Dgelqf Dgeqp3 Dgeqrf Dorglq Dorgqr Dormlq Dormqr

Dgeqrf() Sub Dgeqrf ( M As  Long, N As  Long, A() As  Double, Tau() As  Double, Info As  Long  ) QR factorization Purpose This routine computes a QR factorization of a real m x n matrix A. A = Q * R Parameters [in] M Number of rows of the matrix A. (M >= 0) (If M = 0, returns without computation) [in] N Number of columns of the matrix A. (N >= 0) (If N = 0, returns without computation) [in,out] A() Array A(LA1 - 1, LA2 - 1) (LA1 >= M, LA2 >= N) [in] M x N matrix A. [out] The elements on and above the diagonal of the array contains the min(M, N) x N upper trapezoidal matrix R (R is upper triangular if M >= N); the elements below the diagonal, with the array Tau(), represent the orthogonal matrix Q as a product of min(M, N) elementary reflectors (see Further Details). [out] Tau() Array Tau(LTau - 1) (LTau >= min(M, N)) The scalar factors of the elementary reflectors (see Further Details). [out] Info = 0: Successful exit. = -1: The argument M had an illegal value. (M < 0) = -2: The argument N had an illegal value. (N < 0) = -3: The argument A() is invalid. = -4: The argument Tau() is invalid. Further Details The matrix Q is represented as a product of elementary reflectors Q = H(1) H(2) . . . H(k), where k = min(M, N). 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 to i-1) = 0 and v(i) = 1; v(i+1 to M) is stored on exit in A(i to M-1, i-1), and tau in Tau(i-1). Reference LAPACK Example Program Compute QR factorization of the matrix A, where ( 2.20 -0.11 -0.32 ) A = ( -0.11 2.93 0.81 ) ( -0.32 0.81 2.37 ) Sub Ex_Dgeqrf() Const M = 3, N = 3, K = N Dim A(M - 1, N - 1) As Double, Tau(N - 1) As Double, Info As Long A(0, 0) = 0.2: A(0, 1) = -0.11: A(0, 2) = -0.93 A(1, 0) = -0.32: A(1, 1) = 0.81: A(1, 2) = 0.37 A(2, 0) = -0.8: A(2, 1) = -0.92: A(2, 2) = -0.29 '-- Compute QR factorization of A Call Dgeqrf(M, N, A(), Tau(), Info) Debug.Print "R =" Debug.Print A(0, 0), A(0, 1), A(0, 2) Debug.Print A(1, 1), A(1, 2) Debug.Print A(2, 2) Debug.Print "Info =", Info '-- Compute Q Call Dorgqr(M, N, K, A(), Tau(), Info) Debug.Print "Q =" Debug.Print A(0, 0), A(0, 1), A(0, 2) Debug.Print A(1, 0), A(1, 1), A(1, 2) Debug.Print A(2, 0), A(2, 1), A(2, 2) Debug.Print "Info =", Info End Sub Example Results R = -0.884533775499839 -0.514169173181655 8.18510293279503E-02 -1.11813687057967 -0.635776616242947 -0.821576839520579 Info = 0 Q = -0.226107815823067 0.202352390539057 0.952852961199602 0.361772505316908 -0.890778487093709 0.275016983707865 0.904431263292269 0.406899492472248 0.128206446816755 Info = 0