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



NAME
       SGEES

SYNOPSIS
       SUBROUTINE SGEES (JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS,
           LDVS, WORK, LWORK, BWORK, INFO)



PURPOSE
            SGEES 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.
            The leading columns of Z then form an orthonormal basis for the
            invariant subspace corresponding to the selected eigenvalues.

            A 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 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 complex
                     eigenvalues are selected.
                     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 is set to N+2 (see INFO below).

           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 has been 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 will 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; if
                     JOBVS = 'V', LDVS >= N.

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

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK.  LWORK >= max(1,3*N).
                     For good performance, LWORK must generally be larger.

                     If LWORK = -1, then a workspace query is assumed; the routine
                     only calculates the optimal size of the WORK array, returns
                     this value as the first entry of the WORK array, and no error
                     message related to LWORK is 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 matrix 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                       SGEES(3)