Compute real Schur form T of real nonsymmetric matrix A, and, optionally, the matrix of Schur vectors Z as well as the eigenvalues

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    DGEES 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
            [  a  b  ]
            [  c  a  ]

    where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).


    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 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 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) INTEGER
            The order of the matrix A. N >= 0.

    A       (input/output) DOUBLE PRECISION 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) 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) DOUBLE PRECISION array, dimension (N)
    WI      (output) 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 will appear
            consecutively with the eigenvalue having the positive
            imaginary part first.

    VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)
            If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
            If JOBVS = 'N', VS is not referenced.

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

    WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
            On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

    LWORK   (input) 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   (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 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.


(T, Z, eval_real, eval_imag, info) = dgees(A)

Inputs (1)


Type: Real[:,size(A, 1)]

Description: Square matrix

Outputs (5)


Default Value: A

Type: Real[size(A, 1),size(A, 2)]

Description: Real Schur form with A = Z*T*Z'


Type: Real[size(A, 1),size(A, 1)]

Description: Orthogonal matrix Z of Schur vectors


Type: Real[size(A, 1)]

Description: Real part of the eigenvectors of A


Type: Real[size(A, 1)]

Description: Imaginary part of the eigenvectors of A


Type: Integer