dgebak dgebal dgehrd dhsein dhseqr dorghr dormhr dtrevc3 dtrexc dtrsen dtrsna dtrsyl

dtrsna() void dtrsna ( char  job, char  howmny, int  select[], int  n, int  ldt, double  t[], int  ldvl, double  vl[], int  ldvr, double  vr[], double  s[], double  sep[], int  mm, int *  m, double  work[], int  lwork, int  iwork[], int  liwork, int *  info  ) 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 select[]. [in] select[] Array select[lselect] (lselect >= n) If howmny = 'S', select[] specifies the eigenpairs for which condition numbers are required. To select condition numbers for the eigenpair corresponding to a real eigenvalue w[j], select[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 select[j] or select[j+1] or both, must be set to true. If howmny = 'A', select[] is not referenced. [in] n Order of the matrix T. (n >= 0) (If n = 0, returns without computation) [in] ldt Leading dimension of the two dimensional array t[][]. (ldt >= max(1, n)) [in] t[][] Array t[lt][ldt] (lt >= n) The upper quasi-triangular matrix T in Schur canonical form. [in] ldvl Leading dimension of the two dimensional array vl[][]. (ldvl >= 1 if job = 'V', ldvl >= n if job = 'E' or 'B') [in] vl[][] Array vl[lvl][ldvl] (lvl >= 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 select[]. The eigenvectors must be stored in consecutive rows of vl[][], as returned by dhsein or dtrevc. If job = 'V', vl[][] is not referenced. [in] ldvr Leading dimension of the two dimensional array vr[][]. (ldvr >= 1 if job = 'V', ldvr >= n if job = 'E' or 'B') [in] vr[][] Array vr[lvr][ldvr] (lvr >= 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 select[]. The eigenvectors must be stored in consecutive rows of vr[][], as returned by dhsein or dtrevc. If job = 'V', vr[][] is not referenced. [out] s[] Array s[ls] (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 rows 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] (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] work[] Array work[lwork] Work array. If job = 'E', work[] is not referenced. [in] lwork The dimension of the array work[]. (lwork >= n*(n + 6) if job = 'V' or 'B') [out] iwork[] Array iwork[liwork] Integer work array. If job = 'E', iwork[] is not referenced. [in] liwork The dimension of the array iwork[]. (liwork >= 2*(n - 1) if job = 'V' or 'B') [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') = -4: The argument n had an illegal value (n < 0) = -5: The argument ldt had an illegal value (ldt < max(1, n)) = -7: The argument ldvl had an illegal value (ldvl too small) = -9: The argument ldvr had an illegal value (ldvr too small) = -13: The argument mm had an illegal value (mm < m) = -16: The argument lwork had an illegal value (lwork too small) = -18: The argument liwork had an illegal value (liwork too small) 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