LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dgees()

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

Download DGEES + dependencies [TGZ] [ZIP] [TXT]

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
[in]JOBVS
!>          JOBVS is CHARACTER*1
!>          = 'N': Schur vectors are not computed;
!>          = 'V': Schur vectors are computed.
!>
[in]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).
!>
[in]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).
!>
[in]N
!>          N is INTEGER
!>          The order of the matrix A. N >= 0.
!>
[in,out]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.
!>
[in]LDA
!>          LDA is INTEGER
!>          The leading dimension of the array A.  LDA >= max(1,N).
!>
[out]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.)
!>
[out]WR
!>          WR is DOUBLE PRECISION array, dimension (N)
!>
[out]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.
!>
[out]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.
!>
[in]LDVS
!>          LDVS is INTEGER
!>          The leading dimension of the array VS.  LDVS >= 1; if
!>          JOBVS = 'V', LDVS >= N.
!>
[out]WORK
!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
!>
[in]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.
!>
[out]BWORK
!>          BWORK is LOGICAL array, dimension (N)
!>          Not referenced if SORT = 'N'.
!>
[out]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.