Higher-Order Root-Finding Algorithm and its Applications

Root-finding method is an iterative process that constructs a sequence converging to a solution of an equation. Householder's method is a higher-order method that requires higher order derivatives of the reciprocal of a function and has disadvantages. Firstly, symbolic computations can take a long time, and numerical methods to differentiate a function can accumulate errors. Secondly, the convergence factor existing in the literature is a rough estimate. In this paper, we propose a higher-order root-finding method using only Taylor expansion of a function. It has lower computational complexity with explicit convergence factor, and can be used to numerically implement Householder's method. As an application, we apply the proposed method to compute pre-images of $q$-ary entropy functions, commonly seen in coding theory. Finally, we study basins of attraction using the proposed method and compare them with other root-finding methods.