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-triangular matrix T,

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( * )

DTRSEN reorders 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-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.

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.

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, either 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 eigenvalues 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 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.

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 reordering.

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 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) DOUBLE PRECISION

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) DOUBLE PRECISION array, dimension (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 illegal value

= 1: reordering of T failed because some eigenvalues are too close to separate (the problem is very ill-conditioned); T may have been partially reordered, and WR and WI contain the eigenvalues in the same order as in T; S and SEP (if requested) are set to zero.

DTRSEN first collects the selected eigenvalues by computing 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 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