strsen.f(3)

NAME
strsen.f -

SYNOPSIS

  Functions/Subroutines

subroutine strsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO)
STRSEN

Function/Subroutine Documentation

  subroutine strsen (characterJOB, characterCOMPQ, logical, dimension( * )SELECT, integerN, real, dimension( ldt, * )T, integerLDT, real, dimension( ldq, * )Q, integerLDQ, real, dimension( * )WR, real, dimension( * )WI, integerM, realS, realSEP, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, integerINFO)

STRSEN

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 elements equal and its
     off-diagonal elements of opposite sign.

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 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

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

    T

              T is 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

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

    Q

              Q is 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

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

    WR

              WR is REAL array, dimension (N)

    WI

              WI is 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

              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 REAL 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 >= max(1,N);
              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.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

    LIWORK

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

              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the IWORK array,
              returns this value as the first entry of the IWORK array, and
              no error message related to LIWORK is issued by XERBLA.

    INFO

              INFO is 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

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 * T * Z = ( T11 T12 ) n1
                             (  0  T22 ) n2
                                n1  n2

      where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given
      by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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 314 of file strsen.f.

Author

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