ctrsen(3P)

NAME

ctrsen - reorder 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

SYNOPSIS

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

CHARACTER COMPQ, JOB

INTEGER INFO, LDQ, LDT, LWORK, M, N

REAL S, SEP

LOGICAL SELECT( ∗ )

COMPLEX Q( LDQ, ∗ ), T( LDT, ∗ ), W( ∗ ), WORK( ∗ )

#include <sunperf.h>

void ctrsen(char job, char compq, int ∗select, int n, complex ∗t, int ldt, complex ∗q, int ldq, complex ∗w, int ∗m, float ∗s, float ∗sep, int ∗info) ;

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.

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 the j-th eigenvalue, SELECT(j) must be set to .TRUE..

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

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

Q (input/output) 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 (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

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

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) COMPLEX array, dimension (LWORK)
If JOB = ’N’, WORK is not referenced.

LWORK (input) INTEGER
The dimension of the array WORK. If JOB = ’N’, LWORK >= 1; if JOB = ’E’, LWORK = M∗(N-M); if JOB = ’V’ or ’B’, LWORK >= 2∗M∗(N-M).

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

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’∗T∗Z = ( T11 T12 ) n1
                       ( 0 T22 ) n2
                          n1 n2

where N = n1+n2 and Z’ means the conjugate transpose of Z. 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’, then the reordered 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