ctrsen.f(3)

NAME
ctrsen.f -

SYNOPSIS

  Functions/Subroutines

subroutine ctrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
CTRSEN

Function/Subroutine Documentation

  subroutine ctrsen (characterJOB, characterCOMPQ, logical, dimension( * )SELECT, integerN, complex, dimension( ldt, * )T, integerLDT, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( * )W, integerM, realS, realSEP, complex, dimension( * )WORK, integerLWORK, integerINFO)

CTRSEN

Purpose:

     CTRSEN 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.

Parameters:
    JOB

              JOB is 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

              COMPQ is CHARACTER*1
              = 'V': update the matrix Q of Schur vectors;
              = 'N': do not update Q.

    SELECT

              SELECT is 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

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is COMPLEX 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

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

    Q

              Q is COMPLEX 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

              LDQ is INTEGER
              The leading dimension of the array Q.
              LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

    W

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

    M

              M is INTEGER
              The dimension of the specified invariant subspace.
              0 <= M <= N.

    S

              S is 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

              SEP is 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

              WORK is COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is 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

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

Further Details:

      CTRSEN 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

Definition at line 264 of file ctrsen.f.

Author

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