Dgelqf Dgeqp3 Dgeqrf Dorglq Dorgqr Dormlq Dormqr

Dgelqf() Sub Dgelqf ( M As  Long, N As  Long, A() As  Double, Tau() As  Double, Info As  Long  ) LQ factorization Purpose This routine computes a LQ factorization of a real m x n matrix A. A = L * Q 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 below the diagonal of the array contains the M x min(M, N) lower trapezoidal matrix L (L is lower triangular if M <= N); the elements above the diagonal, with the array Tau(), represent the orthogonal matrix Q as a product of 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(k) . . . H(2) H(1), 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 N) is stored on exit in A(i-1, i to N-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_Dgelqf() 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 LQ factorization of A Call Dgelqf(M, N, A(), Tau(), Info) Debug.Print "L =" Debug.Print A(0, 0) Debug.Print A(1, 0), A(1, 1) Debug.Print A(2, 0), A(2, 1), A(2, 2) Debug.Print "Info =", Info '-- Compute Q Call Dorglq(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 L = -0.957601169589929 0.51921406926948 -0.791085804620857 -0.220237826244838 0.609478035393939 -1.07262846515618 Info = 0 Q = -0.208855216922558 0.114870369307407 0.971176758689895 0.267429186186539 -0.948516183365263 0.169701739147835 0.940670573973608 0.295164103761576 0.167382863850482 Info = 0