WZgels WZgels2 WZgelss WZgelss2 WZgelsy WZgelsy2

WZgels() Function WZgels ( M As  Long, N As  Long, A As  Variant, B As  Variant, Optional Nrhs As  Long = 1, Optional Trans As  String = "N", Optional Cov As  String = "N"  ) Solution to overdetermined or underdetermined linear equations Ax = b for complex matrices (full rank) (complex number representation in Excel format) Purpose WZgels solves overdetermined or underdetermined complex linear systems involving an M x N matrix A, or its conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. The following options are provided: If Trans = "N" and M >= N: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||. If Trans = "N" and M < N: find the minimum norm solution of an underdetermined system A * X = B. If Trans = "T" and M >= N: find the minimum norm solution of an underdetermined system A^H * X = B. If Trans = "T" and M < N: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A^H*X ||. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M x Nrhs right hand side matrix B and the N x Nrhs solution matrix 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 If Trans = "N" and M > N (N+2 x Nrhs (Cov = "N"), N+2 x Nrhs+1 (Cov = "D") or N+2 x Nrhs+N (Cov = "C")) Column 1 to Nrhs Column Nrhs+1 (if Cov = "D") Columns Nrhs+1 to Nrhs+N (if Cov = "C") Rows 1 to N Least squares solution vector x Variance (diagonal elements of variance-covariance matrix) Variance-covariance matrix Row N+1 Residual sum of squares for the solution 0 0 Row N+2 Return code (column 1) 0 0 If Trans = "T" and M < N (M+2 x Nrhs) Column 1 to Nrhs Rows 1 to M Least squares solution vector x Row M+1 Residual sum of squares for the solution Row M+2 Return code (column 1) If Trans = "N" and M <= N (N+1 x Nrhs) Column 1 to Nrhs Rows 1 to N Minimum norm solution vector x Row N+1 Return code (column 1) Id Trans = "T" and M >= N (M+1 x Nrhs) Column 1 to Nrhs Rows 1 to M Minimum norm solution vector x Row M+1 Return code (column 1) Return code = 0: Successful exit = i > 0: The i-th diagonal element of the triangular factor of A is zero. (A does not have full rank) Parameters [in] M Number of rows of the matrix A. (M >= 1) [in] N Number of columns of the matrix A. (N >= 1) [in] A (M x N) M x N coefficient matrix A. (should be full rank) [in] B (M x Nrhs if Trans = "N", N x Nrhs if Trans = "T") Right hand side matrix B. [in] Nrhs (Optional) Number of columns of right hand side matrix B. (Nrhs >= 1) (default = 1) [in] Trans (Optional) = "N": Solve Ax = b = "T": Solve (A^T)x = b (default = "N") [in] Cov (Optional) = "N": Do not compute variance-covariance matrix. = "D": Compute variance. (diagonal elements of variance-covariance matrix) (effective if Trans = "N" and M >= N) = "C": Compute variance-covariance matrix. (effective if Trans = "N" and M >= N) (default = "N") Reference LAPACK Example Compute the least squares solution of the overdetermined linear equations Ax = b and its variance, 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 ) ( 1.7126-0.6648i ) B = ( 0.8697+0.7604i ) ( -2.1048-1.6171i ) ( -0.9297+0.1252i )