dgees.f - Man Page



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

Function/Subroutine Documentation

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  


 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
         [  a  b  ]
         [  c  a  ]

 where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).


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


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


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


          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 is DOUBLE PRECISION array, dimension (N)


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


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


          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) contains the optimal 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 is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.


          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.


December 2016

Definition at line 218 of file dgees.f.


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Referenced By

The man page dgees(3) is an alias of dgees.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK