ztrsen(3)

NAME

ZTRSEN - reorder the Schur factorization of a complex ma

trix A = Q*T*Q**H, so that a selected cluster of eigenvalues ap

pears in the leading positions on the diagonal of the upper tri

angular matrix T, and the leading columns of Q form an orthonor

mal basis of the corresponding right invariant subspace

SYNOPSIS

SUBROUTINE ZTRSEN( 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
    DOUBLE         PRECISION S, SEP
    LOGICAL        SELECT( * )
    COMPLEX*16     Q( LDQ, * ), T( LDT, * ), W( * ), WORK(
* )

PURPOSE

ZTRSEN reorders the Schur factorization of a complex ma

trix A = Q*T*Q**H, so that a selected cluster of eigenvalues ap

pears in the leading positions on the diagonal of the upper tri

angular matrix T, and the leading columns of Q form an orthonor

mal basis of the corresponding right invariant subspace. Option

ally 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*16 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*16 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 invari

ant 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*16 array, dimension (N)

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) DOUBLE PRECISION

If JOB = 'E' or 'B', S is a lower bound on the re

ciprocal condition number for the selected cluster of eigenval

ues. 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) DOUBLE PRECISION

If JOB = 'V' or 'B', SEP is the estimated recipro

cal condition number of the specified invariant subspace. If M =

0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP is not refer

enced.

WORK (workspace/output) COMPLEX*16 array, dimension

(LWORK)

If JOB = 'N', WORK is not referenced. Otherwise,

on exit, if INFO = 0, WORK(1) returns the optimal LWORK.

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).

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 (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value

FURTHER DETAILS

ZTRSEN first collects the selected eigenvalues by comput

ing a unitary transformation Z to move them to the top left cor

ner of T. In other words, the selected eigenvalues are the eigen

values 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 condi

tion 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 re

turned 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

LAPACK version 3.0 15 June 2000

ZTRSEN(3)