This information is part of the Modelica Standard Library maintained by the Modelica Association.

Lapack documentation
    Purpose
    =======

    DTRSEN reorders the real Schur factorization of a real matrix
    A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
    the leading diagonal blocks of the upper quasi-triangular matrix T,
    and the leading columns of Q form an orthonormal basis of the
    corresponding right invariant subspace.

    Optionally the routine computes the reciprocal condition numbers of
    the cluster of eigenvalues and/or the invariant subspace.

    T must be in Schur canonical form (as returned by DHSEQR), that is,
    block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
    2-by-2 diagonal block has its diagonal elements equal and its
    off-diagonal elements of opposite sign.

    Arguments
    =========

    JOB     (input) CHARACTER*1
            Specifies whether condition numbers are required for the
            cluster of eigenvalues (S) or the invariant subspace (SEP):
            = 'N': none;
            = 'E': for eigenvalues only (S);
            = 'V': for invariant subspace only (SEP);
            = 'B': for both eigenvalues and invariant subspace (S and
                   SEP).

    COMPQ   (input) CHARACTER*1
            = 'V': update the matrix Q of Schur vectors;
            = 'N': do not update Q.

    SELECT  (input) LOGICAL array, dimension (N)
            SELECT specifies the eigenvalues in the selected cluster. To
            select a real eigenvalue w(j), SELECT(j) must be set to
            .TRUE.. To select a complex conjugate pair of eigenvalues
            w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
            either SELECT(j) or SELECT(j+1) or both must be set to
            .TRUE.; a complex conjugate pair of eigenvalues must be
            either both included in the cluster or both excluded.

    N       (input) INTEGER
            The order of the matrix T. N >= 0.

    T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
            On entry, the upper quasi-triangular matrix T, in Schur
            canonical form.
            On exit, T is overwritten by the reordered matrix T, again in
            Schur canonical form, with the selected eigenvalues in the
            leading diagonal blocks.

    LDT     (input) INTEGER
            The leading dimension of the array T. LDT >= max(1,N).

    Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
            On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
            On exit, if COMPQ = 'V', Q has been postmultiplied by the
            orthogonal transformation matrix which reorders T; the
            leading M columns of Q form an orthonormal basis for the
            specified invariant subspace.
            If COMPQ = 'N', Q is not referenced.

    LDQ     (input) INTEGER
            The leading dimension of the array Q.
            LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

    WR      (output) DOUBLE PRECISION array, dimension (N)
    WI      (output) DOUBLE PRECISION array, dimension (N)
            The real and imaginary parts, respectively, of the reordered
            eigenvalues of T. The eigenvalues are stored in the same
            order as on the diagonal of T, with WR(i) = T(i,i) and, if
            T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
            WI(i+1) = -WI(i). Note that if a complex eigenvalue is
            sufficiently ill-conditioned, then its value may differ
            significantly from its value before reordering.

    M       (output) INTEGER
            The dimension of the specified invariant subspace.
            0 < = M <= N.

    S       (output) DOUBLE PRECISION
            If JOB = 'E' or 'B', S is a lower bound on the reciprocal
            condition number for the selected cluster of eigenvalues.
            S cannot underestimate the true reciprocal condition number
            by more than a factor of sqrt(N). If M = 0 or N, S = 1.
            If JOB = 'N' or 'V', S is not referenced.

    SEP     (output) DOUBLE PRECISION
            If JOB = 'V' or 'B', SEP is the estimated reciprocal
            condition number of the specified invariant subspace. If
            M = 0 or N, SEP = norm(T).
            If JOB = 'N' or 'E', SEP is not referenced.

    WORK    (workspace/output) DOUBLE PRECISION 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.
            If JOB = 'N', LWORK >= max(1,N);
            if JOB = 'E', LWORK >= max(1,M*(N-M));
            if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

            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.

    IWORK   (workspace) 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.
            If JOB = 'N' or 'E', LIWORK >= 1;
            if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).

            If LIWORK = -1, then a workspace query is assumed; the
            routine only calculates the optimal size of the IWORK array,
            returns this value as the first entry of the IWORK array, and
            no error message related to LIWORK is issued by XERBLA.

    INFO    (output) INTEGER
            = 0: successful exit
            < 0: if INFO = -i, the i-th argument had an illegal value
            = 1: reordering of T failed because some eigenvalues are too
                 close to separate (the problem is very ill-conditioned);
                 T may have been partially reordered, and WR and WI
                 contain the eigenvalues in the same order as in T; S and
                 SEP (if requested) are set to zero.

    Further Details
    ===============

    DTRSEN first collects the selected eigenvalues by computing an
    orthogonal transformation Z to move them to the top left corner of T.
    In other words, the selected eigenvalues are the eigenvalues of T11
    in:

                  Z'*T*Z = ( T11 T12 ) n1
                           (  0  T22 ) n2
                              n1  n2

    where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
    of Z span the specified invariant subspace of T.

    If T has been obtained from the real Schur factorization of a matrix
    A = Q*T*Q', then the reordered real Schur factorization of A is given
    by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
    the corresponding invariant subspace of A.

    The reciprocal condition number of the average of the eigenvalues of
    T11 may be returned in S. S lies between 0 (very badly conditioned)
    and 1 (very well conditioned). It is computed as follows. First we
    compute R so that

                           P = ( I  R ) n1
                               ( 0  0 ) n2
                                 n1 n2

    is the projector on the invariant subspace associated with T11.
    R is the solution of the Sylvester equation:

                          T11*R - R*T22 = T12.

    Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
    the two-norm of M. Then S is computed as the lower bound

                        (1 + F-norm(R)**2)**(-1/2)

    on the reciprocal of 2-norm(P), the true reciprocal condition number.
    S cannot underestimate 1 / 2-norm(P) by more than a factor of
    sqrt(N).

    An approximate error bound for the computed average of the
    eigenvalues of T11 is

                           EPS * norm(T) / S

    where EPS is the machine precision.

    The reciprocal condition number of the right invariant subspace
    spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
    SEP is defined as the separation of T11 and T22:

                       sep( T11, T22 ) = sigma-min( C )

    where sigma-min(C) is the smallest singular value of the
    n1*n2-by-n1*n2 matrix

       C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

    I(m) is an m by m identity matrix, and kprod denotes the Kronecker
    product. We estimate sigma-min(C) by the reciprocal of an estimate of
    the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
    cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

    When SEP is small, small changes in T can cause large changes in
    the invariant subspace. An approximate bound on the maximum angular
    error in the computed right invariant subspace is

                        EPS * norm(T) / SEP

(To, Qo, wr, wi, m, s, sep, info) = dtrsen(job, compq, select, T, Q)

job	Default Value: "N" Type: String Description: Specifies the usage of a condition number
compq	Default Value: "V" Type: String Description: Is "V" if Schur vector matrix is to be updated
select	Type: Boolean[:] Description: Specifies the eigenvalues to reorder
T	Type: Real[:,:] Description: Real Schur form to be reordered
Q	Type: Real[:,size(T, 2)] Description: Matrix of the Schur vectors

To	Default Value: T Type: Real[:,:] Description: Reordered Schur form
Qo	Default Value: Q Type: Real[:,:] Description: Reordered Schur vectors
wr	Type: Real[size(T, 2)] Description: Reordered eigenvalues, real part
wi	Type: Real[size(T, 2)] Description: Reordered eigenvalues, imaginary part
m	Type: Integer Description: Dimension of the invariant sub space spanned bei the selected eigenvalues
s	Type: Real Description: Lower bound of the reciprocal condition number. Not referenced for job==V
sep	Type: Real Description: Estimated reciprocal condition number of the specified invariant subspace
info	Type: Integer