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 complex Schur form (S,T),

SYNOPSIS

       SUBROUTINE CGGESX( 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 )

           CHARACTER      JOBVSL, JOBVSR, SENSE, SORT

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

           LOGICAL        BWORK( * )

           INTEGER        IWORK( * )

           REAL           RCONDE( 2 ), RCONDV( 2 ), RWORK( * )

           COMPLEX        A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ),  VSR(
                          LDVSR, * ), WORK( * )

           LOGICAL        SELCTG

           EXTERNAL       SELCTG

PURPOSE

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

        A       (input/output) COMPLEX 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 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 array, dimension (N)
                BETA    (output) COMPLEX 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) COMPLEX 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 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) REAL 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) REAL 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    (workspace/output) COMPLEX 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.
                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   (workspace) REAL array, dimension ( 8*N )
                Real workspace.

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

        LIWORK  (input) INTEGER
                The dimension of the array WORK.
                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   (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 CHGEQZ
                =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 CTGSEN.

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