gees(3)

NAME
gees - gees: Schur form

SYNOPSIS

  Functions
subroutine cgees (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)
CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgees (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)
DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgees (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)
SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgees (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)
ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Detailed Description

Function Documentation

  subroutine cgees (character jobvs, character sort, external select, integer n, complex, dimension( lda, * ) a, integer lda, integer sdim, complex, dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     CGEES computes for an N-by-N complex nonsymmetric matrix A, the
     eigenvalues, the Schur form T, and, optionally, the matrix of Schur
     vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
     Optionally, it also orders the eigenvalues on the diagonal of the
     Schur form so that selected eigenvalues are at the top left.
     The leading columns of Z then form an orthonormal basis for the
     invariant subspace corresponding to the selected eigenvalues.
     A complex matrix is in Schur form if it is upper triangular.

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 one COMPLEX argument
              SELECT must be declared EXTERNAL in the calling subroutine.
              If SORT = 'S', SELECT is used to select eigenvalues to order
              to the top left of the Schur form.
              IF SORT = 'N', SELECT is not referenced.
              The eigenvalue W(j) is selected if SELECT(W(j)) is true.

    N

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

    A

              A is COMPLEX array, dimension (LDA,N)
              On entry, the N-by-N matrix A.
              On exit, A has been overwritten by its 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 for which
                             SELECT is true.

    W

              W is COMPLEX array, dimension (N)
              W contains the computed eigenvalues, in the same order that
              they appear on the diagonal of the output Schur form T.

    VS

              VS is COMPLEX array, dimension (LDVS,N)
              If JOBVS = 'V', VS contains the unitary 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; if
              JOBVS = 'V', LDVS >= N.

    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.  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

              RWORK is REAL array, dimension (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.
              > 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 W
                          contain those eigenvalues which have converged;
                          if JOBVS = 'V', VS contains the matrix 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 195 of file cgees.f.

  subroutine dgees (character jobvs, character sort, external select, integer n, double precision, dimension( lda, * ) a, integer lda, integer sdim, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvs, * ) vs, integer ldvs, double precision, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     DGEES 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.
     The leading columns of Z then form an orthonormal basis for the
     invariant subspace corresponding to the selected eigenvalues.
     A 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 DOUBLE PRECISION 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 complex
              eigenvalues are selected.
              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 is set to N+2 (see INFO below).

    N

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

    A

              A is DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the N-by-N matrix A.
              On exit, A has been 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 DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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 will appear
              consecutively with the eigenvalue having the positive
              imaginary part first.

    VS

              VS is DOUBLE PRECISION 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; if
              JOBVS = 'V', LDVS >= N.

    WORK

              WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,3*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.

    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 matrix 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 214 of file dgees.f.

  subroutine sgees (character jobvs, character sort, external select, integer n, real, dimension( lda, * ) a, integer lda, integer sdim, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvs, * ) vs, integer ldvs, real, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     SGEES 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.
     The leading columns of Z then form an orthonormal basis for the
     invariant subspace corresponding to the selected eigenvalues.
     A 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 complex
              eigenvalues are selected.
              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 is set to N+2 (see INFO below).

    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 has been 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 will 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; if
              JOBVS = 'V', LDVS >= N.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= max(1,3*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.

    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 matrix 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 214 of file sgees.f.

  subroutine zgees (character jobvs, character sort, external select, integer n, complex*16, dimension( lda, * ) a, integer lda, integer sdim, complex*16, dimension( * ) w, complex*16, dimension( ldvs, * ) vs, integer ldvs, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

     ZGEES computes for an N-by-N complex nonsymmetric matrix A, the
     eigenvalues, the Schur form T, and, optionally, the matrix of Schur
     vectors Z.  This gives the Schur factorization A = Z*T*(Z**H).
     Optionally, it also orders the eigenvalues on the diagonal of the
     Schur form so that selected eigenvalues are at the top left.
     The leading columns of Z then form an orthonormal basis for the
     invariant subspace corresponding to the selected eigenvalues.
     A complex matrix is in Schur form if it is upper triangular.

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 one COMPLEX*16 argument
              SELECT must be declared EXTERNAL in the calling subroutine.
              If SORT = 'S', SELECT is used to select eigenvalues to order
              to the top left of the Schur form.
              IF SORT = 'N', SELECT is not referenced.
              The eigenvalue W(j) is selected if SELECT(W(j)) is true.

    N

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

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the N-by-N matrix A.
              On exit, A has been overwritten by its 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 for which
                             SELECT is true.

    W

              W is COMPLEX*16 array, dimension (N)
              W contains the computed eigenvalues, in the same order that
              they appear on the diagonal of the output Schur form T.

    VS

              VS is COMPLEX*16 array, dimension (LDVS,N)
              If JOBVS = 'V', VS contains the unitary 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; if
              JOBVS = 'V', LDVS >= 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.  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

              RWORK is DOUBLE PRECISION array, dimension (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.
              > 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 W
                          contain those eigenvalues which have converged;
                          if JOBVS = 'V', VS contains the matrix 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 195 of file zgees.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.