NAG Library Routine Document

F08WNF (ZGGEV)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08WNF (ZGGEV) computes for a pair of n by n complex nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.

2 Specification

SUBROUTINE F08WNF (

JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

INTEGER	N, LDA, LDB, LDVL, LDVR, LWORK, INFO
REAL (KIND=nag_wp)	RWORK(max(1,8*N))
COMPLEX (KIND=nag_wp)	A(LDA,), B(LDB,), ALPHA(N), BETA(N), VL(LDVL,), VR(LDVR,), WORK(max(1,LWORK))
CHARACTER(1)	JOBVL, JOBVR

The routine may be called by its LAPACK name zggev.

3 Description

A generalized eigenvalue for a pair of matrices A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.

The right generalized eigenvector vj corresponding to the generalized eigenvalue λj of A,B satisfies

A vj = λj B vj .

The left generalized eigenvector uj corresponding to the generalized eigenvalues λj of A,B satisfies

ujH A = λj ujH B ,

where ujH is the conjugate-transpose of uj.

All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem Ax=λBx, where A and B are complex, square matrices, are determined using the QZ algorithm. The complex QZ algorithm consists of three stages:

A is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time B is reduced to upper triangular form.
A is further reduced to triangular form while the triangular form of B is maintained and the diagonal elements of B are made real and non-negative. This is the generalized Schur form of the pair A,B .
This routine does not actually produce the eigenvalues λj, but instead returns αj and βj such that
λj=αj/βj, j=1,2,…,n.
The division by βj becomes your responsibility, since βj may be zero, indicating an infinite eigenvalue.
If the eigenvectors are required they are obtained from the triangular matrices and then transferred back into the original coordinate system.

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Wilkinson J H (1979) Kronecker's canonical form and the QZ algorithm Linear Algebra Appl. 28 285–303

5 Parameters

1: JOBVL – CHARACTER(1)Input

On entry: if JOBVL='N', do not compute the left generalized eigenvectors.

If JOBVL='V', compute the left generalized eigenvectors.

Constraint: JOBVL='N' or 'V'.

2: JOBVR – CHARACTER(1)Input

On entry: if JOBVR='N', do not compute the right generalized eigenvectors.

If JOBVR='V', compute the right generalized eigenvectors.

Constraint: JOBVR='N' or 'V'.

3: N – INTEGERInput

On entry: n, the order of the matrices A and B.

Constraint: N≥0.

4: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least max1,N.

On entry: the matrix A in the pair A,B.

On exit: A has been overwritten.

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08WNF (ZGGEV) is called.

Constraint: LDA≥max1,N.

6: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array B must be at least max1,N.

On entry: the matrix B in the pair A,B.

On exit: B has been overwritten.

7: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08WNF (ZGGEV) is called.

Constraint: LDB≥max1,N.

8: ALPHA(N) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: see the description of BETA.

9: BETA(N) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: ALPHAj/BETAj, for j=1,2,…,N, will be the generalized eigenvalues.

Note: the quotients ALPHAj/BETAj may easily overflow or underflow, and BETAj may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with A2 in magnitude, and maxβj will always be less than and usually comparable with B2.

10: VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayOutput

Note: the second dimension of the array VL must be at least max1,N if JOBVL='V', and at least 1 otherwise.

On exit: if JOBVL='V', the left generalized eigenvectors uj are stored one after another in the columns of VL, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have real part+imag. part=1.

If JOBVL='N', VL is not referenced.

11: LDVL – INTEGERInput

On entry: the first dimension of the array VL as declared in the (sub)program from which F08WNF (ZGGEV) is called.

Constraints:

if JOBVL='V', LDVL≥max1,N;
otherwise LDVL≥1.

12: VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayOutput

Note: the second dimension of the array VR must be at least max1,N if JOBVR='V', and at least 1 otherwise.

On exit: if JOBVR='V', the right generalized eigenvectors vj are stored one after another in the columns of VR, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have real part+imag. part=1.

If JOBVR='N', VR is not referenced.

13: LDVR – INTEGERInput

On entry: the first dimension of the array VR as declared in the (sub)program from which F08WNF (ZGGEV) is called.

Constraints:

if JOBVR='V', LDVR≥max1,N;
otherwise LDVR≥1.

14: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if INFO=0, the real part of WORK1 contains the minimum value of LWORK required for optimal performance.

15: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08WNF (ZGGEV) is called.

If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.

Suggested value: for optimal performance, LWORK must generally be larger than the minimum; increase workspace by, say, nb×N, where nb is the optimal block size.

Constraint: LWORK≥max1,2×N.

16: RWORK(max1,8×N) – REAL (KIND=nag_wp) arrayWorkspace

17: INFO – INTEGEROutput

On exit: INFO=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

INFO<0: If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

INFO=1 to N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHAj and BETAj should be correct for j=INFO+1,…,N.

INFO=N+1: Unexpected error returned from F08XSF (ZHGEQZ).

INFO=N+2: Error returned from F08YXF (ZTGEVC).

7 Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices A+E and B+F, where

E,F F = Oε A,B F ,

and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.

Note: interpretation of results obtained with the QZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in Wilkinson (1979), in relation to the significance of small values of αj and βj. It should be noted that if αj and βj are both small for any j, it may be that no reliance can be placed on any of the computed eigenvalues λi=αi/βi. You are recommended to study Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.