SRC/cggesx.f(3)

NAME
SRC/cggesx.f

SYNOPSIS

  Functions/Subroutines
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)
CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Function/Subroutine Documentation

  subroutine cggesx (character jobvsl, character jobvsr, character sort, external selctg, character sense, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, real, dimension( 2 ) rconde, real, dimension( 2 ) rcondv, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

CGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

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.
    !>

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 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

    !>          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

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

    A

    !>          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

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

    B

    !>          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

    !>          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 array, dimension (N)
    !>

    BETA

    !>          BETA is 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

    !>          VSL is COMPLEX 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 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.
    !>

    RCONDE

    !>          RCONDE is 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

    !>          RCONDV is 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

    !>          WORK is COMPLEX 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 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

    !>          RWORK is REAL array, dimension ( 8*N )
    !>          Real workspace.
    !>

    IWORK

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

    LIWORK

    !>          LIWORK is 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

    !>          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 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 326 of file cggesx.f.

Author

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