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## sgeesx()

subroutine sgeesx | ( | character | jobvs,
|  | character | sort,
|  | external | select,
|  | character | sense,
|  | integer | n,
|  | real, dimension( lda, * ) | a,
|  | integer | lda,
|  | integer | sdim,
|  | real, dimension( * ) | wr,
|  | real, dimension( * ) | wi,
|  | real, dimension( ldvs, * ) | vs,
|  | integer | ldvs,
|  | real | rconde,
|  | real | rcondv,
|  | real, dimension( * ) | work,
|  | integer | lwork,
|  | integer, dimension( * ) | iwork,
|  | integer | liwork,
|  | logical, dimension( * ) | bwork,
|  | integer | info )

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

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

Purpose:

      !>
      !> SGEESX 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;
      !> computes a reciprocal condition number for the average of the
      !> selected eigenvalues (RCONDE); and computes a reciprocal condition
      !> number for the right invariant subspace corresponding to the
      !> selected eigenvalues (RCONDV).  The leading columns of Z form an
      !> orthonormal basis for this invariant subspace.
      !>
      !> For further explanation of the reciprocal condition numbers RCONDE
      !> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
      !> these quantities are called s and sep respectively).
      !>
      !> A real 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 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
      !>          are.  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 may be set to N+3 (see INFO below).
      !>

  [in]

  SENSE

      !>          SENSE is CHARACTER*1
      !>          Determines which reciprocal condition numbers are computed.
      !>          = 'N': None are computed;
      !>          = 'E': Computed for average of selected eigenvalues only;
      !>          = 'V': Computed for selected right invariant subspace only;
      !>          = 'B': Computed for both.
      !>          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
      !>

  [in]

  N

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

  [in,out]

  A

      !>          A is REAL array, dimension (LDA, N)
      !>          On entry, the N-by-N matrix A.
      !>          On exit, A is 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 REAL array, dimension (N)
      !>

  [out]

  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 appear consecutively with the
      !>          eigenvalue having the positive imaginary part first.
      !>

  [out]

  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.
      !>

  [in]

  LDVS

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

  [out]

  RCONDE

      !>          RCONDE is REAL
      !>          If SENSE = 'E' or 'B', RCONDE contains the reciprocal
      !>          condition number for the average of the selected eigenvalues.
      !>          Not referenced if SENSE = 'N' or 'V'.
      !>

  [out]

  RCONDV

      !>          RCONDV is REAL
      !>          If SENSE = 'V' or 'B', RCONDV contains the reciprocal
      !>          condition number for the selected right invariant subspace.
      !>          Not referenced if SENSE = 'N' or 'E'.
      !>

  [out]

  WORK

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

  [in]

  LWORK

      !>          LWORK is INTEGER
      !>          The dimension of the array WORK.  LWORK >= max(1,3*N).
      !>          Also, if SENSE = 'E' or 'V' or 'B',
      !>          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
      !>          selected eigenvalues computed by this routine.  Note that
      !>          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
      !>          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
      !>          'B' this may not be large enough.
      !>          For good performance, LWORK must generally be larger.
      !>
      !>          If LWORK = -1, then a workspace query is assumed; the routine
      !>          only calculates upper bounds on the optimal sizes of the
      !>          arrays WORK and IWORK, returns these values as the first
      !>          entries of the WORK and IWORK arrays, and no error messages
      !>          related to LWORK or LIWORK are issued by XERBLA.
      !>

  [out]

  IWORK

      !>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
      !>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
      !>

  [in]

  LIWORK

      !>          LIWORK is INTEGER
      !>          The dimension of the array IWORK.
      !>          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
      !>          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
      !>          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
      !>          may not be large enough.
      !>
      !>          If LIWORK = -1, then a workspace query is assumed; the
      !>          routine only calculates upper bounds on the optimal sizes of
      !>          the arrays WORK and IWORK, returns these values as the first
      !>          entries of the WORK and IWORK arrays, and no error messages
      !>          related to LWORK or LIWORK are 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 transformation 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.