ztrsen(3F)

NAME
     ZTRSEN - reorder 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
             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) COMPLEX*16 array, dimension (LWORK)
             If JOB = 'N', WORK is not referenced.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  If JOB = 'N', LWORK >= 1; if
             JOB = 'E', LWORK = M*(N-M); if JOB = 'V' or 'B', LWORK >=
             2*M*(N-M).

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

     where N = n1+n2 and Z' means the conjugate transpose of Z. 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', then the reordered 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