gees - Man Page

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

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.

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.

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.

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.