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

Dhseqr() Sub Dhseqr ( Job As  String, Compz As  String, N As  Long, Ilo As  Long, Ihi As  Long, H() As  Double, Wr() As  Double, Wi() As  Double, Z() As  Double, Info As  Long  ) 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,out] H() Array H(LH1 - 1, LH2 - 1) (LH1 >= N, LH2 >= 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 - 1) (LWr >= N) [out] Wi() Array Wi(LWi - 1) (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+1, i)*H(i, i+1)) and Wi(i+1) = -Wi(i). [in,out] Z() Array Z(LZ1 - 1, LZ2 - 1) (LZ1 >= N, LZ2 >= 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] 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 H() is invalid. = -7: The argument Wr() is invalid. = -8: The argument Wi() is invalid. = -9: The argument Z() is invalid. = 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 Example Program Compute all eigenvalues and of the general matrix A, where ( 0.20 -0.11 -0.93 ) A = ( -0.32 0.81 0.37 ) ( -0.80 -0.92 -0.29 ) See examples of Dtrevc3 and Dhsein which also compute eigenvectors. Sub Ex_Dgehrd_Dhseqr() Const N = 3 Dim A(N - 1, N - 1) As Double, Tau(N - 2) As Double Dim Wr(N - 1) As Double, Wi(N - 1) As Double, Z() As Double Dim Ilo As Long, Ihi As Long, Info As Long A(0, 0) = 0.2: A(0, 1) = -0.11: A(0, 2) = -0.93 A(1, 0) = -0.32: A(1, 1) = 0.81: A(1, 2) = 0.37 A(2, 0) = -0.8: A(2, 1) = -0.92: A(2, 2) = -0.29 Ilo = 1: Ihi = N Call Dgehrd(N, Ilo, Ihi, A(), Tau(), Info) If Info <> 0 Then Debug.Print "Error in Dgehrd: Info =", Info Exit Sub End If Call Dhseqr("E", "N", N, Ilo, Ihi, A(), Wr(), Wi(), Z(), Info) If Info <> 0 Then Debug.Print "Error in Dhseqr: Info =", Info Exit Sub End If Debug.Print "Eigenvalues (r) =", Wr(0), Wr(1), Wr(2) Debug.Print "Eigenvalues (i) =", Wi(0), Wi(1), Wi(2) End Sub Example Results Eigenvalues (r) = -0.904130023851345 0.812065011925673 0.812065011925673 Eigenvalues (i) = 0 0.48915757543818 -0.48915757543818