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

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( * )

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.

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 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) 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/output) COMPLEX 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 illegal value

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(n), T11 ) - kprod( transpose(T22), I(n) )

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