Extended interval Newton method based on the precise quotient set

Abstract

The interval Newton method can be used for computing an enclosure of a single simple zero of a smooth function in an interval domain. It can practically be extended to allow computing enclosures of all zeros in a given interval. This paper deals with the extended interval Newton method. An essential operation of the method is division by an interval that contains zero (extended interval division). This operation has been studied by many researchers in recent decades, but inconsistency in the research has occurred again and again. This paper adopts the definition of extended interval division redefined in recent documents (Kulisch in Arithmetic operations for floating-point intervals, 2009; Pryce in P1788: IEEE standard for interval arithmetic version 02.2, 2010). The result of the division is called the precise quotient set. Earlier definitions differ in the overestimation of the quotient set in particular cases, causing inefficiency in Newton’s method and even leading to redundant enclosures of a zero. The paper reviews and compares some extended interval quotient sets defined during the last few decades. As a central theorem, we present the fundamental properties of the extended interval Newton method based on the precise quotient set. On this basis, we develop an algorithm and a convenient program package for the extended interval Newton method. Statements on its convergence are also given. We then demonstrate the performance of the algorithm through nine carefully selected very sensitive numerical examples and show that it can compute correct enclosures of all zeros of the functions with high efficiency, particularly in cases where earlier methods are less effective.