zgges(3S)

NAME
     ZGGES - compute for a pair of N-by-N complex nonsymmetric matrices (A,B),
     the generalized eigenvalues, the generalized complex Schur form (S, T),
     and optionally left and/or right Schur vectors (VSL and VSR)

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

         CHARACTER     JOBVSL, JOBVSR, SORT

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

         LOGICAL       BWORK( * )

         DOUBLE        PRECISION RWORK( * )

         COMPLEX*16    A( LDA, * ), ALPHA( * ), 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
     ZGGES computes for a pair of N-by-N complex nonsymmetric matrices (A,B),
     the generalized eigenvalues, the generalized complex Schur form (S, T),
     and optionally left and/or right Schur vectors (VSL and VSR). This gives
     the generalized Schur factorization
             (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

     where (VSR)**H is the conjugate-transpose of VSR.

     Optionally, it also orders the eigenvalues so that a selected cluster of
     eigenvalues appears in the leading diagonal blocks of the upper
     triangular matrix S and the upper triangular matrix T. The leading
     columns of VSL and VSR then form an unitary basis for the corresponding
     left and right eigenspaces (deflating subspaces).
     (If only the generalized eigenvalues are needed, use the driver ZGGEV
     instead, which is faster.)

     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, and even for both being zero.

     A pair of matrices (S,T) is in generalized complex Schur form if S and T
     are upper triangular and, in addition, the diagonal elements of T are
     non-negative real numbers.


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 DELZTG).

     DELZTG  (input) LOGICAL FUNCTION of two COMPLEX*16 arguments
             DELZTG must be declared EXTERNAL in the calling subroutine.  If
             SORT = 'N', DELZTG is not referenced.  If SORT = 'S', DELZTG is
             used to select eigenvalues to sort to the top left of the Schur
             form.  An eigenvalue ALPHA(j)/BETA(j) is selected if
             DELZTG(ALPHA(j),BETA(j)) is true.

             Note that a selected complex eigenvalue may no longer satisfy
             DELZTG(ALPHA(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+2
             (See INFO below).

     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.  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) COMPLEX*16 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 DELZTG is true.

     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 ZGGES. 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.  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.  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.

             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.
     RWORK   (workspace) DOUBLE PRECISION array, dimension (8*N)

     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 ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N.  >
             N:  =N+1: other than QZ iteration failed in ZHGEQZ
             =N+2: after reordering, roundoff changed values of some complex
             eigenvalues so that leading eigenvalues in the Generalized Schur
             form no longer satisfy DELZTG=.TRUE.  This could also be caused
             due to scaling.  =N+3: reordering falied in ZTGSEN.

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

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