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## zgeesx()

subroutine zgeesx | ( | character | jobvs,
|  | character | sort,
|  | external | select,
|  | character | sense,
|  | integer | n,
|  | complex*16, dimension( lda, * ) | a,
|  | integer | lda,
|  | integer | sdim,
|  | complex*16, dimension( * ) | w,
|  | complex*16, dimension( ldvs, * ) | vs,
|  | integer | ldvs,
|  | double precision | rconde,
|  | double precision | rcondv,
|  | complex*16, dimension( * ) | work,
|  | integer | lwork,
|  | double precision, dimension( * ) | rwork,
|  | logical, dimension( * ) | bwork,
|  | integer | info )

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

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

Purpose:

      !>
      !> ZGEESX 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;
      !> 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 complex matrix is in Schur form if it is upper triangular.
      !>

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 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.
      !>          An eigenvalue W(j) is selected if SELECT(W(j)) is true.
      !>

  [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 COMPLEX*16 array, dimension (LDA, N)
      !>          On entry, the N-by-N matrix A.
      !>          On exit, A is overwritten by its 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 for which
      !>                         SELECT is true.
      !>

  [out]

  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.
      !>

  [out]

  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.
      !>

  [in]

  LDVS

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

  [out]

  RCONDE

      !>          RCONDE is DOUBLE PRECISION
      !>          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 DOUBLE PRECISION
      !>          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 COMPLEX*16 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(1,2*N).
      !>          Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
      !>          where SDIM is the number of selected eigenvalues computed by
      !>          this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
      !>          that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the
      !>          array WORK, returns this value as the first entry of the WORK
      !>          array, and no error message related to LWORK is issued by
      !>          XERBLA.
      !>

  [out]

  RWORK

      !>          RWORK is DOUBLE PRECISION array, dimension (N)
      !>

  [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 W
      !>                   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.