Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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)

SYNOPSIS

       SUBROUTINE ZGGES( JOBVSL, JOBVSR, SORT, SELCTG, 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       SELCTG

           EXTERNAL      SELCTG

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 SELCTG).

        SELCTG  (external 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.
                An eigenvalue ALPHA(j)/BETA(j) is selected if
                SELCTG(ALPHA(j),BETA(j)) is true.
                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+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 SELCTG 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 (MAX(1,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 SELCTG=.TRUE.  This could also
                be caused due to scaling.
                =N+3: reordering falied in ZTGSEN.

 LAPACK driver routine (version 3.2)        April 2011                             ZGGES(3lapack)