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# zgegs

```
NAME
ZGEGS - a pair of N-by-N complex nonsymmetric matrices A, B

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
For a pair of N-by-N complex nonsymmetric matrices A, B:

compute the generalized eigenvalues (alpha, beta)
compute the complex Schur form (A,B)
compute the left and/or right Schur vectors (VSL and VSR)

The last action is optional -- see the description of JOBVSL
and JOBVSR below.  (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 interpre-
tation 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 uni-
tary 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 diago-
nal 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) )

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 number of rows and columns in 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 vec-
tors 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 vec-
tors 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 gen-
eralized 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 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).

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 usu-
ally 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)
```