ZGEGV - routine is deprecated and has been replaced by routine ZGGEV

SUBROUTINE ZGEGV(

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

CHARACTER JOBVL, JOBVR

INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N

DOUBLE PRECISION RWORK( * )

COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

This routine is deprecated and has been replaced by routine ZGGEV. ZGEGV computes for a pair of N-by-N complex nonsymmetric matrices A and B, the generalized eigenvalues (alpha, beta), and optionally, the left and/or right generalized eigenvectors (VL and VR).

A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. A good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press)

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - w B) r = 0 . A left generalized eigenvector is a vector l such that l**H * (A - w B) = 0, where l**H is the

conjugate-transpose of l.

Note: this routine performs "full balancing" on A and B -- see "Further Details", below.

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigenvectors;

= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized eigenvectors;

= 'V': compute the right generalized eigenvectors.

N (input) INTEGER

The order of the matrices A, B, VL, and VR. N >= 0.

A (input/output) COMPLEX*16 array, dimension (LDA, N)

On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed. On exit, the contents will have been destroyed. (For a description of the contents of A on exit, see "Further Details", below.)

LDA (input) INTEGER

The leading dimension of A. LDA >= max(1,N).

B (input/output) COMPLEX*16 array, dimension (LDB, N)

On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed. On exit, the contents will have been destroyed. (For a description of the contents of B on exit, see "Further Details", below.)

LDB (input) INTEGER

The leading dimension of B. LDB >= max(1,N).

ALPHA (output) COMPLEX*16 array, dimension (N)

BETA (output) COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues.

Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

VL (output) COMPLEX*16 array, dimension (LDVL,N)

If JOBVL = 'V', the left generalized eigenvectors. (See "Purpose", above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVL = 'N'.

LDVL (input) INTEGER

The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.

VR (output) COMPLEX*16 array, dimension (LDVR,N)

If JOBVR = 'V', the right generalized eigenvectors. (See "Purpose", above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is MAX( 2*N, N*(NB+1) ).

If LWORK = -1, then 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 by XERBLA.

RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)

INFO (output) INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

=1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems:

=N+1: error return from ZGGBAL

=N+2: error return from ZGEQRF

=N+3: error return from ZUNMQR

=N+4: error return from ZUNGQR

=N+5: error return from ZGGHRD

=N+6: error return from ZHGEQZ (other than failed iteration) =N+7: error return from ZTGEVC

=N+8: error return from ZGGBAK (computing VL)

=N+9: error return from ZGGBAK (computing VR)

=N+10: error return from ZLASCL (various calls)

Balancing

---------

This driver calls ZGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that start out zero.)

After the eigenvalues and eigenvectors of the balanced matrices have been computed, ZGGBAK transforms the eigenvectors back to what they would have been (in perfect arithmetic) if they had not been balanced.

Contents of A and B on Exit

-------- -- - --- - -- ----

If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the complex Schur form[*] of the "balanced" versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct.

[*] In other words, upper triangular form.