A Special Interval Newton Step for Solving Nonlinear Equations

Abstract

The problem of finding the zeros of a nonlinear equation has been extensively and thoroughly studied. Point methods constitute a significant and extensive category of techniques, allowing for the efficient finding of zeros with arbitrary precision under specific conditions. However, the limitation of these methods is that they typically yield a single zero. An alternative approach employs Interval Analysis, leveraging its properties to provide reliable and with certainty inclusions of all zeros within a given search interval. Interval methods, such as the Interval Newton method, exhibit quadratic convergence to the corresponding inclusions when monotonicity and simple zeros exist. Nonetheless, there exist pathological cases, like the existence of multiple zeros, where the obtained inclusions cannot be bounded with arbitrary precision, necessitating the adoption of bisection schemes to refine the search interval. These schemes not only increase computational time and cost but also result in a higher number of enclosures, enclosing sometimes the same zero more than once. The main objective of this work is to enhance the applicability of Interval Newton method in cases where no efficient alternative are available. Thus, in this paper, the Interval Newton method is studied and an adjusted perturbation technique is proposed to address the cases where multiple zeros exist. In particular, the given function is vertically shifted. Then, the Interval Newton operator is applied once to this shifted function. The resulting enclosures are then used to efficiently partition the search interval. The successful application of the Interval Newton method is expected to improve overall performance and reduce reliance on bisection schemes. Experimental results on a set of problems demonstrate the effectiveness of the proposed technique.