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dtrsen


 NAME
      DTRSEN - reorder the real Schur factorization of a real
      matrix A = Q*T*Q**T, so that a selected cluster of eigen-
      values 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 eigen-
      values 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 diag-
      onal 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 sub-
              space (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 vec-
              tors.  On exit, if COMPQ = 'V', Q has been postmul-
              tiplied by the orthogonal transformation matrix
              which reorders T; the leading M columns of Q form an
              orthonormal basis for the specified invariant sub-
              space.  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 signifi-
              cantly 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 sub-
              space. If M = 0 or N, SEP = norm(T).  If JOB = 'N'
              or 'E', SEP is not referenced.

      WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK)

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

      IWORK   (workspace) INTEGER
              IF JOB = 'N' or 'E', IWORK is not referenced.

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

      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 factori-
      zation 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 eigen-
      values of T11 may be returned in S. S lies between 0 (very
      badly conditioned) and 1 (very well conditioned). It is com-
      puted 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 condi-
      tion 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 sub-
      space 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 recipro-
      cal 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