SRC/strsen.f(3)

NAME
SRC/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 (character job, character compq, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldq, * ) q, integer ldq, real, dimension( * ) wr, real, dimension( * ) wi, integer m, real s, real sep, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

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.

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 312 of file strsen.f.

Author

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