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

NAME

       LAPACK-3  - 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,

SYNOPSIS

       SUBROUTINE DTRSEN( JOB,  COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK,
                          IWORK, LIWORK, INFO )

           CHARACTER      COMPQ, JOB

           INTEGER        INFO, LDQ, LDT, LIWORK, LWORK, M, N

           DOUBLE         PRECISION S, SEP

           LOGICAL        SELECT( * )

           INTEGER        IWORK( * )

           DOUBLE         PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), WR( * )

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 elemnts 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 * T * Z = ( T11 T12 ) n1
                               (  0  T22 ) n2
                                  n1  n2
        where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given
        by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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

 LAPACK routine (version 3.3.1)             April 2011                            DTRSEN(3lapack)