SRC/zgges3.f(3)

NAME
SRC/zgges3.f

SYNOPSIS

  Functions/Subroutines
subroutine zgges3 (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)
ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Function/Subroutine Documentation

  subroutine zgges3 (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

     ZGGES3 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.

Parameters
    JOBVSL

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

    JOBVSR

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

    SORT

              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

              SELCTG is a 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

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

    A

              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

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

    B

              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

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

    SDIM

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

    ALPHA

              ALPHA is COMPLEX*16 array, dimension (N)

    BETA

              BETA is 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 ZGGES3. 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

              VSL is COMPLEX*16 array, dimension (LDVSL,N)
              If JOBVSL = 'V', VSL will contain the left Schur vectors.
              Not referenced if JOBVSL = 'N'.

    LDVSL

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

    VSR

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

    LDVSR

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

    WORK

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

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              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

              RWORK is DOUBLE PRECISION array, dimension (8*N)

    BWORK

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

    INFO

              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 ZLAQZ0
                    =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.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 266 of file zgges3.f.

Author

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