A new class of root-finding methods in $${\mathbb {R}}^n$$ R n : the inexact tensor-free Chebyshev–Halley class

Abstract

A new class of methods for solving systems of nonlinear equations that can attain cubic convergence rate, named the inexact tensor-free Chebyshev–Halley class, is introduced. As its name implies, the methods belonging to this class do not require second-order derivative information (a third-order tensor of size $$n\times n\times n$$ ) and find the next iterate by approximately solving two linear systems. This new class can be seen as a generalization of the Chebyshev–Halley class. No norm reduction in the quadratic model is needed to ensure convergence as required by Steihaug and Suleiman (Appl Numer Math 67:230–242, 2013). This gives more flexibility for choosing the step than the Steihaug and Suleiman algorithm does. In addition, in the convergence analysis section, we show that, depending on reasonable assumptions, the methods of this class can have superlinear, quadratic, superquadratic, or cubic convergence rates. We presented numerical evidence that demonstrates significant improvement when utilizing the proposed inexact tensor-free methods when compared to the Steihaug and Suleiman algorithm. This new class of methods also enlarges the Inexact Newton step by an additional step. The role of this extra step is to accelerate the convergence of the Inexact Newton method, so that cubic convergence rate is achieved by the inexact tensor-free Chebyshev–Halley methods. For this reason, we also tested them numerically against Newton-GMRES and the results have shown that the extra-cost due to the computation of the further step is more than counterbalanced by the gain in convergence rate.