Dsbgv Dsbgvd Dsbgvx Dspgv Dspgvd Dspgvx Dsygv Dsygvd Dsygvx

Dspgvd() Sub Dspgvd ( IType As  Long, Jobz As  String, Uplo As  String, N As  Long, Ap() As  Double, Bp() As  Double, W() As  Double, Z() As  Double, Info As  Long  ) (Divide and conquer driver) Generalized eigenvalue problem of symmetric matrices in packed form Purpose This routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x = lambda*B*x, A*Bx = lambda*x, or B*A*x = lambda*x. Here A and B are assumed to be symmetric, stored in packed form, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. Parameters [in] Itype Specifies the problem type to be solved: = 1: A*x = lambda*B*x. = 2: A*B*x = lambda*x. = 3: B*A*x = lambda*x. [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,out] Ap() Array Ap(LAp - 1) (LAp >= N(N + 1)/2) [in] The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array Ap() as follows. Uplo = "U": Ap(i + j*(j + 1)/2) = Aij for 0 <= i <= j <= N - 1. Uplo = "L": Ap((i + j*(2*N - j - 1)/2) = Aij for 0 <= j < = i <= N - 1. [out] The contents of Ap() are destroyed. [in,out] Bp() Array Bp(LBp - 1) (LBp >= N(N + 1)/2) [in] The upper or lower triangle of the symmetric positive definite matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array Bp() as follows. Uplo = "U": Bp(i + j*(j + 1)/2) = Bij for 0 <= i <= j <= N - 1. Uplo = "L": Bp((i + j*(2*N - j - 1)/2) = Bij for 0 <= j < = i <= N - 1. [out] The triangular factor U or L from the Cholesky factorization B = U^T*U or B = L*L^T, in the same storage format as B. [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. The eigenvectors are normalized as follows:   Itype = 1 or 2: Z^T*B*Z = I   Itype = 3: Z^T*inv(B)*Z = I Jobz = "N": Z() is not referenced. [out] Info = 0: Successful exit = -1: The argument Itype had an illegal value (Itype < 1 or Itype > 3) = -2: The argument Jobz had an illegal value (Jobz <> "V" nor "N") = -3: The argument Uplo had an illegal value (Uplo <> "U" nor "L") = -4: The argument N had an illegal value (N < 0) = -5: The argument Ap() is invalid. = -6: The argument Bp() is invalid. = -7: The argument W() is invalid. = -8: The argument Z() is invalid. = i (0 < i <= N): Dspevd failed to converge. i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. = i (N < i <= 2n): The leading minor of order i-N of 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 symmetric-definite eigenproblem of the form Ax = λBx, where A is a symmetric matrix and B is a symmetric positive definite matrix. ( 0.54 -0.90 -0.94 ) ( 1.18 0.54 -1.22 ) A = ( -0.90 0.70 1.04 ) B = ( 0.54 0.60 -0.71 ) ( -0.94 1.04 1.65 ) ( -1.22 -0.71 1.66 ) Sub Ex_Dspgvd() Const N = 3 Dim Ap(N * (N + 1) / 2) As Double, Bp(N * (N + 1) / 2) As Double Dim W(N - 1) As Double, Z(N - 1, N - 1) As Double Dim Info As Long Ap(0) = 0.54 Ap(1) = -0.9: Ap(3) = 0.7 Ap(2) = -0.94: Ap(4) = 1.04: Ap(5) = 1.65 Bp(0) = 1.18 Bp(1) = 0.54: Bp(3) = 0.6 Bp(2) = -1.22: Bp(4) = -0.71: Bp(5) = 1.66 Call Dspgvd(1, "V", "L", N, Ap(), Bp(), W(), Z(), Info) Debug.Print "Eigenvalues =", W(0), W(1), W(2) Debug.Print "Eigenvectors =" Debug.Print Z(0, 0), Z(0, 1), Z(0, 2) Debug.Print Z(1, 0), Z(1, 1), Z(1, 2) Debug.Print Z(2, 0), Z(2, 1), Z(2, 2) Debug.Print "Info =", Info End Sub Example Results Eigenvalues = -0.297963342573455 0.510423243055614 7.37370278804149 Eigenvectors = 1.17970064313729 1.46384155786189 9.13803098094182E-02 0.497345303675336 -0.442269445774235 -1.71612038711754 0.524910948248221 1.35130854246605 -0.947378600715302 Info = 0