Solution of an Overdetermined System of Linear Equations in $L_2$, $L_∞$, $L_P$ Norm Using L.S. Techniques

Abstract

A lot of curve fitting problems of experiment data lead to solution of an overdetermined system of linear equations. But it is not clear prior to that whether the data are exact or contaminated with errors of an unknown nature. Consequently we need to use not only $L_2$-solution of the system but also $L_{\infty}$- or $L_p$-solution.
In this paper, we propose a universal algorithm called the Directional Perturbation Least Squares (DPLS) Algorithm, which can give optimal solutions of an overdetermined system of linear equations in $L_2$, $L_{\infty}$,$L_p (1\leq p＜2)$ norms using only L.S. techniques (in $\S$2). Theoretical principle of the algorithm is given in $\S$ 3. Two examples are given in the end.