Thresholds of the inner steps in multi-step Newton method (Q1657024)

Summary: We investigate the efficiency of multi-step Newton method (the classical Newton method in which the first derivative is re-evaluated periodically after \(m\) steps) for solving nonlinear equations, \(F(x) = 0\), \(F:D \subseteq \mathbb R^n \rightarrow \mathbb R^n\). We highlight the following property of multi-step Newton method with respect to some other Newton-type method: for a given \(n\), there exist thresholds of \(m\), that is an interval \((m_i, m_s)\), such that for \(m\) inside of this interval, the efficiency index of multi-step Newton method is better than that of other Newton-type method. We also search for optimal values of \(m\).