ZGEGS - routine is deprecated and has been replaced by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This routine is deprecated and has been replaced by rou
tine 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 option
ally 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 inter
pretation for beta=0, and even for both being zero. A
good beginning reference is the book, "Matrix Computa
tions", 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 general
ized 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 general
ized 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 com
plex 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'.
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 gen
erally 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 ille
gal 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 (comput
ing VSL)
=N+8: error return from ZGGBAK (computing VSR)
=N+9: error return from ZLASCL (various places)
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