dgees dgees_r dgeesx dgeesx_r dgeev dgeevx

dgeesx() void dgeesx ( char  jobvs, char  sort, int(*)(double, double)  select, char  sense, int  n, int  lda, double  a[], int *  sdim, double  wr[], double  wi[], int  ldvs, double  vs[], double *  rconde, double *  rcondv, double  work[], int  lwork, int  iwork[], int  liwork, int  bwork[], int *  info  ) (Expert driver) Schur factorization of a general matrix Purpose This routine computes for an n x n real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*Z^T. Optionally, it also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of Z form an orthonormal basis for this invariant subspace. For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8.1 of the LAPACK Users' Guide Third Edition (where these quantities are called s and sep respectively). A real matrix is in real Schur form if it is upper quasi-triangular with 1 x 1 and 2 x 2 blocks. 2 x 2 blocks will be standardized in the form [ a b ] [ c a ] where b*c < 0. The eigenvalues of such a block are a+-sqrt(bc). The matrix A is reduced to upper Hessenberg form first. Then the eigenvalues and the Schur factorization are computed by QR method. Parameters [in] jobvs = 'N': Schur vectors are not computed. = 'V': Schur vectors are computed. [in] sort Specifies whether or not to order the eigenvalues on the diagonal of the Schur form. = 'N': Eigenvalues are not ordered. = 'S': Eigenvalues are ordered (see select). [in] select sort = 'S': select is used to select eigenvalues to sort to the top left of the Schur form.   Eigenvalues wr[j]+-wi[j]*i are selected if select(wr[j], wi[j]) is true (= 1); i.e., if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected.   Note that a selected complex eigenvalue may no longer satisfy select(wr[j], wi[j]) = true after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case info is set to n+2 (see info below). sort = 'N': select is not referenced. [in] sense Determines which reciprocal condition numbers are computed. = 'N': None are computed. = 'E': Computed for average of selected eigenvalues only. = 'V': Computed for selected right invariant subspace only. = 'B': Computed for both. If sense = 'E', 'V' or 'B', sort must equal 'S'. [in] n Order of the matrix A. (n >= 0) (If n = 0, returns without computation) [in] lda Leading dimension of the two dimensional array a[][]. (lda >= max(1, n)) [in,out] a[][] Array a[la][lda] (la >= n) [in] n x n matrix A. [out] a[][] is overwritten by its real Schur form T. [in] sdim sort = 'N': sdim = 0. sort = 'S': sdim = number of eigenvalues (after sorting) for which select is true. (Complex conjugate pairs for which select is true for either eigenvalue count as 2.) [out] wr[] Array wr[lwr] (lwr >= n) [out] wi[] Array wi[lwi] (lwi >= n) wr[] and wi[] contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form T. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first. [in] ldvs Leading dimension of the two dimensional array vs[][]. (ldvs >= 1 if jobvs = 'N', ldvs >= n if jobvs = 'V') [out] vs[][] Array vs[lvs][ldvs] (lvs >= n) jobvs = 'V': vs[][] contains the orthogonal matrix Z of Schur vectors. jobvs = 'N': vs[][] is not referenced. [out] rconde sense = 'E' or 'B': rconde contains the reciprocal condition number for the average of the selected eigenvalues. sense = 'N' or 'V': Not referenced. [out] rcondv sense = 'V' or 'B': rcondv contains the reciprocal condition number for the selected right invariant subspace. sense = 'N' or 'E': Not referenced. [out] work[] Array work[lwork] Work array. On exit, if info = 0, work[0] returns the optimal lwork. [in] lwork The dimension of the array work[]. (lwork >= max(1, 3*n) if sense = 'N', lwork >= max(1, n+2*sdim*(n-sdim)) if sense = 'E', 'V' or 'B' where sdim is the number of selected eigenvalues computed by this routine. Note that n+2*sdim*(n-sdim) <= n+n*n/2). Note also that an error is only returned if lwork < max(1, 3*n), but if sense = 'E' or 'V' or 'B', this may not be large enough.) For good performance, lwork must generally be larger. If lwork = -1, then a workspace query is assumed. The routine only calculates upper bounds on the optimal size of the arrays work[] and iwork[], and returns these values in work[0] and iwork[0]. [out] iwork[] Array iwork[liwork] Integer work array. On exit, if info = 0, iwork[0] returns the optimal liwork. [in] liwork The dimension of the array iwork[]. (liwork >= 1 if sense = 'N' or 'E', liwork >= sdim*(n-sdim) if sense = 'V' or 'B'. Note that sdim*(n-sdim) <= n*n/4). Note also that an error is only returned if liwork < 1, but if sense = 'V' or 'B' this may not be large enough.) If lwork = -1, then a workspace query is assumed. The routine only calculates upper bounds on the optimal size of the arrays work[] and iwork[], and returns these values in work[0] and iwork[0]. [out] bwork[] Array bwork[lbwork] (lbwork >= n) Logical type (integer) work array. Not referenced if sort = 'N'. [out] info = 0: Successful exit = -1: The argument jobvs had an illegal value (jobvs != 'V' nor 'N') = -2: The argument sort had an illegal value (sort != 'S' nor 'N') = -4: The argument sense had an illegal value (sense != 'E', 'V', 'B' nor 'N') = -5: The argument n had an illegal value (n < 0) = -6: The argument lda had an illegal value (lda < max(1, n)) = -11: The argument ldvs had an illegal value (ldvs too small) = -16: The argument lwork had an illegal value (lwork too small) = -18: The argument liwork had an illegal value (liwork too small) = i (0 < i <= n): The QR algorithm failed to compute all the eigenvalues. Elements 0 to ilo-2 and i to n-1 of wr[] and wi[] contain those eigenvalues which have converged. If jobvs = 'V', vs[][] contains the matrix which reduces A to its partially converged Schur form. = n+1: The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned) = n+2: After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy select=true. This could also be caused by underflow due to scaling Reference LAPACK