ZGEGS - routine is deprecated and has been replaced by routine ZGGES

SUBROUTINE ZGEGS(

JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO )

CHARACTER JOBVSL, JOBVSR

INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

DOUBLE PRECISION RWORK( * )

COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )

This routine is deprecated and has been replaced by routine ZGGES. ZGEGS computes for a pair of N-by-N complex nonsymmetric matrices A, B: the generalized eigenvalues (alpha, beta), the complex Schur form (A, B), and optionally left and/or right Schur vectors (VSL and VSR).

(If only the generalized eigenvalues are needed, use the driver ZGEGV instead.)

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)

The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one unitary matrix and both on the right by another unitary matrix, these two unitary matrices being chosen so as to bring the pair of matrices into upper triangular form with the diagonal elements of B being non-negative real numbers (this is also called complex Schur form.)

The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the unitary matrices

which reduce A and B to Schur form:

Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) )

JOBVSL (input) CHARACTER*1

= 'N': do not compute the left Schur vectors;

= 'V': compute the left Schur vectors.

JOBVSR (input) CHARACTER*1

= 'N': do not compute the right Schur vectors;

= 'V': compute the right Schur vectors.

N (input) INTEGER

The order of the matrices A, B, VSL, and VSR. 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) Schur vectors are to be computed. On exit, the generalized Schur form of A.

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) Schur vectors are to be computed. On exit, the generalized Schur form of B.

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. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by ZGEGS. The BETA(j) will be non-negative real.

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).

VSL (output) COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors. (See "Purpose", above.) Not referenced if JOBVSL = 'N'.

LDVSL (input) INTEGER

The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N.

VSR (output) COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors. (See "Purpose", above.) Not referenced if JOBVSR = 'N'.

LDVSR (input) INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= 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 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) DOUBLE PRECISION array, dimension (3*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. (A,B) are not in Schur form, 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 ZGGBAK (computing VSL)

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

=N+9: error return from ZLASCL (various places)