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cosolver/FM/FMsamples/EigenSystemFM.f95
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EigenSystemFM.f95
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! This program finds the eigenvalues and eigenvectors for a random 20x20 real matrix.
! The peculiar algorithm used is not a very good way to do this, but is a test for several
! of the FM sample subroutines.
! Some trace output is written to the screen as the program runs, and the eigenvalues and
! eigenvectors are written to file EigenSystem.out at the end.
MODULE EIGEN_VARIABLES
USE FMZM
! COEFFS holds the coefficients for the characteristic polynomial of a matrix.
! N_COEFFS is the number of coefficients, and NDIG_OF_COEFFS is their precision.
TYPE (FM), ALLOCATABLE :: COEFFS(:)
INTEGER, SAVE :: N_COEFFS, NDIG_OF_COEFFS
END MODULE EIGEN_VARIABLES
PROGRAM ETEST
USE FMZM
IMPLICIT NONE
TYPE (FM), ALLOCATABLE :: A(:,:)
TYPE (ZM), ALLOCATABLE :: EIGENVECTORS(:,:), EIGENVALUES(:)
TYPE (FM) :: ERR
INTEGER :: SEED(7)
INTEGER :: J, K, KU, KPRT, N
DOUBLE PRECISION :: XVAL
REAL :: TIME1, TIME2
N = 20
K = 40
KPRT = 1
KU = 6
CALL FM_SET(K)
ALLOCATE(A(N,N),EIGENVECTORS(N,N),EIGENVALUES(N),STAT=J)
IF (J /= 0) STOP
CALL FM_SETVAR(" KSWIDE = 110 ")
! Generate an NxN matrix of random integers between -100 and +100.
CALL FM_RANDOM_SEED_SIZE(J)
SEED(1:J) = 13
SEED(1) = N
SEED(2) = K
CALL FM_RANDOM_SEED_PUT(SEED(1:J))
DO J = 1, N
DO K = 1, N
CALL FM_RANDOM_NUMBER(XVAL)
A(J,K) = INT(200*XVAL - 100)
ENDDO
ENDDO
! Find the Eigenvalues and Eigenvectors.
CALL CPU_TIME(TIME1)
CALL FM_EIGENSYSTEM(A,N,EIGENVECTORS,EIGENVALUES,KPRT,KU)
CALL CPU_TIME(TIME2)
! Write the solution.
OPEN(22,FILE='EigenSystem.out')
CALL FM_SETVAR(" KW = 22 ")
DO J = 1, N
WRITE (22,*) ' '
WRITE (22,"(A,I3,A)") ' Eigenvalue ',J,':'
CALL ZM_PRINT(EIGENVALUES(J))
WRITE (22,*) ' '
WRITE (22,"(A,I3,A)") ' Eigenvector ',J,':'
DO K = 1, N
CALL ZM_PRINT(EIGENVECTORS(K,J))
ENDDO
ENDDO
WRITE (22,*) ' '
WRITE (22,*) ' '
! Find the norm of the residual, A X - c X for each eigenvalue c and eigenvector X.
ERR = 0
DO J = 1, N
ERR = ERR + SUM(ABS(MATMUL(TO_ZM(A(1:N,1:N)),EIGENVECTORS(1:N,J)) - &
EIGENVALUES(J)*EIGENVECTORS(1:N,J)))
ENDDO
WRITE (*,*) ' '
WRITE (*,"(A,ES16.7)") ' Norm of the residual = ',TO_DP(ERR/N**2)
WRITE (*,*) ' '
WRITE (*,*) ' '
WRITE (*,"(A,F9.2,A)") ' Time to solve the system: ',TIME2-TIME1,' seconds.'
WRITE (*,*) ' '
WRITE (*,*) ' The eigenvalues and eigenvectors are in file EigenSystem.out.'
WRITE (*,*) ' '
STOP
END PROGRAM ETEST
SUBROUTINE FM_EIGENSYSTEM(A,N,EIGENVECTORS,EIGENVALUES,KPRT,KU)
USE FMVALS
USE FMZM
USE EIGEN_VARIABLES
IMPLICIT NONE
! Find the eigensystem for real NxN matrix A.
! EIGENVALUES(1:N) is returned with the (complex) eigenvalues.
! EIGENVECTORS(1:N,J) is returned with the (complex) eigenvector corresponding to EIGENVALUES(J),
! for J = 1, 2, ..., N.
! A is a type (fm) multiprecision real array,
! EIGENVALUES and EIGENVECTORS are type (zm) multiprecision complex arrays.
! KPRT controls printing within the routine:
! KPRT = 0 for no output
! KPRT = 1 for some trace output.
! KU is the unit number for output.
! This routine does not handle cases with eigenvalues of multiplicity greater than 1, since
! some of those cases do not have a full set of N eigenvectors.
! The algorithm used here is not the best way to compute eigensystems. It is not even a good
! way to do it. But it was easy to code using existing FM routines for finding determinants
! and all the roots of a polynomial, and it has a certain Rube Goldberg charm.
! Algorithm: 1. Generate the N+1 matrices A - lambda*I for lambda = 1, 2, ..., N+1
! 2. Use FM_FACTOR_LU to find the N+1 determinants for these matrices
! 3. Use FM_LIN_SOLVE to find the N+1 coefficients of the Nth degree characteristic
! polynomial for A, using the points on the curve found in step 2.
! 4. Use ZM_ROOTS to find the N (complex) roots of this polynomial.
! These roots are the eigenvalues.
! 5. For each eigenvalue, do a few iterations of the inverse power method to find
! the corresponding eigenvector.
INTEGER :: I, J, JMAX, K, KPRT, KU, KWARN_SAVE, LAMBDA, N, NDSAVE, N_FOUND
TYPE (FM) :: A(N,N)
TYPE (ZM) :: EIGENVALUES(N), EIGENVECTORS(N,N)
TYPE (FM), SAVE :: DET, P, TOL
TYPE (FM), ALLOCATABLE :: A1(:,:), A2(:,:), EQN(:,:), IDENTITY(:,:), RHS(:)
TYPE (ZM), ALLOCATABLE :: Z1(:,:), Z2(:,:), ZX(:), ZT(:), LIST_OF_ROOTS(:)
TYPE (ZM), SAVE :: T, ZDET
INTEGER, ALLOCATABLE :: KSWAP(:)
TYPE (ZM), EXTERNAL :: EIGEN_POLY
CALL FM_ENTER_USER_ROUTINE
! Raise precision slightly.
NDSAVE = NDIG
NDIG = NDIG + NGRD52
KWARN_SAVE = KWARN
KWARN = 0
ALLOCATE(A1(N,N),A2(N,N),EQN(N+1,N+1),IDENTITY(N,N),Z1(N,N),Z2(N,N),ZX(N),ZT(N),RHS(N+1), &
COEFFS(N+1),LIST_OF_ROOTS(N),KSWAP(N),STAT=J)
IF (J /= 0) THEN
WRITE (KU,"(/' Error in FM_EIGENSYSTEM. Unable to allocate arrays with N = ',I8/)") N
STOP
ENDIF
! Copy A to A1 with higher precision, and generate an identity matrix.
110 TOL = TO_FM(MBASE)**(-NDSAVE)
IDENTITY = 0
DO I = 1, N
DO J = 1, N
CALL FM_EQU(A(I,J),A1(I,J),NDSAVE,NDIG)
ENDDO
IDENTITY(I,I) = 1
ENDDO
! Generate the N+1 (real) N x N matrices A - lambda*I for lambda = 1, 2, ..., N+1,
! and find the N+1 determinants for these matrices
DO LAMBDA = 1, N+1
A2 = A1 - LAMBDA*IDENTITY
CALL FM_FACTOR_LU(A2,N,DET,KSWAP)
P = 1
DO J = N+1, 1, -1
EQN(LAMBDA,J) = P
P = LAMBDA*P
ENDDO
RHS(LAMBDA) = DET
ENDDO
N_COEFFS = N + 1
! Find the (real) coefficients of the characteristic polynomial for A.
CALL FM_LIN_SOLVE(EQN,COEFFS,RHS,N_COEFFS,DET)
NDIG_OF_COEFFS = NDIG
IF (DET == 0) THEN
WRITE (KU,"(/' Error in FM_EIGENSYSTEM. Zero determinant -- no unique ',"// &
"'characteristic polynomial.'/)")
STOP
ENDIF
! Use ZM_ROOTS to find all the (complex) roots of this polynomial.
! These roots (in LIST_OF_ROOTS) are the eigenvalues.
CALL ZM_ROOTS(N,EIGEN_POLY,1,N_FOUND,LIST_OF_ROOTS,KPRT,KU)
IF (N_FOUND /= N) THEN
IF (KPRT > 0) THEN
WRITE (KU,"(/' In FM_EIGENSYSTEM, N_FOUND /= N. N = ',"// &
"I3,'. N_FOUND =',I3/)") N,N_FOUND
WRITE (KU,"(/' Increase precision and try again.'/)")
ENDIF
NDIG = 2*NDIG
GO TO 110
ENDIF
! Sort the eigenvalues so they have decreasing magnitude, and check that there were
! no multiple roots.
DO I = 2, N
JMAX = I-1
DO J = I, N
IF (ABS(LIST_OF_ROOTS(J)) > ABS(LIST_OF_ROOTS(JMAX))) JMAX = J
ENDDO
T = LIST_OF_ROOTS(I-1)
LIST_OF_ROOTS(I-1) = LIST_OF_ROOTS(JMAX)
LIST_OF_ROOTS(JMAX) = T
ENDDO
DO I = 2, N
IF (LIST_OF_ROOTS(I-1) == LIST_OF_ROOTS(I)) THEN
WRITE (KU,"(/' Error in FM_EIGENSYSTEM. eigenvalues of multiplicity more than 1'/)")
CALL ZM_PRINT(LIST_OF_ROOTS(I))
STOP
ENDIF
ENDDO
! For each eigenvalue, do a few iterations of the inverse power method to find
! the corresponding eigenvector.
! Since the A matrix is real, conjugate eigenvalues have conjugate eigenvectors,
! so some eigenvectors may not need the inverse power method.
DO I = 1, N
IF (I > 1) THEN
IF (LIST_OF_ROOTS(I) == CONJG(LIST_OF_ROOTS(I-1))) THEN
DO K = 1, N
CALL ZM_EQU(EIGENVECTORS(K,I-1),T,NDSAVE,NDIG)
CALL ZM_EQU(CONJG(T),EIGENVECTORS(K,I),NDIG,NDSAVE)
ENDDO
CALL ZM_EQU(EIGENVALUES(I-1),T,NDSAVE,NDIG)
CALL ZM_EQU(CONJG(T),EIGENVALUES(I),NDIG,NDSAVE)
CYCLE
ENDIF
ENDIF
P = (1 + EPSILON(TO_FM(1))**0.3D0)
Z1 = A1
DO J = 1, N
Z1(J,J) = Z1(J,J) - P*LIST_OF_ROOTS(I)
ENDDO
CALL ZM_INVERSE(Z1,N,Z2,ZDET)
ZX = 1
DO J = 1, 6
ZX = MATMUL(Z2,ZX)
ZX = ZX / SQRT(ABS(DOT_PRODUCT(ZX,ZX)))
ZT = MATMUL(TO_ZM(A1),ZX) - LIST_OF_ROOTS(I)*ZX
P = SQRT(ABS(DOT_PRODUCT(ZT,ZT)))
IF (P < TOL) THEN
DO K = 1, N
CALL ZM_EQU(ZX(K),EIGENVECTORS(K,I),NDIG,NDSAVE)
ENDDO
CALL ZM_EQU(LIST_OF_ROOTS(I),EIGENVALUES(I),NDIG,NDSAVE)
EXIT
ENDIF
IF (J == 6) THEN
IF (KPRT > 0) THEN
WRITE (KU, &
"(/' In FM_EIGENSYSTEM inverse iteration did not converge for I = ',"// &
"I3,'.'/' Error =',ES15.7,' was not less than Tol =',ES15.7)") I, &
TO_DP(P),TO_DP(TOL)
WRITE (KU,"(/' Increase precision and try again.'/)")
ENDIF
NDIG = 2*NDIG
GO TO 110
ENDIF
ENDDO
ENDDO
CALL FM_DEALLOCATE(A1)
CALL FM_DEALLOCATE(A2)
CALL FM_DEALLOCATE(EQN)
CALL FM_DEALLOCATE(IDENTITY)
CALL FM_DEALLOCATE(Z1)
CALL FM_DEALLOCATE(Z2)
CALL FM_DEALLOCATE(ZX)
CALL FM_DEALLOCATE(ZT)
CALL FM_DEALLOCATE(RHS)
CALL FM_DEALLOCATE(COEFFS)
CALL FM_DEALLOCATE(LIST_OF_ROOTS)
DEALLOCATE(A1,A2,EQN,IDENTITY,Z1,Z2,ZX,ZT,RHS,COEFFS,LIST_OF_ROOTS,KSWAP)
NDIG = NDSAVE
KWARN = KWARN_SAVE
CALL FM_EXIT_USER_ROUTINE
END SUBROUTINE FM_EIGENSYSTEM
FUNCTION EIGEN_POLY(X,NF)
USE EIGEN_VARIABLES
USE FMVALS
USE FMZM
IMPLICIT NONE
! X is the argument to the function.
! NF is the function number.
INTEGER :: J, NF
TYPE (ZM) :: EIGEN_POLY, X
TYPE (FM), SAVE :: D
CALL FM_ENTER_USER_FUNCTION(EIGEN_POLY)
IF (NF == 1) THEN
CALL FM_EQU(COEFFS(1),D,NDIG_OF_COEFFS,NDIG)
EIGEN_POLY = D
DO J = 2, N_COEFFS
CALL FM_EQU(COEFFS(J),D,NDIG_OF_COEFFS,NDIG)
EIGEN_POLY = EIGEN_POLY*X + D
ENDDO
ELSE
EIGEN_POLY = 3*X - 2
ENDIF
CALL FM_EXIT_USER_FUNCTION(EIGEN_POLY)
END FUNCTION EIGEN_POLY
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