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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( * ) 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 sub- space (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 ele- ments. 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 vec- tors. On exit, if COMPQ = 'V', Q has been postmul- tiplied by the unitary transformation matrix which reorders T; the leading M columns of Q form an orthonormal basis for the specified invariant sub- space. 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 sub- space. 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 sub- space 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 eigen- values of T11 may be returned in S. S lies between 0 (very badly conditioned) and 1 (very well conditioned). It is com- puted 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 sub- space 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 recipro- cal 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