Solving systems of nonlinear equations using the Lyapunov direct method (Q1173902)

The paper is devoted to the issue of finding the value of the parameter \(k\) which guarantees the fastest convergence of the steepest descent method \(x_{k+1}=x_ k-k J^ T_ k f_ k\), \(k>0\), for solving a system of algebraic or transcendental equations of the form: (1) \(f(x)=0\). The author shows the way of a direct Lyapunov method to this problem and presents a broad list of Lyapunov functions which are suitable to some of the forms of the system (1). Numerical examples are presented with estimates of the convergence rate and time evolution.