Efficient high-order methods based on golden ratio for nonlinear systems (Q621000)

The authors consider the problem of finding a real solution of the nonlinear system \(F(x)=0\), with \(n\) equations and \(n\) unknowns. They derive new iterative methods with order of convergence four or higher, by composing iteratively golden ratio methods with a modified Newton's method. They prove that if the iterative method \(x^{(k+1)}=M(x^{(k)})\), where \(M:\mathbb{R}^n\rightarrow\mathbb{R}^n\) is a fixed point function, converges to the solution \(\overline{x}\) of \(F(x)=0\) with convergence order \(p\), then under some assumptions the new iterative process \(x^{(k+1)}=M(x^{(k)})-[F'(x^{(k)})]^{-1}F(M(x^{(k)}))\) converges to \(\overline{x}\) with order \(p+1\). Finally, they present several numerical tests which confirm the theoretical results and they conclude that the new methods are very efficient, specifically in the case of large systems.