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NAME DTRSEN - reorder the real Schur factorization of a real matrix A = Q*T*Q**T, so that a selected cluster of eigen- values appears in the leading diagonal blocks of the upper quasi-triangular 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 matrix A = Q*T*Q**T, so that a selected cluster of eigen- values 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 diag- onal blocks; each 2-by-2 diagonal block has its diagonal elements 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 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 a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select a complex conjugate pair of eigenvalues 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 .TRUE.; a complex conjugate pair of eigenvalues must be 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 vec- tors. On exit, if COMPQ = 'V', Q has been postmul- tiplied by the orthogonal 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. 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 signifi- cantly 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 sub- space. If M = 0 or N, SEP = norm(T). If JOB = 'N' or 'E', SEP is not referenced. WORK (workspace) DOUBLE PRECISION array, dimension (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). IWORK (workspace) INTEGER 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). 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. FURTHER DETAILS 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 factori- zation 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