An efficient method based on progressive interpolation for solving non-linear equations

The root-finding problem of a univariate polynomial is a fundamental and long-studied problem, which has wide applications in mathematics, engineering, computer science, and natural sciences. This paper presents a progressive interpolation based method for solving a simple root within a given interval, which is of convergence order 3⋅2n−33⋅2n−3 and needs nn functional evaluations of the given function f(t)f(t). The new method can ensure the convergence and achieve a better efficiency index. It needs none of the evaluations of the derivatives of f(t)f(t). Numerical examples show the convergence order of the progressive method.