Zhbgv Zhbgvd Zhbgvx Zhegv Zhegvd Zhegvx Zhpgv Zhpgvd Zhpgvx

Zhbgvd() Sub Zhbgvd ( Jobz As  String, Uplo As  String, N As  Long, Ka As  Long, Kb As  Long, Ab() As  Complex, Bb() As  Complex, W() As  Double, Z() As  Complex, Info As  Long  ) (Divide and conquer driver) Generalized eigenvalue problem of Hermitian band matrices Purpose This routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized Hermitian-definite banded eigenproblem, of the form A*x = lambda*B*x. Here A and B are assumed to be Hermitian and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. Parameters [in] Jobz = "N": Compute eigenvalues only. = "V": Compute eigenvalues and eigenvectors. [in] Uplo = "U": Upper triangles of A and B are stored. = "L": Lower triangles of A and B are stored. [in] N Order of the matrices A and B. (N >= 0) (If N = 0, returns without computation) [in] Ka Number of super-diagonals of the matrix A if Uplo = "U" or the number of sub-diagonals if Uplo = "L". (Ka >= 0) [in] Kb Number of super-diagonals of the matrix B if Uplo = "U" or the number of sub-diagonals if Uplo = "L". (Kb >= 0) [in,out] Ab() Array Ab(LAb1 - 1, LAb2 - 1) (LAb1 >= Ka + 1, LAb2 >= N) [in] The upper or lower triangle of the Hermitian band matrix A, stored in the first Ka+1 columns of the array. The j-th column of A is stored in the j-th row of the array ab as follows. Uplo = "U": Ab(Ka + i - j, j) = Aij for max(0, j - Ka - 1) <= i <= j <= N - 1. Uplo = "L": Ab(i - j, j) = Aij for 0 <= j <= i <= min(N - 1, j + Ka - 1). [out] The contents of Ab() are destroyed. [in,out] Bb() Array Bb(LBb1 - 1, LBb2 - 1) (LBb1 >= Kb + 1, LBb2 >= N) [in] The upper or lower triangle of the Hermitian positive definite band matrix B, stored in the first Kb+1 columns of the array. The j-th column of B is stored in the j-th row of the array bb as follows. Uplo = "U": Bb(Kb + i - j, j) = Bij for max(0, j - Kb - 1) <= i <= j <= N - 1. Uplo = "L": Bb(i - j, j) = Bij for 0 <= j <= i <= min(N - 1, j + Kb - 1). [out] The factor S from the split Cholesky factorization B = S^H*S, as returned by Dpbstf. [out] W() Array W(LW - 1) (LW >= N) If Info = 0, the eigenvalues in ascending order. [out] Z() Array Z(LZ1 - 1, LZ2 - 1) (LZ1 >= N, LZ2 >= N) Jobz = "V": If Info = 0, Z() contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z^H*B*Z = I. Jobz = "N": Z() is not referenced. [out] Info = 0: Successful exit = -1: The argument Jobz had an illegal value (Jobz <> "V" nor "N") = -2: The argument Uplo had an illegal value (Uplo <> "U" nor "L") = -3: The argument N had an illegal value (N < 0) = -4: The argument Ka had an illegal value (Ka < 0) = -5: The argument Kb had an illegal value (Kb < 0 or Kb > Ka) = -6: The argument Ab() is invalid. = -7: The argument Bb() is invalid. = -8: The argument W() is invalid. = -9: The argument Z() is invalid. = i (0 < i <= N): The algorithm failed to converge. i off-diagonal elements of an intermediate tridiagonal form did not converge to zero = i (i > N): Zpbstf returned Info = i-N. B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed. Reference LAPACK Example Program Compute the eigenvalues and the eigenvectors of a generalized Hermitian-definite eigenproblem of the form Ax = λBx, where A is an Hermitian band matrix and B is an Hermitian positive definite band matrix. ( 0.20 -0.11+0.93i 0 ) A = ( -0.11-0.93i -0.32 -0.80+0.92i ) ( 0 -0.80-0.92i -0.29 ) ( 2.20 -0.11+0.93i 0 ) B = ( -0.11-0.93i 2.32 -0.80+0.92i ) ( 0 -0.80-0.92i 2.29 ) Sub Ex_Zhbgvd() Const N = 3, Ka = 1, Kb = 1 Dim AB(Ka, N - 1) As Complex, Bb(Kb, N - 1) As Complex Dim W(N - 1) As Double, Z(N - 1, N - 1) As Complex, Info As Long AB(0, 0) = Cmplx(0.2): AB(0, 1) = Cmplx(-0.32): AB(0, 2) = Cmplx(-0.29) AB(1, 0) = Cmplx(-0.11, -0.93): AB(1, 1) = Cmplx(-0.8, -0.92) Bb(0, 0) = Cmplx(2.2): Bb(0, 1) = Cmplx(2.32): Bb(0, 2) = Cmplx(2.29) Bb(1, 0) = Cmplx(-0.11, -0.93): Bb(1, 1) = Cmplx(-0.8, -0.92) Call Zhbgvd("V", "L", N, Ka, Kb, AB(), Bb(), W(), Z(), Info) 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 "Info =", Info End Sub Example Results Eigenvalues = -2.33466883563969 4.53015245998351E-03 0.35977634455495 Eigenvectors = 0.47826114359381 -6.39634991743266E-18 0.507837765924616 -8.10231908112231E-19 9.59938458849389E-02 0.811584333390847 1.21596344669029E-02 0.102804182311088 -0.441771484124975 0.486432071889564 0.281208537371776 -0.309637123223847 0.332163577636529 2.56485488839366E-18 -3.84923101537441E-02 -0.325434985845291 -0.154385502431493 0.169992999811227 Info = 0