On the condition number of linear least squares problems in a weighted Frobenius norm

Let A be an m x n, m greater than or equal to n full rank real matrix and b a real vector of size m. We give in this paper an explicit formula for the condition number of the linear least squares problem (LLSP) defined by min //Ax - b//(2), x epsilon R(n). Let alpha and beta be two positive real numbers, we choose the weighted Frobenius norm //[alpha A, beta b]//(F) on the data and the usual Euclidean norm on the solution. A straightforward generalization of the backward error of [9] to this norm is also provided. This allows us to carry out a first-order estimate of the forward error for the LLSP with this norm. This enables us to perform a complete backward error analysis in the chosen norms. Finally, some numerical results are presented in the last section on matrices from the collection of [5]. Three algorithms have been tested: the QR factorization, the Normal Equations (NE), the Semi-Normal Equations (SNE).