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

Zhsein() Sub Zhsein ( Side As  String, Eigsrc As  String, Initv As  String, Selct() As  Boolean, N As  Long, H() As  Complex, W() As  Complex, Vl() As  Complex, Vr() As  Complex, Mm As  Long, M As  Long, Ifaill() As  Long, Ifailr() As  Long, Info As  Long  ) Eigenvectors of complex Hessenberg matrix by inverse iteration method Purpose This routine uses inverse iteration to find specified right and/or left eigenvectors of a complex upper Hessenberg matrix H. The right eigenvector x and the left eigenvector y of the matrix H corresponding to an eigenvalue w are defined by: H * x = w * x, y^H * H = w * y^H where y^H denotes the conjugate transpose of the vector y. Parameters [in] Side = "R": Compute right eigenvectors only. = "L": Compute left eigenvectors only. = "B": Compute both right and left eigenvectors. [in] Eigsrc Specifies the source of eigenvalues supplied in (Wr(), Wi()). = "Q": The eigenvalues were found using Zhseqr; thus, if H has zero subdiagonal elements, and so is block-triangular, then the j-th eigenvalue can be assumed to be an eigenvalue of the block containing the j-th row/column. This property allows this routine to perform inverse iteration on just one diagonal block. = "N": No assumptions are made on the correspondence between eigenvalues and diagonal blocks. In this case, this routine must always perform inverse iteration using the whole matrix H. [in] Initv = "N": No initial vectors are supplied; = "U": User-supplied initial vectors are stored in the arrays vl and/or vr. [in] Selct() Array Selct(LSelct - 1) (LSelct >= N) Specifies the eigenvectors to be computed. To select the eigenvector corresponding to the eigenvalue W(j), Selct(j) must be set to True. [in] N Order of the matrix H. (N >= 0) (If N = 0, returns without computation) [in] H() Array H(LH1 - 1, LH2 - 1) (LH1 >= N, LH2 >= N) The upper Hessenberg matrix H. If a NaN is detected in H(), the routine will return with Info = -7. [in] W() Array W(LW - 1) (LW >= N) [in] The eigenvalues of H. [out] The real part of W() may have been altered since close eigenvalues are perturbed slightly in searching for independent eigenvectors. [in,out] Vl() Array Vl(LVl1 - 1, LVl2 - 1) (LVl1 >= N, LVl2 >= MM) [in] If Initv = "U" and Side = "L" or "B", Vl() must contain starting vectors for the inverse iteration for the left eigenvectors. The starting vector for each eigenvector must be in the same column(s) in which the eigenvector will be stored. [out] If Side = "L" or "B", the left eigenvectors specified by Selct() will be stored consecutively in the columns of Vl(), in the same order as their eigenvalues. If Side = "R", Vl() is not referenced. [in,out] Vr() Array Vr(LVr1 - 1, LVr2 - 1) (LVr1 >= N, LVr2 >= MM) [in] If Initv = "U" and Side = "R" or "B", Vr() must contain starting vectors for the inverse iteration for the right eigenvectors. The starting vector for each eigenvector must be in the same column(s) in which the eigenvector will be stored. [out] If Side = "R" or "B", the right eigenvectors specified by Selct() will be stored consecutively in the columns of Vr(), in the same order as their eigenvalues. If Side = "L", Vr() is not referenced. [in] MM The number of columns in the arrays Vl() and/or Vr(). (MM >= M) [out] M The number of columns in the arrays Vl() and/or Vr() required to store the eigenvectors (= the number of True elements in Selct()). [out] Ifaill() Array Ifaill(LIfaill - 1) (LIfaill >= MM) If Side = "L" or "B", Ifaill(i) = j > 0 if the left eigenvector in the i-th column of Vl() (corresponding to the eigenvalue W(j)) failed to converge. Ifaill(i) = 0 if the eigenvector converged satisfactorily. If Side = "R", Ifaill() is not referenced. [out] Ifailr() Array Ifailr(LIfailr - 1) (LIfailr >= MM) If Side = "R" or "B", Ifailr(i) = j > 0 if the right eigenvector in the i-th column of Vr() (corresponding to the eigenvalue W(j)) failed to converge. Ifailr(i) = 0 if the eigenvector converged satisfactorily. If Side = "L", Ifailr() is not referenced. [out] Info = 0: Successful exit. = -1: The argument Side had an illegal value. (Side <> "R", "L" nor "B") = -2: The argument Eigsrc had an illegal value. (Eigsrc <> "Q" nor "N") = -3: The argument Initv had an illegal value. (Initv <> "N" nor "U") = -4: The argument Selct() is invalid. = -5: The argument N had an illegal value. (N < 0) = -6: The argument H() is invalid. = -7: The argument W() is invalid. = -8: The argument Vl() is invalid. = -9: The argument Vr() is invalid. = -10: The argument MM had an illegal value. (MM < M) = -12: The argument Ifaill() is invalid. = -13: The argument Ifailr() is invalid. = i > 0: i is the number of eigenvectors which failed to converge. See Ifaill() and Ifailr() for further details. Reference LAPACK Example Program Compute all eigenvalues and eigenvectors 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 and those eigenvectors by Zhsein and Zunmhr. Sub Ex_Zgehrd_Zhseqr_Zhsein() Const N = 3 Dim A(N - 1, N - 1) As Complex, Tau(N - 2) As Complex, H(N - 1, N - 1) As Complex Dim W(N - 1) As Complex, Z() As Complex, Selct(N - 1) As Boolean Dim Vl(N - 1, N - 1) As Complex, Vr(N - 1, N - 1) As Complex, M As Long Dim Ilo As Long, Ihi As Long, Ifaill(N - 1) As Long, Ifailr(N - 1) As Long Dim I As Long, J 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 For I = 0 To N - 1 For J = 0 To N - 1 H(I, J) = A(I, J) Next Next Call Zhseqr("E", "N", N, Ilo, Ihi, H(), W(), Z(), Info) If Info <> 0 Then Debug.Print "Error in Zhseqr: Info =", Info Exit Sub End If For I = 0 To N - 1 Selct(I) = True Next Call Zhsein("B", "Q", "N", Selct(), N, A(), W(), Vl(), Vr(), N, M, Ifaill(), Ifailr(), Info) If Info <> 0 Then Debug.Print "Error in Zhsein: Info =", Info Exit Sub End If Call Zunmhr("L", "N", N, N, Ilo, Ihi, A(), Tau(), Vr(), Info) If Info <> 0 Then Debug.Print "Error in Zunmhr: Info =", Info Exit Sub End If Call Zunmhr("L", "N", N, N, Ilo, Ihi, A(), Tau(), Vl(), Info) If Info <> 0 Then Debug.Print "Error in Zunmhr: 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)) Debug.Print "Eigenvectors (L) =" Debug.Print Creal(Vl(0, 0)), Cimag(Vl(0, 0)), Creal(Vl(0, 1)), Cimag(Vl(0, 1)) Debug.Print Creal(Vl(1, 0)), Cimag(Vl(1, 0)), Creal(Vl(1, 1)), Cimag(Vl(1, 1)) Debug.Print Creal(Vl(2, 0)), Cimag(Vl(2, 0)), Creal(Vl(2, 1)), Cimag(Vl(2, 1)) Debug.Print Creal(Vl(0, 2)), Cimag(Vl(0, 2)) Debug.Print Creal(Vl(1, 2)), Cimag(Vl(1, 2)) Debug.Print Creal(Vl(2, 2)), Cimag(Vl(2, 2)) Debug.Print "Eigenvectors (R) =" Debug.Print Creal(Vr(0, 0)), Cimag(Vr(0, 0)), Creal(Vr(0, 1)), Cimag(Vr(0, 1)) Debug.Print Creal(Vr(1, 0)), Cimag(Vr(1, 0)), Creal(Vr(1, 1)), Cimag(Vr(1, 1)) Debug.Print Creal(Vr(2, 0)), Cimag(Vr(2, 0)), Creal(Vr(2, 1)), Cimag(Vr(2, 1)) Debug.Print Creal(Vr(0, 2)), Cimag(Vr(0, 2)) Debug.Print Creal(Vr(1, 2)), Cimag(Vr(1, 2)) Debug.Print Creal(Vr(2, 2)), Cimag(Vr(2, 2)) Debug.Print "M =", M End Sub Example Results Eigenvalues = -1.15894122423918 -0.50662892448174 1.05593587167591 0.900255855387815 0.21300535256327 1.29637306909393 Eigenvectors (L) = 0 -0.871083495245762 0.2 -0.8 0.267351466048342 -0.480647278057965 -0.03692440974696 0.616773162002434 -0.285413896331544 0.542078734362481 0.166244018883372 -0.368066071419648 -2.22091099794888E-02 0.344241204682076 7.27399270176405E-02 -0.5924933721717 0.347217078108291 -0.4230226508624 Eigenvectors (R) = 0.690958738203183 -0.309041261796817 -0.507806593830091 0.492193406169909 0.634809967097297 -0.421369311149223 0.229049609768182 -0.190619676572438 -0.363524659234702 0.113190794408639 -0.763294421913407 0.107049283139747 -0.111157316336775 -2.52863433046769E-02 -0.481933704163733 0.142931109108698 -0.552705878022003 3.27787741584283E-02 M = 3