned N x 1 solution vector.
! DET is returned as the determinant of A.
! Nonzero DET means a solution was found.
! DET = 0 is returned if the system is singular.
! A,X,B,DET are all type (zm) complex multiprecision variables.
INTEGER :: N
TYPE (ZM) :: A(N,N), B(N), X(N), DET
TYPE (FM), SAVE :: TOL
TYPE (ZM), ALLOCATABLE :: A1(:,:), A2(:,:), B1(:), R1(:), X1(:)
INTEGER, ALLOCATABLE :: KSWAP(:)
INTEGER :: I, J, NDSAVE
CALL FM_ENTER_USER_ROUTINE
ALLOCATE(A1(N,N),A2(N,N),B1(N),R1(N),X1(N),KSWAP(N),STAT=J)
IF (J /= 0) THEN
WRITE (*,"(/' Error in ZM_LIN_SOLVE. Unable to allocate arrays with N = ',I8/)") N
STOP
ENDIF
TOL = EPSILON(TO_FM(1))/MBASE/TO_FM(10)**10
NDSAVE = NDIG
NDIG = 2*NDIG
! Copy A and B to A1 and B1 with higher precision.
110 CALL FM_EQU_R1(TOL,NDSAVE,NDIG)
DO I = 1, N
DO J = 1, N
CALL ZM_EQU(A(I,J),A1(I,J),NDSAVE,NDIG)
CALL ZM_EQ(A1(I,J),A2(I,J))
ENDDO
CALL ZM_EQU(B(I),B1(I),NDSAVE,NDIG)
ENDDO
! Solve the system.
CALL ZM_FACTOR_LU(A1,N,DET,KSWAP)
IF (DET == 0 .OR. IS_UNKNOWN(DET)) THEN
IF (KWARN > 0) THEN
WRITE (KW,"(/' Error in ZM_LIN_SOLVE. The matrix is singular.'/)")
ENDIF
IF (KWARN >= 2) STOP
X1 = TO_ZM(' UNKNOWN ')
GO TO 120
ENDIF
CALL ZM_SOLVE_LU(A1,N,B1,X1,KSWAP)
! Do an iterative refinement.
R1 = MATMUL(A2,X1) - B1
CALL ZM_SOLVE_LU(A1,N,R1,B1,KSWAP)
X1 = X1 - B1
! Check for accuracy at the user's precision.
IF (SQRT( ABS(DOT_PRODUCT( B1 , B1 )) ) > TOL) THEN
NDIG = 2*NDIG
GO TO 110
ENDIF
! Round and return X and DET.
120 DO I = 1, N
CALL ZM_EQU(X1(I),X(I),NDIG,NDSAVE)
ENDDO
CALL ZMEQU_R1(DET%MZM,NDIG,NDSAVE)
NDIG = NDSAVE
CALL FM_DEALLOCATE(A1)
CALL FM_DEALLOCATE(A2)
CALL FM_DEALLOCATE(B1)
CALL FM_DEALLOCATE(R1)
CALL FM_DEALLOCATE(X1)
DEALLOCATE(A1,A2,B1,R1,X1,KSWAP)
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE ZM_LIN_SOLVE
SUBROUTINE ZM_FACTOR_LU(A,N,DET,KSWAP)
USE FMZM
IMPLICIT NONE
! Gauss elimination to factor the NxN matrix A (LU decomposition).
! The time is proportional to N**3.
! Once this factorization has been done, a linear system A x = b
! with the same coefficient matrix A and Nx1 vector b can be solved
! for x using routine ZM_SOLVE_LU in time proportional to N**2.
! DET is returned as the determinant of A.
! Nonzero DET means there is a unique solution.
! DET = 0 is returned if the system is singular.
! KSWAP is a list of row interchanges made by the partial pivoting strategy during the
! elimination phase.
! After returning, the values in matrix A have been replaced by the multipliers
! used during elimination. This is equivalent to factoring the A matrix into
! a lower triangular matrix L times an upper triangular matrix U.
INTEGER :: N
INTEGER :: JCOL, JDIAG, JMAX, JROW, KSWAP(N)
TYPE (ZM) :: A(N,N), DET
TYPE (ZM), SAVE :: AMAX, AMULT, TEMP
CALL FM_ENTER_USER_ROUTINE
DET = 1
KSWAP(1:N) = 1
IF (N <= 0) THEN
DET = 0
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
IF (N == 1) THEN
KSWAP(1) = 1
DET = A(1,1)
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
! Do the elimination phase.
! JDIAG is the current diagonal element below which the elimination proceeds.
DO JDIAG = 1, N-1
! Pivot to put the element with the largest absolute value on the diagonal.
AMAX = ABS(A(JDIAG,JDIAG))
JMAX = JDIAG
DO JROW = JDIAG+1, N
IF (ABS(A(JROW,JDIAG)) > ABS(AMAX)) THEN
AMAX = ABS(A(JROW,JDIAG))
JMAX = JROW
ENDIF
ENDDO
! If AMAX is zero here then the system is singular.
IF (AMAX == 0.0) THEN
DET = 0
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
! Swap rows JDIAG and JMAX unless they are the same row.
KSWAP(JDIAG) = JMAX
IF (JMAX /= JDIAG) THEN
DET = -DET
DO JCOL = JDIAG, N
TEMP = A(JDIAG,JCOL)
A(JDIAG,JCOL) = A(JMAX,JCOL)
A(JMAX,JCOL) = TEMP
ENDDO
ENDIF
DET = DET * A(JDIAG,JDIAG)
! For JROW = JDIAG+1, ..., N, eliminate A(JROW,JDIAG) by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG / A(JDIAG,JDIAG)
DO JROW = JDIAG+1, N
IF (A(JROW,JDIAG) == 0) CYCLE
AMULT = A(JROW,JDIAG)/A(JDIAG,JDIAG)
! Save the multiplier for use later by ZM_SOLVE_LU.
A(JROW,JDIAG) = AMULT
DO JCOL = JDIAG+1, N
A(JROW,JCOL) = A(JROW,JCOL) - AMULT*A(JDIAG,JCOL)
ENDDO
ENDDO
ENDDO
DET = DET * A(N,N)
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE ZM_FACTOR_LU
SUBROUTINE ZM_SOLVE_LU(A,N,B,X,KSWAP)
USE FMZM
IMPLICIT NONE
! Solve a linear system A x = b.
! A is the NxN coefficient matrix, after having been factored by ZM_FACTOR_LU.
! B is the Nx1 right-hand-side vector.
! X is returned with the solution of the linear system.
! KSWAP is a list of row interchanges made by the partial pivoting strategy during the
! elimination phase in ZM_FACTOR_LU.
! Time for this call is proportional to N**2.
INTEGER :: N, KSWAP(N)
TYPE (ZM) :: A(N,N), B(N), X(N)
INTEGER :: J, JDIAG, JMAX
TYPE (ZM), SAVE :: TEMP
CALL FM_ENTER_USER_ROUTINE
IF (N <= 0) THEN
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
IF (N == 1) THEN
X(1) = B(1) / A(1,1)
CALL FM_EXIT_USER_ROUTINE
RETURN
ENDIF
DO J = 1, N
X(J) = B(J)
ENDDO
! Do the elimination phase operations only on X.
! JDIAG is the current diagonal element below which the elimination proceeds.
DO JDIAG = 1, N-1
! Pivot to put the element with the largest absolute value on the diagonal.
JMAX = KSWAP(JDIAG)
! Swap rows JDIAG and JMAX unless they are the same row.
IF (JMAX /= JDIAG) THEN
TEMP = X(JDIAG)
X(JDIAG) = X(JMAX)
X(JMAX) = TEMP
ENDIF
! For JROW = JDIAG+1, ..., N, eliminate A(JROW,JDIAG) by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG / A(JDIAG,JDIAG)
! After factoring, A(JROW,JDIAG) is the original A(JROW,JDIAG) / A(JDIAG,JDIAG).
DO J = JDIAG+1, N
X(J) = X(J) - A(J,JDIAG) * X(JDIAG)
ENDDO
ENDDO
! Do the back substitution.
DO JDIAG = N, 1, -1
! Divide row JDIAG by the diagonal element.
X(JDIAG) = X(JDIAG) / A(JDIAG,JDIAG)
! Zero above the diagonal in column JDIAG by replacing row JROW by
! row JROW - A(JROW,JDIAG) * row JDIAG
! For JROW = 1, ..., JDIAG-1.
IF (JDIAG == 1) EXIT
DO J = 1, JDIAG-1
X(J) = X(J) - A(J,JDIAG) * X(JDIAG)
ENDDO
ENDDO
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE ZM_SOLVE_LU
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