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



NAME
       CGGES

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



PURPOSE
            CGGES 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
            CGGEV 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)
                     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 a 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.
                     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)
                     N is INTEGER
                     The order of the matrices A, B, VSL, and VSR.  N >= 0.

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

           B         (input/output)
                     B is 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)
                     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 array, dimension (N)

           BETA      (output)
                     BETA is COMPLEX 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 CGGES. 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)
                     VSL is COMPLEX 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 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.

           WORK      (output)
                     WORK is COMPLEX 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.  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     (output)
                     RWORK is REAL array, dimension (8*N)

           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 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 falied in CTGSEN.



LAPACK routine                  31 October 2017                       CGGES(3)