Zgebak Zgebal Zgehrd Zhsein Zhseqr Ztrevc3 Ztrexc Ztrsen Ztrsna Ztrsyl Zunghr Zunmhr

Zhseqr() Sub Zhseqr ( Job As  String, Compz As  String, N As  Long, Ilo As  Long, Ihi As  Long, H() As  Complex, W() As  Complex, Z() As  Complex, Info As  Long  ) Eigenvalues and Schur factorization of complex 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^H, where T is an upper quasi-triangular matrix (the Schur form), and Z is the unitary matrix of Schur vectors. Optionally Z may be postmultiplied into an input unitary 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 unitary matrix Q: A = Q*H*Q^H = (QZ)*T*(QZ)^H. 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 unitary 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 Zgebal, and then passed to Zgehrd when the matrix output by Zgebal 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 triangular matrix T from the Schur decomposition (the Schur form). 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] W() Array W(LW - 1) (LW >= N) The computed eigenvalues. If Job = "S", the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H(), with W(i) = H(i, 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 unitary 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 unitary matrix generated by Zunghr after the call to Zgehrd 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 W() is invalid. = -8: The argument Z() is invalid. = i > 0: Failed to compute all of the eigenvalues. Elements 1〜Ilo-1 and i+1〜N of W() 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 unitary 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 unitary matrix in (*) (regardless of the value of Job.) If Compz = "I", then (final value of Z()) = U where U is the unitary matrix in (*) (regardless of the value of Job.) If Compz = "N", then Z() is not accessed. Reference LAPACK Example Program Compute all eigenvalues of the general matrix A, where ( 0.20-0.11i -0.93-0.32i 0.81+0.37i ) A = ( -0.80-0.92i -0.29+0.86i 0.64+0.51i ) ( 0.71+0.59i -0.15+0.19i 0.20+0.94i ) Reduces to Hessenberg form by Zgehrd, then computes eigenvalues by Zhseqr. See examples of Ztrevc3 and Zhsein which also compute eigenvectors. Sub Ex_Zgehrd_Zhseqr() Const N = 3 Dim A(N - 1, N - 1) As Complex, Tau(N - 2) As Complex Dim W(N - 1) As Complex, Z() As Complex Dim Ilo As Long, Ihi As Long, Info As Long A(0, 0) = Cmplx(0.2, -0.11): A(0, 1) = Cmplx(-0.93, -0.32): A(0, 2) = Cmplx(0.81, 0.37) A(1, 0) = Cmplx(-0.8, -0.92): A(1, 1) = Cmplx(-0.29, 0.86): A(1, 2) = Cmplx(0.64, 0.51) A(2, 0) = Cmplx(0.71, 0.59): A(2, 1) = Cmplx(-0.15, 0.19): A(2, 2) = Cmplx(0.2, 0.94) Ilo = 1: Ihi = N Call Zgehrd(N, Ilo, Ihi, A(), Tau(), Info) If Info <> 0 Then Debug.Print "Error in Zgehrd: Info =", Info Exit Sub End If Call Zhseqr("E", "N", N, Ilo, Ihi, A(), W(), Z(), Info) If Info <> 0 Then Debug.Print "Error in Zhseqr: Info =", Info Exit Sub End If Debug.Print "Eigenvalues =" Debug.Print Creal(W(0)), Cimag(W(0)), Creal(W(1)), Cimag(W(1)) Debug.Print Creal(W(2)), Cimag(W(2)) End Sub Example Results Eigenvalues = -1.15894122423918 -0.50662892448174 1.05593587167591 0.900255855387815 0.21300535256327 1.29637306909393