Geometric interpretation of some Cauchy related methods

Abstract

This paper shows that a class of methods for solving linear equations, including the Cauchy–Barzilai–Borwein method, can be interpreted by means of a simple geometric object, the Bezier parabola. This curve is built from the current iterate using a transformation characterizing the system to be solved. The localization of the next iterates in the plane of the parabola sheds some light on the behavior of the methods and provides some new understanding of their relative efficiency.