ctrsen-f(3)

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

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

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

 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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