
ZGEGS(l) ) ZGEGS(l)
NAME
ZGEGS  routine is deprecated and has been replaced by routine ZGGES
SYNOPSIS
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( * )
PURPOSE
This routine is deprecated and has been replaced by routine ZGGES. ZGEGS computes for a
pair of NbyN 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 nonnegative 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) )
ARGUMENTS
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 nonnegative 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 ith 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)
LAPACK version 3.0 15 June 2000 ZGEGS(l) 
