Ddisna Dopgtr Dopmtr Dorgtr Dormtr Dpteqr Dsbtrd Dsptrd Dstebz Dstedc Dstein Dstemr Dsteqr Dsterf Dsytrd

Dsptrd() Sub Dsptrd ( Uplo As  String, N As  Long, Ap() As  Double, D() As  Double, E() As  Double, Tau() As  Double, Info As  Long  ) Reduces a real symmetric matrix stored in packed form to tridiagonal form Purpose This routine reduces a real symmetric matrix A stored in packed form to real symmetric tridiagonal form T by an orthogonal similarity transformation: Q^T * A * Q = T. Parameters [in] Uplo = "U": Upper triangle of A is stored. = "L": Lower triangle of A is stored. [in] N Order of the matrix A. (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] = a[j][i] for 0 <= i <= j <= N - 1. Uplo = "L": ap[(i + j*(2*N - j - 1)/2] = a[j][i] for 0 <= j < = i <= N - 1. [out] If Uplo = "U", the diagonal and first superdiagonal of Ap() are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array Tau(), represent the orthogonal matrix Q as a product of elementary reflectors. If Uplo = "L", the diagonal and first subdiagonal of Ap() are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array Tau(), represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details. [out] D() Array D(LD - 1) (LD >= N) The diagonal elements of the tridiagonal matrix T: D(i) = Tii. [out] E() Array E(LE - 1) (LE >= N - 1) The off-diagonal elements of the tridiagonal matrix T: E(i) = T(i, i+1) if Uplo = "U", E(i) = T(i+1, i) if Uplo = "L". [out] Tau() Array Tau(LTau - 1) (LTau >= N - 1) The scalar factors of the elementary reflectors (see Further Details). [out] Info = 0: Successful exit. = -1: The argument Uplo had an illegal value (Uplo != "U" nor "L") = -2: The argument N had an illegal value. (N < 0) = -3: The argument Ap() is invalid. = -4: The argument D() is invalid. = -5: The argument E() is invalid. = -6: The argument Tau() is invalid. Further Details If Uplo = "U", the matrix Q is represented as a product of elementary reflectors Q = H(N-1) . . . H(2) H(1). Each H(i) has the form H(i) = I - tau*v*v^T where tau is a real scalar, and v is a real vector with v(i+1〜N) = 0 and v(i) = 1. v(1〜i-1) is stored on exit in Ap(), overwriting A(1〜i-1, i+1), and tau in Tau(i-1). If Uplo = "L", the matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(N-1). Each H(i) has the form H(i) = I - tau*v*v^T where tau is a real scalar, and v is a real vector with v(1〜i) = 0 and v(i+1) = 1; v(i+2〜N) is stored on exit in Ap(), overwriting A(i+2〜N, i), and tau in Tau(i-1). Reference LAPACK Example Program Compute all eigenvalues and eigenvectors of the symmetric matrix A, where ( 2.20 -0.11 -0.32 ) A = ( -0.11 2.93 0.81 ) ( -0.32 0.81 2.37 ) Reduces to tridiagonal form by Dsptrd, then Dsterf is applied. Sub Ex_Dsptrd_Dsterf() Const N = 3 Dim Ap(N * (N + 1) / 2) As Double, Info As Long Dim D(N - 1) As Double, E(N - 2) As Double, Tau(N - 2) As Double Ap(0) = 2.2 Ap(1) = -0.11: Ap(3) = 2.93 Ap(2) = -0.32: Ap(4) = 0.81: Ap(5) = 2.37 Call Dsptrd("L", N, Ap(), D(), E(), Tau(), Info) If Info <> 0 Then Debug.Print "Error in Dsptrd: Info =", Info Exit Sub End If Call Dsterf(N, D(), E(), Info) If Info <> 0 Then Debug.Print "Error in Dsterf: Info =", Info Exit Sub End If Debug.Print "Eigenvalues =", D(0), D(1), D(2) End Sub Example Results Eigenvalues = 1.70705954911046 2.22943643244226 3.56350401844728