Dsbev Dsbevd Dsbevx Dspev Dspevd Dspevx Dstev Dstevd Dstevr Dstevx Dsyev Dsyevd Dsyevr Dsyevx

Dstevx() Sub Dstevx ( Jobz As  String, Range As  String, N As  Long, D() As  Double, E() 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) Eigenvalues and eigenvectors of a symmetric tridiagonal matrix Purpose This routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Parameters [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] N Order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in,out] D() Array D(LD - 1) (LD >= N) [in] N diagonal elements of the symmetric tridiagonal matrix A. [out] D() may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues. [in,out] E() Array E(LE - 1) (LE >= N - 1) [in] N-1 subdiagonal elements of the symmetric tridiagonal matrix A stored in elements 0 to n-2 of E(). [out] E() may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues. [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"). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. [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). If an eigenvector fails to converge (Info > 0), 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 Jobz had an illegal value. (Jobz <> "V" nor "N") = -2: The argument Range had an illegal value. (Range <> "A", "V" nor "I") = -3: The argument N had an illegal value. (N < 0) = -4: The argument D() is invalid. = -5: The argument E() is invalid. = -7: The argument Vu had an illegal value. (Vu <= Vl) = -8: The argument Il had an illegal value. (Il < 1 or Il > N) = -9: The argument Iu had an illegal value. (Iu < min(N, Il) or Iu > N) = -12: The argument W() is invalid. = -13: The argument Z() is invalid. = -14: The argument IFail() is invalid. = i > 0: i eigenvectors failed to converge. Their indices are stored in array IFail(). Reference LAPACK Example Program Compute all eigenvalues and eigenvectors of the symmetric tridiagonal matrix A, where ( 2.58 -0.99 0 ) A = ( -0.99 0.69 -0.03 ) ( 0 -0.03 0.18 ) Sub Ex_Dstevx() Const N = 3 Dim D(N - 1) As Double, E(N - 2) As Double, 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 D(0) = 2.58: D(1) = 0.69: D(2) = 0.18 E(0) = -0.99: E(1) = -0.03 Call Dstevx("V", "A", N, D(), E(), 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 = 0.171899161473039 0.274429936398504 3.00367090212846 Eigenvectors = 0.106563041190365 -0.378750155986403 -0.919343590608286 0.259206614996466 -0.882055576996615 0.393433463029321 0.959925126764757 0.280225406466732 -4.18002107893757E-03 M = 3 Info = 0