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

Dtrsna() Sub Dtrsna ( Job As  String, Howmny As  String, Selct() As  Boolean, N As  Long, T() As  Double, Vl() As  Double, Vr() As  Double, S() As  Double, Sep() As  Double, Mm As  Long, M As  Long, Info As  Long  ) Condition numbers for eigenvalues and/or eigenvectors of upper quasi-triangular matrix Purpose This routine estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q^T with Q orthogonal). 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 eigenvalues (S) or eigenvectors (Sep). = "E": For eigenvalues only (S). = "V": For eigenvectors only (Sep). = "B": For both eigenvalues and eigenvectors (S and Sep). [in] Howmny = "A": Compute condition numbers for all eigenpairs. = "S": Compute condition numbers for selected eigenpairs specified by the array Selct(). [in] Selct() Array Selct(LSelct - 1) (LSelct >= N) If Howmny = "S", Selct() specifies the eigenpairs for which condition numbers are required. To select condition numbers for the eigenpair corresponding to a real eigenvalue W(j), Selct(j) must be set to True. To select condition numbers corresponding to a complex conjugate pair of eigenvalues W(j) and W(j+1), either Selct(j) or Selct(j+1) or both, must be set to True. If Howmny = "A", Selct() is not referenced. [in] N Order of the matrix T. (N >= 0) (If N = 0, returns without computation) [in] T() Array T(LT1 - 1, LT2 - 1) (LT1 >= N, LT2 >= N) The upper quasi-triangular matrix T in Schur canonical form. [in] Vl() Array Vl(LVl1 - 1, LVl2 - 1) (LVl1 >= N, LVl2 >= MM) If Job = "E" or "B", Vl() must contain left eigenvectors of T (or of any Q*T*Q^T with Q orthogonal), corresponding to the eigenpairs specified by Howmny and Selct(). The eigenvectors must be stored in consecutive columns of Vl(), as returned by Dhsein or Dtrevc. If Job = "V", Vl() is not referenced. [in] Vr() Array Vr(LVr1 - 1, LVr2 - 1) (LVr1 >= N, LVr2 >= MM) If Job = "E" or "B", Vr() must contain right eigenvectors of T (or of any Q*T*Q^T with Q orthogonal), corresponding to the eigenpairs specified by Howmny and Selct(). The eigenvectors must be stored in consecutive columns of Vr(), as returned by Dhsein or Dtrevc. If Job = "V", Vr() is not referenced. [out] S() Array S(LS - 1) (LS >= MM) If Job = "E" or "B", the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of S() are set to the same value. Thus S(j), Sep(j), and the j-th columns of Vl() and Vr() all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). If Job = "V", S() is not referenced. [out] Sep() Array Sep(LSep - 1) (LSep >= MM) If Job = "V" or "B", the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of Sep() are set to the same value. If the eigenvalues cannot be reordered to compute Sep(j), Sep(j) is set to 0; this can only occur when the true value would be very small anyway. If Job = "E", Sep() is not referenced. [in] MM The number of elements in the arrays S() (if Job = "E" or "B") and/or Sep() (if Job = "V" or "B"). (mm >= M) [out] M The number of elements of the arrays S() and/or Sep() actually used to store the estimated condition numbers. If Howmny = "A", M is set to N. [out] Info = 0: Successful exit. = -1: The argument Job had an illegal value. (Job <> "E", "V" nor "B") = -2: The argument Howmny had an illegal value. (Hownmy <> "A" nor "S") = -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 Vl() is invalid. = -7: The argument Vr() is invalid. = -8: The argument S() is invalid. = -9: The argument Sep() is invalid. = -10: The argument MM had an illegal value. (MM < M) Further Details The reciprocal of the condition number of an eigenvalue lambda is defined as S(lambda) = |v^T*u| / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of T corresponding to lambda; v^T denotes the transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If N = 1, S(lambda) is defined to be 1. An approximate error bound for a computed eigenvalue w(i) is given by eps * norm(T) / S(i) where eps is the machine precision. The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose T = ( lambda c ) ( 0 T22 ) Then the reciprocal condition number is Sep( lambda, T22 ) = sigma-min( T22 - lambda*I ) where sigma-min denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the one-norm of the inverse of T22 - lambda*I. If N = 1, Sep(1) is defined to be abs(T(1,1)). An approximate error bound for a computed right eigenvector vr(i) is given by eps * norm(T) / Sep(i) Reference LAPACK