SRC/sgeesx.f(3)

NAME
SRC/sgeesx.f

SYNOPSIS

  Functions/Subroutines
subroutine sgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)
SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Function/Subroutine Documentation

  subroutine sgeesx (character jobvs, character sort, external select, character sense, integer n, real, dimension( lda, * ) a, integer lda, integer sdim, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     SGEESX computes for an N-by-N real nonsymmetric matrix A, the
     eigenvalues, the real Schur form T, and, optionally, the matrix of
     Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
     Optionally, it also orders the eigenvalues on the diagonal of the
     real Schur form so that selected eigenvalues are at the top left;
     computes a reciprocal condition number for the average of the
     selected eigenvalues (RCONDE); and computes a reciprocal condition
     number for the right invariant subspace corresponding to the
     selected eigenvalues (RCONDV).  The leading columns of Z form an
     orthonormal basis for this invariant subspace.
     For further explanation of the reciprocal condition numbers RCONDE
     and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
     these quantities are called s and sep respectively).
     A real matrix is in real Schur form if it is upper quasi-triangular
     with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
     the form
               [  a  b  ]
               [  c  a  ]
     where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

Parameters
    JOBVS

              JOBVS is CHARACTER*1
              = 'N': Schur vectors are not computed;
              = 'V': Schur vectors are computed.

    SORT

              SORT is CHARACTER*1
              Specifies whether or not to order the eigenvalues on the
              diagonal of the Schur form.
              = 'N': Eigenvalues are not ordered;
              = 'S': Eigenvalues are ordered (see SELECT).

    SELECT

              SELECT is a LOGICAL FUNCTION of two REAL arguments
              SELECT must be declared EXTERNAL in the calling subroutine.
              If SORT = 'S', SELECT is used to select eigenvalues to sort
              to the top left of the Schur form.
              If SORT = 'N', SELECT is not referenced.
              An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
              SELECT(WR(j),WI(j)) is true; i.e., if either one of a
              complex conjugate pair of eigenvalues is selected, then both
              are.  Note that a selected complex eigenvalue may no longer
              satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
              ordering may change the value of complex eigenvalues
              (especially if the eigenvalue is ill-conditioned); in this
              case INFO may be 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 right invariant subspace only;
              = 'B': Computed for both.
              If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

    N

              N is INTEGER
              The order of the matrix A. N >= 0.

    A

              A is REAL array, dimension (LDA, N)
              On entry, the N-by-N matrix A.
              On exit, A is overwritten by its real Schur form T.

    LDA

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

    SDIM

              SDIM is INTEGER
              If SORT = 'N', SDIM = 0.
              If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                             for which SELECT is true. (Complex conjugate
                             pairs for which SELECT is true for either
                             eigenvalue count as 2.)

    WR

              WR is REAL array, dimension (N)

    WI

              WI is REAL array, dimension (N)
              WR and WI contain the real and imaginary parts, respectively,
              of the computed eigenvalues, in the same order that they
              appear on the diagonal of the output Schur form T.  Complex
              conjugate pairs of eigenvalues appear consecutively with the
              eigenvalue having the positive imaginary part first.

    VS

              VS is REAL array, dimension (LDVS,N)
              If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
              vectors.
              If JOBVS = 'N', VS is not referenced.

    LDVS

              LDVS is INTEGER
              The leading dimension of the array VS.  LDVS >= 1, and if
              JOBVS = 'V', LDVS >= N.

    RCONDE

              RCONDE is REAL
              If SENSE = 'E' or 'B', RCONDE contains the reciprocal
              condition number for the average of the selected eigenvalues.
              Not referenced if SENSE = 'N' or 'V'.

    RCONDV

              RCONDV is REAL
              If SENSE = 'V' or 'B', RCONDV contains the reciprocal
              condition number for the selected right invariant subspace.
              Not referenced if SENSE = 'N' or 'E'.

    WORK

              WORK is REAL 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.  LWORK >= max(1,3*N).
              Also, if SENSE = 'E' or 'V' or 'B',
              LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
              selected eigenvalues computed by this routine.  Note that
              N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
              returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
              'B' this may not be large enough.
              For good performance, LWORK must generally be larger.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates upper bounds on the optimal sizes of the
              arrays WORK and IWORK, returns these values as the first
              entries of the WORK and IWORK arrays, and no error messages
              related to LWORK or LIWORK are issued by XERBLA.

    IWORK

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

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
              Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
              only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
              may not be large enough.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates upper bounds on the optimal sizes of
              the arrays WORK and IWORK, returns these values as the first
              entries of the WORK and IWORK arrays, and no error messages
              related to LWORK or LIWORK are 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.
              > 0: if INFO = i, and i is
                 <= N: the QR algorithm failed to compute all the
                       eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                       contain those eigenvalues which have converged; if
                       JOBVS = 'V', VS contains the transformation which
                       reduces A to its partially converged Schur form.
                 = N+1: the eigenvalues could not be reordered because some
                       eigenvalues were too close to separate (the problem
                       is very ill-conditioned);
                 = N+2: after reordering, roundoff changed values of some
                       complex eigenvalues so that leading eigenvalues in
                       the Schur form no longer satisfy SELECT=.TRUE.  This
                       could also be caused by underflow due to scaling.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 278 of file sgeesx.f.

Author

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