dtrsen(3)

NAME

DTRSEN - reorder 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-trian

gular matrix T,

SYNOPSIS

SUBROUTINE  DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ,
WR, WI, M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
    CHARACTER      COMPQ, JOB
    INTEGER        INFO, LDQ, LDT, LIWORK, LWORK, M, N
    DOUBLE         PRECISION S, SEP
    LOGICAL        SELECT( * )
    INTEGER        IWORK( * )
    DOUBLE         PRECISION Q( LDQ, * ), T( LDT, * ), WI(
* ), WORK( * ), WR( * )

PURPOSE

DTRSEN reorders the real Schur factorization of a real ma

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

pears in the leading diagonal blocks of the upper quasi-triangu

lar 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 sub

space.

T must be in Schur canonical form (as returned by DHSEQR),

that is, block upper triangular with 1-by-1 and 2-by-2 diagonal

blocks; each 2-by-2 diagonal block has its diagonal elemnts equal

and its off-diagonal elements of opposite sign.

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 a real eigenvalue w(j), SELECT(j) must be set

to w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, ei

ther SELECT(j) or SELECT(j+1) or both must be set to either both

included in the cluster or both excluded.

N (input) INTEGER

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

T (input/output) DOUBLE PRECISION 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 eigen

values in the leading diagonal blocks.

LDT (input) INTEGER

The leading dimension of the array T. LDT >=

max(1,N).

Q (input/output) DOUBLE PRECISION 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 lead

ing 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.

WR (output) DOUBLE PRECISION array, dimension (N)

WI (output) DOUBLE PRECISION 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 re

ordering.

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) DOUBLE PRECISION array, dimen

sion (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. If JOB = 'N',

LWORK >= max(1,N); 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.

IWORK (workspace) INTEGER array, dimension (LIWORK)

IF JOB = 'N' or 'E', IWORK is not referenced.

LIWORK (input) INTEGER

The dimension of the array IWORK. If JOB = 'N' or

'E', LIWORK >= 1; if JOB = 'V' or 'B', LIWORK >= 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 (output) INTEGER

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

gal value
= 1: reordering of T failed because some eigenval

ues are too close to separate (the problem is very ill-condi

tioned); T may have been partially reordered, and WR and WI con

tain the eigenvalues in the same order as in T; S and SEP (if re

quested) are set to zero.

FURTHER DETAILS

DTRSEN first collects the selected eigenvalues by comput

ing 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*Z = ( T11 T12 ) n1

( 0 T22 ) n2

n1 n2

where N = n1+n2 and Z' 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', then the reordered real Schur factoriza

tion 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

DTRSEN(3)