geesx(3)

NAME
geesx - geesx: Schur form, expert

SYNOPSIS

  Functions

subroutine cgeesx (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)
CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)
DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde, rcondv, work, lwork, iwork, liwork, bwork, info)
SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgeesx (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde, rcondv, work, lwork, rwork, bwork, info)
ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Detailed Description

Function Documentation

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

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

Purpose:

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

    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'.
    !>

    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 is 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, and if
    !>          JOBVS = 'V', LDVS >= N.
    !>

    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'.
    !>

    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'.
    !>

    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).
    !>          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.
    !>

    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 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.

  subroutine dgeesx (character jobvs, character sort, external select, character sense, 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 rconde, double precision rcondv, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

    !>
    !> DGEESX 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
    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
    !>          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).
    !>

    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'.
    !>

    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 is 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 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, and if
    !>          JOBVS = 'V', LDVS >= N.
    !>

    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'.
    !>

    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'.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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(1,3*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.
    !>

    IWORK

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

    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.
    !>

    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 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.

  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

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
    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
    !>          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).
    !>

    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'.
    !>

    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 is 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 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, and if
    !>          JOBVS = 'V', LDVS >= N.
    !>

    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'.
    !>

    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'.
    !>

    WORK

    !>          WORK is REAL 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(1,3*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.
    !>

    IWORK

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

    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.
    !>

    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 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.

  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

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

    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'.
    !>

    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 is 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, and if
    !>          JOBVS = 'V', LDVS >= N.
    !>

    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'.
    !>

    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'.
    !>

    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).
    !>          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.
    !>

    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 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.

Author

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