WDgels WDgesv WDposv WDsyev

WDgels() Function WDgels ( 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 (full rank) Purpose WDgels solves overdetermined or underdetermined real linear systems involving an m x n matrix A, or its 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^T * 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^T*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. 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 ( -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 ) ( 0.3884 ) B = ( 0.1120 ) ( -0.3644 ) ( -0.0002 )