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

NAME

       LAPACK-3  -  reorders  the Schur factorization of a complex matrix A = Q*T*Q**H, so that a
       selected cluster of eigenvalues appears in the leading positions on the  diagonal  of  the
       upper  triangular  matrix T, and the leading columns of Q form an orthonormal basis of the
       corresponding right invariant subspace

SYNOPSIS

       SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK,  INFO
                          )

           CHARACTER      COMPQ, JOB

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

           DOUBLE         PRECISION S, SEP

           LOGICAL        SELECT( * )

           COMPLEX*16     Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )

PURPOSE

       ZTRSEN  reorders  the  Schur  factorization  of  a  complex matrix A = Q*T*Q**H, so that a
       selected cluster of eigenvalues appears in the leading positions on the  diagonal  of  the
       upper  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.

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 the j-th eigenvalue, SELECT(j) must be set to .TRUE..

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

        T       (input/output) COMPLEX*16 array, dimension (LDT,N)
                On entry, the upper triangular matrix T.
                On exit, T is overwritten by the reordered matrix T, with the
                selected eigenvalues as the leading diagonal elements.

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

        Q       (input/output) COMPLEX*16 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
                unitary 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.

        W       (output) COMPLEX*16 array, dimension (N)
                The reordered eigenvalues of T, in the same order as they
                appear on the diagonal of T.

        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) COMPLEX*16 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 >= 1;
                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.

        INFO    (output) INTEGER
                = 0:  successful exit
                < 0:  if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

        ZTRSEN first collects the selected eigenvalues by computing a unitary
        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**H * T * Z = ( T11 T12 ) n1
                               (  0  T22 ) n2
                                  n1  n2
        where N = n1+n2. The first
        n1 columns of Z span the specified invariant subspace of T.
        If T has been obtained from the Schur factorization of a matrix
        A = Q*T*Q**H, then the reordered Schur factorization of A is given by
        A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, 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                            ZTRSEN(3lapack)