dggesx(3S)

NAME
     DGGESX - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
     the generalized eigenvalues, the real Schur form (S,T), and,

SYNOPSIS
     SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, B,
                        LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
                        LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
                        BWORK, INFO )

         CHARACTER      JOBVSL, JOBVSR, SENSE, SORT

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

         LOGICAL        BWORK( * )

         INTEGER        IWORK( * )

         DOUBLE         PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
                        LDB, * ), BETA( * ), RCONDE( 2 ), RCONDV( 2 ), 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
     DGGESX computes for a pair of N-by-N real nonsymmetric matrices (A,B),
     the generalized eigenvalues, the real Schur form (S,T), and, 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; computes a
     reciprocal condition number for the average of the selected eigenvalues
     (RCONDE); and computes a reciprocal condition number for the right and
     left deflating subspaces corresponding to the selected eigenvalues
     (RCONDV). The leading columns of VSL and VSR then form an orthonormal
     basis for the corresponding left and right eigenspaces (deflating
     subspaces).

     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 for 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 a selected complex
             eigenvalue may no longer satisfy
             DELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
             since ordering may change the value of complex eigenvalues
             (especially if the eigenvalue is ill-conditioned), in this case
             INFO is set to N+3.

     SENSE   (input) CHARACTER
             Determines which reciprocal condition numbers are computed.  =
             'N' : None are computed;
             = 'E' : Computed for average of selected eigenvalues only;
             = 'V' : Computed for selected deflating subspaces only;
             = 'B' : Computed for both.  If SENSE = 'E', 'V', or 'B', SORT
             must equal 'S'.

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

     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'.

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

     RCONDE  (output) DOUBLE PRECISION array, dimension ( 2 )
             If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
             reciprocal condition numbers for the average of the selected
             eigenvalues.  Not referenced if SENSE = 'N' or 'V'.

     RCONDV  (output) DOUBLE PRECISION array, dimension ( 2 )
             If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
             reciprocal condition numbers for the selected deflating
             subspaces.  Not referenced if SENSE = 'N' or 'E'.

     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+1)+16.  If SENSE
             = 'E', 'V', or 'B', LWORK >= MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).

     IWORK   (workspace) INTEGER array, dimension (LIWORK)
             Not referenced if SENSE = 'N'.

     LIWORK  (input) INTEGER
             The dimension of the array WORK.  LIWORK >= N+6.

     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.

             Further details ===============

             An approximate (asymptotic) bound on the average absolute error
             of the selected eigenvalues is

             EPS * norm((A, B)) / RCONDE( 1 ).

             An approximate (asymptotic) bound on the maximum angular error in
             the computed deflating subspaces is

             EPS * norm((A, B)) / RCONDV( 2 ).

             See LAPACK User's Guide, section 4.11 for more information.

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

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