The Solution of Nonlinear Systems of Equations by Second Order Systems of O.D.E. and Linearly Implicit A-Stable Techniques

Abstract

In [1], [2], [3], [4], a new method for solving systems of nonlinear equations was proposed and developed. The method associates a system of ordinary differential equations (odes) with the equations whose roots we are interested in and integrates the former numerically. The system of differential equations is inspired by classical mechanics and is of second order. In this paper, we prove a new stability result for this system of differential equations that allows some of its coefficients (the mass coefficient $\mu (t)$ and the friction coefficient $g(t)$) to go to zero as the time t tends to infinity. Some numerical algorithms obtained by integrating numerically the Cauchy problem for the system of o.d.e. by A-stable linearly implicit methods are presented. For these algorithms, local convergence (global for a system of linear equations) and rate of convergence results are proved. If the unknown functions are sufficiently regular, the rate of convergence depends on the way in which the time integration step h approaches infinity during the iteration or on the way in which $\mu (t)$, $g(t)$ tend to zero as $t \to \infty $. Finally, some numerical experience obtained on a set of test problems is presented.