ZGGESX(3)      LAPACK routine of NEC Numeric Library Collection      ZGGESX(3)



NAME
       ZGGESX

SYNOPSIS
       SUBROUTINE ZGGESX (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
           LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV,
           WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)



PURPOSE
            ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
            (A,B), the generalized eigenvalues, the complex 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)**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; 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 complex Schur form if T is
            upper triangular with non-negative diagonal and S is upper
            triangular.




ARGUMENTS
           JOBVSL    (input)
                     JOBVSL is CHARACTER*1
                     = 'N':  do not compute the left Schur vectors;
                     = 'V':  compute the left Schur vectors.

           JOBVSR    (input)
                     JOBVSR is CHARACTER*1
                     = 'N':  do not compute the right Schur vectors;
                     = 'V':  compute the right Schur vectors.

           SORT      (input)
                     SORT is 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 SELCTG).

           SELCTG    (input)
                     SELCTG is procedure) LOGICAL FUNCTION of two COMPLEX*16 arguments
                     SELCTG must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'N', SELCTG is not referenced.
                     If SORT = 'S', SELCTG is used to select eigenvalues to sort
                     to the top left of the Schur form.
                     Note that a selected complex eigenvalue may no longer satisfy
                     SELCTG(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+3 see INFO below).

           SENSE     (input)
                     SENSE is CHARACTER*1
                     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)
                     N is INTEGER
                     The order of the matrices A, B, VSL, and VSR.  N >= 0.

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

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

           SDIM      (output)
                     SDIM is INTEGER
                     If SORT = 'N', SDIM = 0.
                     If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                     for which SELCTG is true.

           ALPHA     (output)
                     ALPHA is COMPLEX*16 array, dimension (N)

           BETA      (output)
                     BETA is COMPLEX*16 array, dimension (N)
                     On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
                     generalized eigenvalues.  ALPHA(j) and BETA(j),j=1,...,N  are
                     the diagonals of the complex Schur form (S,T).  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)
                     VSL is COMPLEX*16 array, dimension (LDVSL,N)
                     If JOBVSL = 'V', VSL will contain the left Schur vectors.
                     Not referenced if JOBVSL = 'N'.

           LDVSL     (input)
                     LDVSL is INTEGER
                     The leading dimension of the matrix VSL. LDVSL >=1, and
                     if JOBVSL = 'V', LDVSL >= N.

           VSR       (output)
                     VSR is COMPLEX*16 array, dimension (LDVSR,N)
                     If JOBVSR = 'V', VSR will contain the right Schur vectors.
                     Not referenced if JOBVSR = 'N'.

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

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

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK.
                     If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
                     LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else
                     LWORK >= MAX(1,2*N).  Note that 2*SDIM*(N-SDIM) <= N*N/2.
                     Note also that an error is only returned if
                     LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may
                     not be large enough.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the bound on the optimal size of the WORK
                     array and the minimum size of the IWORK array, returns these
                     values as the first entries of the WORK and IWORK arrays, and
                     no error message related to LWORK or LIWORK is issued by
                     XERBLA.

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

           IWORK     (output)
                     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
                     On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

           LIWORK    (input)
                     LIWORK is INTEGER
                     The dimension of the array IWORK.
                     If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
                     LIWORK >= N+2.

                     If LIWORK = -1, then a workspace query is assumed; the
                     routine only calculates the bound on the optimal size of the
                     WORK array and the minimum size of the IWORK array, returns
                     these values as the first entries of the WORK and IWORK
                     arrays, and no error message related to LWORK or LIWORK is
                     issued by XERBLA.

           BWORK     (output)
                     BWORK is LOGICAL array, dimension (N)
                     Not referenced if SORT = 'N'.

           INFO      (output)
                     INFO is 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 SELCTG=.TRUE.  This could also
                                 be caused due to scaling.
                           =N+3: reordering failed in ZTGSEN.



LAPACK routine                  31 October 2017                      ZGGESX(3)