Generating optimal derivative free iterative methods for nonlinear equations by using polynomial interpolation

Abstract In this work we show a general procedure to obtain optimal derivative free iterative methods (Kung and Traub (1974)) [2] for nonlinear equations f ( x ) = 0 , applying polynomial interpolation to a generic optimal derivative free iterative method of lower order. Let us consider an optimal method of order q = 2 n − 1 , v = ϕ n ( x ) , that uses n functional evaluations. Performing a Newton step w = v − f ( v ) f ′ ( v ) one obtains a method of order 2 n , that is not optimal because it introduces two new functional evaluations. Instead, we approximate the derivative by using a polynomial of degree n that interpolates n + 1 already known functional values and keeps the order 2 n . We have applied this idea to Steffensen’s method, (Ortega and Rheinboldt (1970)) [3] , obtaining a family of optimal derivative free iterative methods of arbitrary high order. We provide different numerical tests, that confirm the theoretical results and compare the new family with other well known family of similar characteristics.

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