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DGGES(3S)							     DGGES(3S)

NAME
     DGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B),

SYNOPSIS
     SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, DELCTG, N, A, LDA, B, LDB, SDIM,
		       ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
		       LWORK, BWORK, INFO )

	 CHARACTER     JOBVSL, JOBVSR, SORT

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

	 LOGICAL       BWORK( * )

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

	 LOGICAL       DELCTG

	 EXTERNAL      DELCTG

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the
     generalized eigenvalues, the generalized real Schur form (S,T),
     optionally, the left and/or right matrices of Schur vectors (VSL and
     VSR). This gives the generalized Schur factorization

	      (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

     Optionally, it also orders the eigenvalues so that a selected cluster of
     eigenvalues appears in the leading diagonal blocks of the upper quasi-
     triangular matrix S and the upper triangular matrix T.The leading columns
     of VSL and VSR then form an orthonormal basis for the corresponding left
     and right eigenspaces (deflating subspaces).

     (If only the generalized eigenvalues are needed, use the driver DGGEV
     instead, which is faster.)

									Page 1

DGGES(3S)							     DGGES(3S)

     A generalized eigenvalue for a pair of matrices (A,B) is 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 or both being zero.

     A pair of matrices (S,T) is in generalized real Schur form if T is upper
     triangular with non-negative diagonal and S 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 S will be "standardized"
     by making the corresponding elements of T have the form:
	     [	a  0  ]
	     [	0  b  ]

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

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.

     SORT    (input) CHARACTER*1
	     Specifies whether or not to order the eigenvalues on the diagonal
	     of the generalized Schur form.  = 'N':  Eigenvalues are not
	     ordered;
	     = 'S':  Eigenvalues are ordered (see DELCTG);

     DELCTG  (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
	     DELCTG must be declared EXTERNAL in the calling subroutine.  If
	     SORT = 'N', DELCTG is not referenced.  If SORT = 'S', DELCTG is
	     used to select eigenvalues to sort to the top left of the Schur
	     form.  An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
	     DELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one
	     of a complex conjugate pair of eigenvalues is selected, then both
	     complex eigenvalues are selected.

	     Note that in the ill-conditioned case, a selected complex
	     eigenvalue may no longer satisfy DELCTG(ALPHAR(j),ALPHAI(j),
	     BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in
	     this case.

     N	     (input) INTEGER
	     The order of the matrices A, B, VSL, and VSR.  N >= 0.

									Page 2

DGGES(3S)							     DGGES(3S)

     A	     (input/output) DOUBLE PRECISION array, dimension (LDA, N)
	     On entry, the first of the pair of matrices.  On exit, A has been
	     overwritten by its generalized Schur form S.

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

     B	     (input/output) DOUBLE PRECISION array, dimension (LDB, N)
	     On entry, the second of the pair of matrices.  On exit, B has
	     been overwritten by its generalized Schur form T.

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

     SDIM    (output) INTEGER
	     If SORT = 'N', SDIM = 0.  If SORT = 'S', SDIM = number of
	     eigenvalues (after sorting) for which DELCTG is true.  (Complex
	     conjugate pairs for which DELCTG is true for either eigenvalue
	     count as 2.)

     ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
	     ALPHAI  (output) DOUBLE PRECISION array, dimension (N) BETA
	     (output) DOUBLE PRECISION 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, and
	     BETA(j),j=1,...,N are the diagonals of the complex Schur form
	     (S,T) 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) negative.

	     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.	 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) DOUBLE PRECISION array, dimension (LDVSL,N)
	     If JOBVSL = 'V', VSL will contain the left Schur vectors.	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) DOUBLE PRECISION array, dimension (LDVSR,N)
	     If JOBVSR = 'V', VSR will contain the right Schur vectors.	 Not
	     referenced if JOBVSR = 'N'.

									Page 3

DGGES(3S)							     DGGES(3S)

     LDVSR   (input) INTEGER
	     The leading dimension of the matrix VSR. LDVSR >= 1, and if
	     JOBVSR = 'V', LDVSR >= N.

     WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The dimension of the array WORK.  LWORK >= 8*N+16.

	     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.

     BWORK   (workspace) LOGICAL array, dimension (N)
	     Not referenced if SORT = '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 ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for
	     j=INFO+1,...,N.  > N:  =N+1: other than QZ iteration failed in
	     DHGEQZ.
	     =N+2: after reordering, roundoff changed values of some complex
	     eigenvalues so that leading eigenvalues in the Generalized Schur
	     form no longer satisfy DELCTG=.TRUE.  This could also be caused
	     due to scaling.  =N+3: reordering failed in DTGSEN.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.

									Page 4

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