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NAME
     zgegs - compute for a pair of  N-by-N  complex  nonsymmetric
     matrices A,

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( * )



     #include <sunperf.h>

     void zgegs(char jobvsl, char jobvsr,  int  n,  doublecomplex
               *za,  int  lda,  doublecomplex *zb, int ldb, doub-
               lecomplex *zalpha,  doublecomplex  *  beta,  doub-
               lecomplex *vsl, int ldvsl, doublecomplex *vsr, int
               ldvsr, int *info) ;

PURPOSE
     DGEGS 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 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 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'.

     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 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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