WDggglm WDgglse

WDggglm() Function WDggglm ( N As  Long, M As  Long, P As  Long, A As  Variant, B As  Variant, D As  Variant  ) General Gauss-Markov linear model (GLM) problem Purpose WDggglm solves a general Gauss-Markov linear model (GLM) problem: minimize || y ||_2 subject to d = A*x + B*y x where A is an N x M matrix, B is an N x P matrix, and d is a given N vector. It is assumed that M <= N <= M + P, and rank(A) = M and rank(A B) = N Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B) given by A = Q*(R), B = Q*T*Z (0) In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem minimize || inv(B)*(d - A*x) ||_2 x Returns max(M,P)+1 x 2 Column 1 Column 2 Rows 1 to max(M,P) Least squares solution vector x (Rows 1 to M) Residual vector y (Rows 1 to P) Row max(M,P)+1 2-norm of residual sum of squares (||y||2) Return code Return code = 0: Successful exit = 1: The least squares solution could not be computed. The upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M. = 2: The least squares solution could not be computed. The bottom N-M x N-M part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank(A B) < N. Parameters [in] N Number of rows of the matrices A and B. (N >= 1) [in] M Number of columns of the matrix A. (1 <= M <= N) [in] P Number of columns of the matrix B. (N - M <= P) [in] A (N x M) N x M coefficient matrix A of the GLM equation. [in] B (N x P) N x P coefficient matrix B of the GLM equation. [in] D (N) Left hand side vector d of the GLM equation. Reference LAPACK Example Solve a general Gauss-Markov linear model (GLM) problem, i.e. find x which minimizes || y ||_2 subject to d = A*x + B*y, where ( -1.06 0.48 -0.04 ) A = ( -1.19 0.73 -0.24 ) ( 1.97 -0.89 0.56 ) ( 0.68 -0.53 0.08 ) ( 1 0 0 0 ) B = ( 0 1 0 0 ) ( 0 0 1 0 ) ( 0 0 0 1 ) ( 0.3884 ) d = ( 0.1120 ) ( -0.3644 ) ( -0.0002 )