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.

  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.

  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.

  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.

Author

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