Quadratic interval refinement for real roots

Abstract

We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use of arbitrary precision rational arithmetic. Unlike Newton's Iteration our method does not need to evaluate the derivative.

Root refinement for real polynomials using quadratic interval refinement

“…Abbott's QIR method [1,19] is a hybrid of the simple (but inefficient) bisection method with a quadratically converging variant of the secant method. We refer to this method as EQIR, where "E" stands for "exact" in order to distinguish from the variant presented in Section 4 1 .…”

“…We refer to this method as EQIR, where "E" stands for "exact" in order to distinguish from the variant presented in Section 4 1 . Given an isolating interval I = (a, b) for a real root ξ of f , we consider the secant through (a, f (a)) and (b, f (b)) (see also Figure 1).…”

“…What are the problems if f is a bitstream polynomial as in (1), so that f (x 0 ) can only be evaluated up to a certain precision? First of all,…”

“…The combination of root isolation and root refinement, which we call strong root isolation, yields a certified approximation of all roots of the polynomial to an absolute precision of 2 −L or, in other words, to L bits after the binary point in binary representation. Abbott's quadratic interval refinement method [1] (QIR for short) is a hybrid of the bisection and the secant method, which eventually converges quadratically (see Section 3). A straightforward approach for refinement is to apply QIR to each isolating interval until the width becomes small enough.…”

Nearly optimal refinement of real roots of a univariate polynomial

“…For robust approximation of zeros based on bigfloats operations we refer the reader to Sharma et al (2005). A very interesting and efficient algorithm that combines bisection and Newton iterations is the Quadratic Interval Refinement (qir) by Abbott (2014). For a detailed analysis of the Boolean complexity of qir we refer the reader to Kerber (2009).…”