Dgebak Dgebal Dgehrd Dhsein Dhseqr Dorghr Dormhr Dtrevc3 Dtrexc Dtrsen Dtrsna Dtrsyl

Dtrsen() Sub Dtrsen ( Job As  String, Compq As  String, Selct() As  Boolean, N As  Long, T() As  Double, Q() As  Double, Wr() As  Double, Wi() As  Double, M As  Long, S As  Double, Sep As  Double, Info As  Long  ) Reordering of real Schur factorization of real matrix and condition numbers of cluster of eigenvalues and/or invariant subspace Purpose This routine 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 x 1 and 2 x 2 diagonal blocks. Each 2 x 2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. Parameters [in] Job 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). [in] Compq = "V": Update the matrix Q of Schur vectors. = "N": Do not update Q. [in] Selct() Array Selct(LSelct - 1) (LSelct >= N) Selct() specifies the eigenvalues in the selected cluster. To select a real eigenvalue W(j), Selct(j) must be set to True. To select a complex conjugate pair of eigenvalues W(j) and W(j+1), corresponding to a 2 x 2 diagonal block, either Selct(j) or Selct(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. [in] N Order of the matrix T. (N >= 0) (If N = 0, returns without computation) [in,out] T() Array T(LT1 - 1, LT2 - 1) (LT1 >= N, LT2 >= N) [in] The upper quasi-triangular matrix T, in Schur canonical form. [out] T is overwritten by the reordered matrix T, again in Schur canonical form, with the selected eigenvalues in the leading diagonal blocks. [in] Q() Array Q(LQ1 - 1, LQ2 - 1) (LQ1 >= N, LQ2 >= N) If Compq = "V", [in] The matrix Q of Schur vectors. [out] 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. [out] Wr() Array Wr(LWr - 1) (LWr >= N) [out] Wi() Array Wi(LWi - 1) (LWi >= 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 x 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. [out] M The dimension of the specified invariant subspace. (0 < = M <= N) [out] S 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. [out] Sep 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. [out] Info = 0: Successful exit. = -1: The argument Job had an illegal value. (Job <> "N", "E", "V" nor "B") = -2: The argument Compq had an illegal value. (Compq <> "V" nor "N") = -3: The argument Selct() is invalid. = -4: The argument N had an illegal value. (N < 0) = -5: The argument T() is invalid. = -6: The argument Q() is invalid. = -7: The argument Wr() is invalid. = -8: The argument Wi() is invalid. = 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 This routine 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 * T * Z = ( T11 T12 ) n1 ( 0 T22 ) n2 n1 n2 where N = n1 + n2 and Z^T 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^T, then the reordered real Schur factorization of A is given by A = (Q*Z)*(Z^T*T*Z)*(Q*Z)^T, 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 x n1*n2 matrix C = kprod( I(n2), T11 ) - kprod( T22^T, I(n1) ) I(M) is an M x 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 C^(-1). The true reciprocal 1-norm of C^(-1) 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(i) Reference LAPACK