[Paper Review] Quadratic Interval Refinement for Real Roots

John Abbott|arXiv (Cornell University)|Mar 6, 2012

Numerical Methods and Algorithms4 citations

TL;DR

This paper introduces Quadratic Interval Refinement (QIR), a novel root refinement algorithm that combines the robustness of bisection with the quadratic convergence of Newton's method. By using discretized linear interpolation and adaptive refinement factors without requiring derivative evaluations, QIR achieves competitive performance—especially in arbitrary-precision rational arithmetic—while ensuring convergence and explicit error bounds.

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.

Motivation & Objective

• To develop a root refinement algorithm that combines the robustness of bisection with the fast convergence of Newton’s method.
• To eliminate the need for derivative evaluation, which is a major limitation in Newton’s method when derivatives are hard to compute.
• To ensure convergence and explicit error bounds by maintaining intervals containing the root throughout refinement.
• To provide a practical, efficient alternative to Newton’s iteration in arbitrary-precision arithmetic, especially for polynomials and well-behaved functions.
• To address the issue of rapidly growing numerators and denominators in rational arithmetic that plagues standard Newton iteration.

Proposed method

• QIR refines an isolating interval containing a single simple root using a sequence of narrowing steps based on discretized linear interpolation.
• Each narrowing step divides the current interval into N equal sub-intervals and uses linear interpolation to predict the sub-interval containing the root.
• The method evaluates the function at the endpoints of the predicted sub-interval; if a sign change is detected, the interval is updated; otherwise, the step fails.
• The refinement factor N is increased after each successful narrowing step, enabling quadratic convergence as the interval shrinks.
• The algorithm uses arbitrary-precision rational arithmetic, ensuring exact representation of intermediate values and explicit error bounds.
• A special case for N=4 is implemented to improve efficiency and avoid unnecessary precision loss in early iterations.

Experimental results

Research questions

• RQ1 Can a root refinement method be designed that combines the robustness of bisection with the quadratic convergence of Newton’s method?
• RQ2 Is it possible to achieve quadratic convergence without evaluating the derivative of the function?
• RQ3 How does the performance of a derivative-free, interval-based method compare to Newton’s iteration in arbitrary-precision arithmetic?
• RQ4 Can such a method avoid the exponential growth of numerator and denominator sizes seen in rational Newton iteration?
• RQ5 What is the impact of nearby complex roots on the convergence behavior of the new method?

Key findings

• QIR achieves quadratic convergence for simple roots when the function has a valid Taylor expansion around the root, matching Newton’s convergence rate.
• For the polynomial $x^5 - 2$, QIR computed a root with error less than $2^{-32}$ in six iterations, with all intermediate values having at most 50 digits, while Newton’s method produced approximants with over 17,000-digit denominators.
• In experiments, QIR refined the root of $f_2 = x^{50} + (10^{50}x - 1)^3$ to $10^{-10000}$ precision in 6,600 seconds, demonstrating robustness even with roots close to complex conjugate pairs.
• QIR was competitive with GMP’s specialized root functions: for $\sqrt[3]{3}$, it achieved $10^6$ digits in 5.1 seconds versus GMP’s 17 seconds.
• For $\sqrt{5}$, QIR was slower than GMP’s mpz_sqrt (35s vs 8.2s for $10^7$ digits), but remained highly effective for general functions where derivative evaluation is impractical.
• The method avoids the issue of unbounded growth in rational arithmetic by maintaining intervals and using controlled refinement, making it suitable for exact computation environments.