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

NAME
  SGEGS	- a pair of N-by-N real	nonsymmetric matrices A, B

SYNOPSIS

  SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
		    VSL, LDVSL,	VSR, LDVSR, WORK, LWORK, INFO )

      CHARACTER	    JOBVSL, JOBVSR

      INTEGER	    INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N

      REAL	    A( LDA, * ), ALPHAI( * ), ALPHAR( *	), B( LDB, * ),	BETA(
		    * ), VSL( LDVSL, * ), VSR( LDVSR, *	), WORK( * )

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

     compute the generalized eigenvalues (alphar +/- alphai*i, beta)
     compute the real 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
  SGEGV	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 orthogonal matrix and	both on	the
  right	by another orthogonal matrix, these two	orthogonal matrices being
  chosen so as to bring	the pair of matrices into (real) Schur form.

  A pair of matrices A,	B is in	generalized real Schur form if B is upper
  triangular with non-negative diagonal	and A is block upper triangular	with
  1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond to real generalized
  eigenvalues, while 2-by-2 blocks of A	will be	"standardized" by making the
  corresponding	entries	of B have the form:
	  [  a	0  ]
	  [  0	b  ]

  and the pair of corresponding	2-by-2 blocks in A and B will have a complex
  conjugate pair of generalized	eigenvalues.

  The left and right Schur vectors are the columns of VSL and VSR, respec-
  tively, where	VSL and	VSR are	the orthogonal matrices	which reduce A and B
  to Schur form:

  Schur	form of	(A,B) =	( (VSL)**T A (VSR), (VSL)**T 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) REAL 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.  Note:	to avoid overflow,
	  the Frobenius	norm of	the matrix A should be less than the overflow
	  threshold.

  LDA	  (input) INTEGER
	  The leading dimension	of A.  LDA >= max(1,N).

  B	  (input/output) REAL 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.  Note:	to avoid overflow,
	  the Frobenius	norm of	the matrix B should be less than the overflow
	  threshold.

  LDB	  (input) INTEGER
	  The leading dimension	of B.  LDB >= max(1,N).

  ALPHAR  (output) REAL	array, dimension (N)
	  ALPHAI  (output) REAL	array, dimension (N) BETA    (output) REAL
	  array, dimension (N)

	  On exit, (ALPHAR(j) +	ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the
	  generalized eigenvalues.  ALPHAR(j) +	ALPHAI(j)*i, j=1,...,N	and
	  BETA(j),j=1,...,N  are the diagonals of the complex Schur form
	  (A,B)	that would result if the 2-by-2	diagonal blocks	of the real
	  Schur	form of	(A,B) were further reduced to triangular form using
	  2-by-2 complex unitary transformations.  If ALPHAI(j)	is zero, then
	  the j-th eigenvalue is real; if positive, then the j-th and (j+1)-
	  st eigenvalues are a complex conjugate pair, with ALPHAI(j+1)	nega-
	  tive.

	  Note:	the quotients ALPHAR(j)/BETA(j)	and ALPHAI(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,
	  ALPHAR and ALPHAI 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) REAL	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) REAL	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) REAL 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(1,4*N).  For good
	  performance, LWORK must generally be larger.	To compute the
	  optimal value	of LWORK, call ILAENV to get blocksizes	(for SGEQRF,
	  SORMQR, and SORGQR.)	Then compute: NB  -- MAX of the	blocksizes
	  for SGEQRF, SORMQR, and SORGQR The optimal LWORK is  2*N +
	  N*(NB+1).

  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 ALPHAR(j), ALPHAI(j), and	BETA(j)	should be correct for
	  j=INFO+1,...,N.  > N:	 errors	that usually indicate LAPACK prob-
	  lems:
	  =N+1:	error return from SGGBAL
	  =N+2:	error return from SGEQRF
	  =N+3:	error return from SORMQR
	  =N+4:	error return from SORGQR
	  =N+5:	error return from SGGHRD
	  =N+6:	error return from SHGEQZ (other	than failed iteration) =N+7:
	  error	return from SGGBAK (computing VSL)
	  =N+8:	error return from SGGBAK (computing VSR)
	  =N+9:	error return from SLASCL (various places)