STRSEN (3lapack)

reorder 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 STRSEN( 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

      REAL           S, SEP

      LOGICAL        SELECT( * )

      INTEGER        IWORK( * )

      REAL           Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), WR( * )

PURPOSE

  STRSEN 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 SHSEQR), 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 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 either both included in the
          cluster or both excluded.

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

  T       (input/output) REAL 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) REAL 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) REAL array, dimension (N)
          WI      (output) REAL 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) REAL
          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) REAL
          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) REAL 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 array, dimension (LIWORK)
          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

  STRSEN 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