Zhbtrd Zhetrd Zhptrd Zpteqr Zstedc Zstein Zstemr Zsteqr Zungtr Zunmtr Zupgtr Zupmtr

Zstein() Sub Zstein ( N As  Long, D() As  Double, E() As  Double, M As  Long, W() As  Double, Iblock() As  Long, Isplit() As  Long, Z() As  Complex, Ifail() As  Long, Info As  Long  ) Eigenvectors of a symmetric tridiagonal matrix to which a Hermitian matrix was reduced (Inverse iteration method) Purpose This routine computes the eigenvectors of a real symmetric tridiagonal matrix T to which a Hermitian matrix was reduced, corresponding to specified eigenvalues, using inverse iteration. The maximum number of iterations allowed for each eigenvector is specified by an internal parameter Maxits (currently set to 5). Parameters [in] N Order of the tridiagonal matrix. (N >= 0) (If N = 0, returns without computation) [in] D() Array D(LD - 1) (LD >= N) The N diagonal elements of the tridiagonal matrix T. [in] E() Array E(LE - 1) (LE >= N - 1) The (N - 1) off-diagonal elements of the tridiagonal matrix T. [in] M The number of eigenvectors to be found. (0 <= M <= N) [in] W() Array W(LW - 1) (LW >= N) The first M elements of W() contain the eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block. (The output array w[] from Dstebz with Order = "B" is expected here.) [in] Iblock() Array Iblock(LIblock - 1) (LIblock >= N) The submatrix indices associated with the corresponding eigenvalues in W(). Iblock(i) = 1 if eigenvalue w(i) belongs to the first submatrix from the top, = 2 if w(i) belongs to the second submatrix, etc. (The output array Iblock() from Dstebz is expected here.) [in] Isplit() Array Isplit(LIsplit - 1) (LIsplit >= N) The splitting points, at which T breaks up into submatrices. The first submatrix consists of rows/columns 1 to Isplit(0), the second of rows/columns Isplit(0) + 1 through Isplit(1), etc. (The output array Isplit() from Dstebz is expected here.) [out] Z() Array Z(LZ1 - 1, LZ2 - 1) (LZ1 >= N, LZ2 >= M) The computed eigenvectors. The eigenvector associated with the eigenvalue W(i) is stored in the i-th column of Z(). Any vector which fails to converge is set to its current iterate after Maxits iterations. [out] Ifail() Array Ifail(LIfail - 1) (LIfail >= M) On normal exit, all elements of Ifail() are zero. If one or more eigenvectors fail to converge after Maxits iterations, then their indices are stored in array Ifail(). [out] Info = 0: Successful exit. = -1: The argument N had an illegal value. (N < 0) = -2: The argument D() is invalid. = -3: The argument E() is invalid. = -4: The argument M had an illegal value. (M < 0 or M > N) = -5: The argument W() is invalid. = -6: The argument Iblock() is invalid. = -7: The argument Isplit() is invalid. = -8: The argument Z() is invalid. = -9: The argument Ifail() is invalid. = i > 0: Eigenvectors failed to converge in Maxits iterations. Their indices are stored in array ifail(). Reference LAPACK Example Program Compute all eigenvalues and eigenvectors of the Hermitian matrix A, where ( 0.20 -0.11+0.93i 0.81-0.37i ) A = ( -0.11-0.93i -0.32 -0.80+0.92i ) ( 0.81+0.37i -0.80-0.92i -0.29 ) Reduces to real tridiagonal form by Zhetrd, then computes eigenvalues by Dstebz and those eigenvectors by Zstein and Zunmtr. Sub Ex_Zhetrd_Dstebz_Zstein() Const N = 3 Dim A(N - 1, N - 1) As Complex, W(N - 1) As Double, Z(N - 1, N - 1) As Complex Dim D(N - 1) As Double, E(N - 2) As Double, Tau(N - 2) As Complex Dim Vl As Double, Vu As Double, Il As Long, Iu As Long, Abstol As Double Dim Iblock(N - 1) As Long, Isplit(N - 1) As Long, Ifail(N - 1) As Long Dim M As Long, Nsplit As Long, Info As Long A(0, 0) = Cmplx(0.2, 0) A(1, 0) = Cmplx(-0.11, -0.93): A(1, 1) = Cmplx(-0.32, 0) A(2, 0) = Cmplx(0.81, 0.37): A(2, 1) = Cmplx(-0.8, -0.92): A(2, 2) = Cmplx(-0.29, 0) Call Zhetrd("L", N, A(), D(), E(), Tau(), Info) If Info <> 0 Then Debug.Print "Error in Zhetrd: Info =", Info Exit Sub End If Abstol = 0 Call Dstebz("A", "E", N, Vl, Vu, Il, Iu, Abstol, D(), E(), M, Nsplit, W(), Iblock(), Isplit(), Info) If Info <> 0 Then Debug.Print "Error in Dstebz: Info =", Info Exit Sub End If Call Zstein(N, D(), E(), M, W(), Iblock(), Isplit(), Z(), Ifail(), Info) If Info <> 0 Then Debug.Print "Error in Zstein: Info =", Info Exit Sub End If Call Zunmtr("L", "L", "N", N, N, A(), Tau(), Z(), Info) If Info <> 0 Then Debug.Print "Error in Zunmtr: Info =", Info Exit Sub End If Debug.Print "Eigenvalues =", W(0), W(1), W(2) Debug.Print "Eigenvectors =" Debug.Print Creal(Z(0, 0)), Cimag(Z(0, 0)), Creal(Z(0, 1)), Cimag(Z(0, 1)) Debug.Print Creal(Z(1, 0)), Cimag(Z(1, 0)), Creal(Z(1, 1)), Cimag(Z(1, 1)) Debug.Print Creal(Z(2, 0)), Cimag(Z(2, 0)), Creal(Z(2, 1)), Cimag(Z(2, 1)) Debug.Print Creal(Z(0, 2)), Cimag(Z(0, 2)) Debug.Print Creal(Z(1, 2)), Cimag(Z(1, 2)) Debug.Print Creal(Z(2, 2)), Cimag(Z(2, 2)) Debug.Print "M =", M, "Nsplit =", Nsplit End Sub Example Results Eigenvalues = -2.05348849668514 0.124622388617308 1.51886610806783 Eigenvectors = -0.449276526719113 0 -0.654793596518192 0 0.227247885813611 -0.597641779578735 -0.519997178670921 3.19846835072554E-02 0.621236109316912 -5.83009495222983E-02 -0.204907317474214 0.507777757881846 0.607779522934083 0 -0.392237107311198 -0.407323787101333 0.23846608290599 0.503959683819116 M = 3 Nsplit = 1