Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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

SYNOPSIS

       SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS,  RCONDE,
                          RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO )

           CHARACTER      JOBVS, SENSE, SORT

           INTEGER        INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM

           REAL           RCONDE, RCONDV

           LOGICAL        BWORK( * )

           INTEGER        IWORK( * )

           REAL           A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( * )

           LOGICAL        SELECT

           EXTERNAL       SELECT

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).

ARGUMENTS

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

        SORT    (input) 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  (external procedure) 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   (input) 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       (input) INTEGER
                The order of the matrix A. N >= 0.

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

        SDIM    (output) 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      (output) REAL array, dimension (N)
                WI      (output) 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      (output) 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    (input) INTEGER
                The leading dimension of the array VS.  LDVS >= 1, and if
                JOBVS = 'V', LDVS >= N.

        RCONDE  (output) 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  (output) 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    (workspace/output) REAL array, dimension (MAX(1,LWORK))
                On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

        LWORK   (input) 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   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
                On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

        LIWORK  (input) 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   (workspace) LOGICAL array, dimension (N)
                Not referenced if SORT = 'N'.

        INFO    (output) 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.

 LAPACK driver routine (version 3.2.2)      April 2011                            SGEESX(3lapack)