WZggglm WZggglm2 WZgglse WZgglse2

WZggglm() Function WZggglm ( 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 of complex matrices (complex number representation in Excel format) Purpose WZggglm 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 To represent complex numbers in Excel cells, complex number format in Excel (e.g. 2.5+1i) is used. Worksheet function Complex can be used to input complex numbers into cells. 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 ( -0.82+0.83i 0.18-0.94i -0.18-0.12i ) A = ( -0.76-0.24i 0.57-0.16i -0.08-0.27i ) ( 1.90+0.26i -0.98+0.54i 0.21+0.28i ) ( 0.50-0.30i -0.31+0.37i 0.22+0.19i ) ( 1 0 0 0 ) B = ( 0 1 0 0 ) ( 0 0 1 0 ) ( 0 0 0 1 ) ( 1.7126-0.6648i ) d = ( 0.8697+0.7604i ) ( -2.1048-1.6171i ) ( -0.9297+0.1252i )