SGEESX(3)      LAPACK routine of NEC Numeric Library Collection      SGEESX(3)



NAME
       SGEESX

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



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

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

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

           SDIM      (output)
                     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        (output)
                     WR is REAL array, dimension (N)

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

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

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

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

           INFO      (output)
                     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.



LAPACK routine                  31 October 2017                      SGEESX(3)