Computes all of the eigenvalues and eigenvectors of a generalized complex eigensystem Az = lBz.
A — Complex matrix of order N. (Input)
B — Complex matrix of order N. (Input)
ALPHA — Complex vector of length N. Ultimately, alpha(i)/betav(i) (for i = 1, …, n), will be the eigenvalues of the system in decreasing order of magnitude. (Output)
BETAV — Complex vector of length N. (Output)
EVEC — Complex
matrix of order N.
(Output)
The J-th eigenvector,
corresponding to ALPHA(J)/BETAV(J), is stored in the
J-th column.
Each vector is normalized to have Euclidean length equal to the value one.
N — Order of the
matrices A and
B.
(Input)
Default: N = size
(A,2).
LDA — Leading
dimension of A
exactly as specified in the dimension statement of the calling
program. (Input)
Default: LDA = size
(A,1).
LDB — Leading
dimension of B
exactly as specified in the dimension statement of the calling
program. (Input)
Default: LDB = size
(B,1).
LDEVEC — Leading
dimension of EVEC exactly as
specified in the dimension statement of the calling program.
(Input)
Default: LDEVEC = size
(EVEC,1).
Generic: CALL GVCCG (A, B, ALPHA, BETAV, EVEC [,…])
Specific: The specific interface names are S_GVCCG and D_GVCCG.
Single: CALL GVCCG (N, A, LDA, B, LDB, ALPHA, BETAV, EVEC, LDEVEC)
Double: The double precision name is DGVCCG.
Routine GVCCG computes the eigenvalues and eigenvectors of the generalized eigensystem Ax = lBx. Here, A and B, are complex matrices of order n. The eigenvalues for this problem can be infinite; so instead of returning l, GVCCG returns α and β. If β is nonzero, then l = α/β.
The routine GVCCG uses the QZ algorithm described by Moler and Stewart (1973). The implementation is based on routines of Garbow (1978). Some timing results are given in Hanson et al. (1990).
1. Workspace may be explicitly provided, if desired, by use of G6CCG/DG6CCG. The reference is:
CALL G6CCG (N, A, LDA, B, LDB, ALPHA, BETAV, EVEC, LDEVEC, ACOPY, BCOPY, CWK, WK, IWK)
The additional arguments are as follows:
ACOPY — Complex work array of length N2. A and ACOPY may be the same in which case the first N2 elements of A will be destroyed.
BCOPY — Complex work array of length N2. B and BCOPY may be the same in which case the first N2 elements of B will be destroyed.
CWK — Complex work array of length N.
WK — Real work array of length N.
IWK — Integer work array of length N.
2. Informational error
Type Code
4 1 The iteration for an eigenvalue failed to converge.
3. The success of this routine can be checked using GPICG.
In this example, DATA statements are used to set A and B. The eigenvalues and eigenvectors are computed and printed. The performance index is also computed and printed. This serves as a check on the computations. For more details, see routine GPICG.
USE IMSL_LIBRARIES
IMPLICIT NONE
INTEGER LDA, LDB, LDEVEC, N
PARAMETER (N=3, LDA=N, LDB=N, LDEVEC=N)
!
INTEGER I, NOUT
REAL PI
COMPLEX A(LDA,N), ALPHA(N), B(LDB,N), BETAV(N), EVAL(N), &
EVEC(LDEVEC,N)
!
! Define values of A and B
! A = ( 1+0i 0.5+i 0+5i )
! (-10+0i 2+i 0+0i )
! ( 5+i 1+0i 0.5+3i )
!
! B = ( 0.5+0i 0+0i 0+0i )
! ( 3+3i 3+3i 0+i )
! ( 4+2i 0.5+i 1+i )
!
! Declare variables
DATA A/(1.0,0.0), (-10.0,0.0), (5.0,1.0), (0.5,1.0), (2.0,1.0), &
(1.0,0.0), (0.0,5.0), (0.0,0.0), (0.5,3.0)/
DATA B/(0.5,0.0), (3.0,3.0), (4.0,2.0), (0.0,0.0), (3.0,3.0), &
(0.5,1.0), (0.0,0.0), (0.0,1.0), (1.0,1.0)/
! Compute eigenvalues
CALL GVCCG (A, B, ALPHA, BETAV, EVEC)
EVAL = ALPHA/BETAV
! Compute performance index
PI = GPICG(N,A,B,ALPHA,BETAV,EVEC)
! Print results
CALL UMACH (2, NOUT)
CALL WRCRN ('EVAL', EVAL, 1, N, 1)
CALL WRCRN ('EVEC', EVEC)
WRITE (NOUT, '(/,A,F6.3)') ' Performance index = ', PI
END
EVAL
1 2 3
( -8.18,-25.38) ( 2.18, 0.61) ( 0.12, -0.39)
EVEC
1 2 3
1 (-0.3267,-0.1245) (-0.3007,-0.2444) ( 0.0371, 0.1518)
2 ( 0.1767, 0.0054) ( 0.8959, 0.0000) ( 0.9577, 0.0000)
3 ( 0.9201, 0.0000) (-0.2019, 0.0801) (-0.2215, 0.0968)
Performance index = 0.709
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