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Local convergence comparison between frozen Kurchatov and Schmidt–Schwetlick–Kurchatov solvers with applications

Abstract:

In this work we are going to use the Kurchatov–Schmidt–Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by x∗ of operator F. We define F as F:D⊆B1⟶B2 where B1 and B2 stand for Banach spaces, D⊆B1 be a convex set and F be a differentiable mapping according to Fréchet. Under these conditions, for all n=0,1,2,… and 0≤i≤m−1 using Taylor expansion, KSSLS and KLS, when B1=B2 and high order derivatives and divided differences not appearing in these solvers, the results obtained are the restart of the utilization of these iterative solvers. Moreover, we show under the same set of conditions that the local convergence radii are the same, the uniqueness balls coincide but the error estimates on ‖xn−x∗‖ differ. It is worth noticing our results improve the corresponding ones (Grau-Sánchez et al., 2011; Kurchatov, 1971 and Shakno, 2009). Finally, we apply our theoretical results to some numerical examples in order to prove the improvement.