ZGEGS(l)	     LAPACK driver routine (version 1.1)	     ZGEGS(l)

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
  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 multi-
  plying 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 diago-
  nal elements of B being non-negative real numbers (this is also called com-
  plex Schur form.)

  The left and right Schur vectors are the columns of VSL and VSR, respec-
  tively, 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 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).

  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)