Dsbgv Dsbgvd Dsbgvx Dspgv Dspgvd Dspgvx Dsygv Dsygvd Dsygvx

Dspgvx() Sub Dspgvx ( IType As  Long, Jobz As  String, Range As  String, Uplo As  String, N As  Long, Ap() As  Double, Bp() As  Double, Vl As  Double, Vu As  Double, Il As  Long, Iu As  Long, AbsTol As  Double, M As  Long, W() As  Double, Z() As  Double, IFail() As  Long, Info As  Long  ) (Expert driver) Generalized eigenvalue problem of symmetric matrices in packed form (expert driver) Purpose This routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form Ax = λBx, ABx = λx or BAx = λx. Here A and B are assumed to be symmetric, stored in packed form, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Parameters [in] IType Specifies the problem type to be solved: = 1: Ax = λBx. = 2: ABx = λx. = 3: BAx = λx. [in] Jobz = "N": Compute eigenvalues only. = "V": Compute eigenvalues and eigenvectors. [in] Range = "A": All eigenvalues will be found. = "V": All eigenvalues in the half-open interval (Vl, Vu] will be found. = "I": The Il-th through Iu-th eigenvalues will be found. [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] N x N symmetric matrix A in packed form. The upper or lower triangular part is to be stored in accordance with Uplo. [out] The contents of Ap() are destroyed. [in,out] Bp() Array Bp(LBp - 1) (LBp >= N(N + 1)/2) [in] N x N symmetric positive definite matrix B in packed form. The upper or lower triangular part is to be stored in accordance with Uplo. [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. [in] Vl Range = "V": The lower bound of the interval to be searched for eigenvalues. (Vl < Vu) Range = "A" or "I": Not referenced. [in] Vu Range = "V": The upper bound of the interval to be searched for eigenvalues. (Vl < Vu) Range = "A" or "I": Not referenced. [in] Il Range = "I": The index of the smallest eigenvalue to be returned. (1 <= Il <= Iu <= N, if N > 0; Il = 1 and Iu = 0 if N = 0) Range = "A" or "V": Not referenced. [in] Iu Range = "I": The index of the largest eigenvalues to be returned. (1 <= Il <= Iu <= N, if N > 0; Il = 1 and Iu = 0 if N = 0) Range = "A" or "V": Not referenced. [in] AbsTol The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a, b] of width less than or equal to AbsTol + eps * max(|a|, |b|), where eps is the machine precision. If AbsTol is less than or equal to zero, then eps*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2*Dlamch("S"), not zero. If this routine returns with Info > 0, indicating that some eigenvectors did not converge, try setting abstol to 2*Dlamch("S"). [out] M The total number of eigenvalues found. (0 <= M <= N) If Range = "A", M = N, and if Range = "I", M = Iu - Il + 1. [out] W() Array W(LW - 1) (LW >= N) On normal exit, the first M elements contain the selected eigenvalues in ascending order. [out] Z() Array Z(LZ1 - 1, LZ2 - 1) (LZ1 >= N, LZ2 >= M) Jobz = "V": If Info = 0, the first M columns of Z() contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z() holding the eigenvector associated with W(i).   The eigenvectors are normalized as follows:     IType = 1 or 2: Z^T*B*Z = I     IType = 3: Z^T*inv(B)*Z = I   If an eigenvector fails to converge, then that column of Z() contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFail(). Jobz = "N": Z() is not referenced. Note: The user must ensure that at least max(1, M) columns are supplied in the array Z(); if Range = "V", the exact value of M is not known in advance and an upper bound must be used. [out] IFail() Array IFail(LIFail - 1) (LIFail >= N) Jobz = "V": If Info = 0, the first M elements of IFail() are zero. If Info > 0, then IFail() contains the indices of the eigenvectors that failed to converge. Jobz = "N": IFail() 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 Range had an illegal value. (Range <> "A", "V" nor "I") = -4: The argument Uplo had an illegal value. (Uplo <> "U" nor "L") = -5: The argument N had an illegal value. (N < 0) = -6: The argument Ap() is invalid. = -7: The argument Bp() is invalid. = -9: The argument Vu had an illegal value. (Vu <= Vl) = -10: The argument Il had an illegal value. (Il < 1 or Il > N) = -11: The argument Iu had an illegal value. (Iu < min(N, Il) or Iu > N) = -14: The argument W() is invalid. = -15: The argument Z() is invalid. = -16: The argument IFail() is invalid. = i (0 < i <= N): Dsyevx failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFail(). = i (i > N): 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_Dspgvx() Const N = 3 Dim Ap(N * (N + 1) / 2) As Double, Bp(N * (N + 1) / 2) As Double Dim W(N - 1) As Double Dim Vl As Double, Vu As Double, Il As Long, Iu As Long, AbsTol As Double Dim M As Long, Z(N - 1, N - 1) As Double, IFail(N - 1) As Long, 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 AbsTol = 0 Call Dspgvx(1, "V", "A", "L", N, Ap(), Bp(), Vl, Vu, Il, Iu, AbsTol, M, W(), Z(), IFail(), 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 "M =", M, "Info =", Info End Sub Example Results Eigenvalues = -0.297963342573455 0.510423243055614 7.37370278804149 Eigenvectors = 1.17970064313729 1.46384155786189 9.13803098094184E-02 0.497345303675336 -0.442269445774234 -1.71612038711754 0.524910948248221 1.35130854246605 -0.947378600715302 M = 3 Info = 0