WZggglm WZggglm2 WZgglse WZgglse2

WZgglse2() Function WZgglse2 ( M As  Long, N As  Long, P As  Long, A As  Variant, B As  Variant, C As  Variant, D As  Variant  ) Linear equality-constrained least squares (LSE) problem of complex matrices (complex numbers in pairs of cells) Purpose WZgglse2 solves the linear equality-constrained least squares (LSE) problem: minimize || c - Ax ||_2 subject to B*x = d where A is an M x N matrix, B is a P x N matrix, c is a given M-vector, and d is a given P-vector. It is assumed that P <= N <= M + P, and rank(B) = P and rank( (A) ) = N ( (B) ) These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by B = (0 R)*Q, A = Z*T*Q To represent complex numbers, a real part and an imaginary part are stored in a pair of adjacent cells (a real part in a left cell, and an imaginary part in a right cell). The computed results are stored in the same way. Returns N+1 x 2 Column 1 Column 2 Rows 1 to N Least squares solution vector x (real part) Least squares solution vector x (imaginary part) Row N+1 2-norm of residual sum of squares Return code Return code = 0: Successful exit = 1: The least squares solution could not be computed. The upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < P. = 2: The least squares solution could not be computed. The N-P x N-P part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that rank((A^T B^T)^T) < N. Parameters [in] M Number of rows of the matrix A. (M >= 1) [in] N Number of columns of the matrices A and B. (N >= 1) [in] P Number of rows of the matrix B. (1 <= P <= N <= M + P) [in] A (M x 2N) M x N coefficient matrix A of the least squares part of the LSE problem. [in] B (P x 2N) P x N coefficient matrix B of the constrained equation. [in] C (M x 2) Right hand side vector c for the least squares part of the LSE problem. [in] D (P x 2) Right hand side vector d for the constrained equation. Reference LAPACK Example Solve the linear equality-constrained least squares (LSE) problem, i.e. minimize || c - Ax ||_2 subject to B*x = d, 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 ) ( 0.57-0.91i -0.28-0.45i 0.25+0.91i ) B = ( 0.83+0.46i 0.63-0.19i -0.69+0.09i ) ( 0.24-1.33i -0.56-0.67i 0.90+1.25i ) ( 1.7126-0.6648i ) c = ( 0.8697+0.7604i ) ( -2.1048-1.6171i ) ( -0.9297+0.1252i ) ( -1.5111+0.3107i ) d = ( -0.0941-1.2737i ) ( -1.5579+1.0462i )