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

dhseqr() void dhseqr ( char  job, char  compz, int  n, int  ilo, int  ihi, int  ldh, double  h[], double  wr[], double  wi[], int  ldz, double  z[], double  work[], int  lwork, int *  info  ) Eigenvalues and Schur factorization of Hessenberg matrix by QR method Purpose This routine computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z^T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q^T = (QZ)*T*(QZ)^T. Parameters [in] job = 'E': Compute eigenvalues only. = 'S': Compute eigenvalues and the Schur form T. [in] compz = 'N': No Schur vectors are computed. = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned. = 'V': Z must contain an orthogonal matrix Q on entry, and the product Q*Z is returned. [in] n Order of the matrix H. (n >= 0) (If n = 0, returns without computation) [in] ilo [in] ihi It is assumed that H is already upper triangular in rows and columns 1〜ilo-1 and ihi+1〜n. ilo and ihi are normally set by a previous call to dgebal, and then passed to dgehrd when the matrix output by dgebal is reduced to Hessenberg form. Otherwise they should be set to 1 and n respectively. (1 <= ilo <= ihi <= n, if n > 0. ilo = 1 and ihi = 0, if n = 0) [in] ldh Leading dimension of the two dimensional array h[][]. (ldh >= max(1, n)) [in,out] h[][] Array h[lh][ldh] (lh >= n) [in] The upper Hessenberg matrix H. [out] If info = 0 and job = 'S', then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form). 2 x 2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i, i) = H(i+1, i+1) and H(i+1, i)*H(i, i+1) < 0. If info = 0 and job = 'E', the contents of H are unspecified on exit. (The output value of H when info > 0 is given under the description of info below.) [out] wr[] Array wr[lwr] (lwr >= n) [out] wi[] Array wi[lwi] (lwi >= n) The real and imaginary parts, respectively, of the computed eigenvalues. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of wr[] and wi[], say the i-th and (i+1)th, with wi[i] > 0 and wi[i+1] < 0. If job = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in h[][], with wr[i] = h[i][i] and, if h[i〜i+1][i〜i+1] is a 2 x 2 diagonal block, wi[i] = sqrt(-h[i][i+1]*h[i+1][i]) and wi[i+1] = -wi[i]. [in] ldz Leading dimension of the two dimensional array z[][]. (ldz >= max(1, n) if compz = 'I' or 'V', ldz >= 1 if compz = 'N') [in,out] z[][] Array z[lz][ldz] (lz >= n) compz = 'I': [in] z[][] need not be set. [out] if info = 0, z[][] contains the orthogonal matrix Z of the Schur vectors of H. compz = 'V': [in] z[][] must contain an n x n matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ilo〜ihi, ilo〜ihi). [out] If info = 0, z[][] contains Q*Z. Normally Q is the orthogonal matrix generated by dorghr after the call to dgehrd which formed the Hessenberg matrix H. (The output value of z[][] when info > 0 is given under the description of info below.) compz = 'N': z[][] is 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, n) is sufficient and delivers very good and sometimes optimal performance. However, lwork as large as 11*n may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size. If lwork = -1, then a workspace query is assumed. The routine only calculates the optimal size of the work[] array, and returns the value in work[0]. [out] info = 0: Successful exit = -1: The argument job had an illegal value. (job != 'E' nor 'S') = -2: The argument compz had an illegal value. (compz != 'N', 'I' nor 'V') = -3: The argument n had an illegal value (n < 0) = -4: The argument ilo had an illegal value (ilo < 1 or ilo > n) = -5: The argument ihi had an illegal value (ihi < min(ilo, n) or ihi > n) = -6: The argument ldh had an illegal value (ldh < max(1, n)) = -10: The argument ldz had an illegal value (ldz < 1 or (compz = 'I' or 'V' and ldz < max(1, n))) = -13: The argument lwork had an illegal value (lwork too small) = i > 0: Failed to compute all of the eigenvalues. Elements 1〜ilo-1 and i+1〜n of wr[] and wi[] contain those eigenvalues which have been successfully computed. (Failures are rare.) If job = 'E', the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ilo through i of the final output value of h[][]. If job = 'S', then (initial value of h[][]) * U = U * (final value of h[][]) --- (*) where U is an orthogonal matrix. The final value of h[][] is upper Hessenberg and quasi-triangular in rows and columns i+1 through ihi. If compz = 'V', then (final value of z[][]) = (initial value of z[][]) * U where U is the orthogonal matrix in (*) (regardless of the value of job.) If compz = 'I', then (final value of z[][]) = U where U is the orthogonal matrix in (*) (regardless of the value of job.) If compz = 'N', then z[][] is not accessed. Reference LAPACK