Highly Efficient Derivative Free Numerical Method for Solving Nonlinear Scalar Equations by Multiple Precision Arithmetic

Abstract

Abstract. A numerical method for solving the real root of nonlinear real-valued scalar equation by multiple-precision arithmetic is developed. The method is an extension of inverse interpolation methods. The method proceeds to the next step with only half the cost of the Newton method, since the method is derivative free and requires only one function evaluation per step. The rate of convergence of the method is shown to be quadratic or super-quadratic, depending on the equation. As a result the method attains the efficiency index 2 or more, which is the largest among the existing ones. Numerical experiments using a multiple-precision library show the efficiency of the method.