[Paper Review] A study on new computational local orders of convergence

Miquel Grau-Sánchez, Miquel Noguera+2 more|arXiv (Cornell University)|Feb 20, 2012

Iterative Methods for Nonlinear Equations13 references11 citations

TL;DR

This paper introduces four new computational local orders of convergence—CLOC, ACLOC, ECLOC, and PCLOC—for one-point iterative methods with memory, enabling accurate estimation of convergence order without requiring the exact root. The ECLOC variant, based on Aitken’s extrapolation, provides the most accurate approximation of the theoretical order, outperforming others in numerical experiments using adaptive multi-precision arithmetic.

ABSTRACT

Four new variants of the Computational Order of Convergence (COC) of a one-point iterative method with memory for solving nonlinear equations are presented. Furthermore, the way to approximate the new variants to the local order of convergence is analyzed. Three of the new definitions given here do not involve the unknown root. Numerical experiments using adaptive arithmetic with multiple precision and a stopping criteria are implemented without using any known root.

Motivation & Objective

• To develop new computational methods for estimating the local order of convergence of iterative solvers for nonlinear equations.
• To eliminate the need for the exact root in convergence order estimation, which is impractical in real-world applications.
• To evaluate and compare the accuracy and efficiency of four new variants: CLOC, ACLOC, ECLOC, and PCLOC.
• To provide practical, root-free alternatives for numerical verification of convergence order in adaptive multi-precision arithmetic.

Proposed method

• Proposes CLOC as a variant of COC using logarithmic ratios of consecutive error terms, but still requiring the exact root.
• Introduces ACLOC using differences between consecutive iterates, avoiding the need for the root.
• Develops ECLOC by applying Aitken’s δ² procedure to three consecutive iterates to estimate the root and compute error-based convergence order.
• Defines PCLOC based on the logarithmic ratio of function values |f(xₙ)| / |f(xₙ₋₁)|, offering a low-computation alternative.
• Employs adaptive multi-precision arithmetic in Maple to simulate high-accuracy computations with stopping criteria based on digit precision.
• Uses a fixed stopping threshold of η = 2200 digits, selecting the last valid iterate I as the final approximation to compute the four orders.

Experimental results

Research questions

• RQ1 Can a computational local order of convergence be defined without requiring knowledge of the exact root?
• RQ2 How accurately do the new variants ACLOC, ECLOC, and PCLOC approximate the theoretical order of convergence compared to CLOC?
• RQ3 Which of the proposed methods provides the most stable and accurate estimation of convergence order in high-precision settings?
• RQ4 What is the computational cost and numerical reliability of each variant under adaptive multi-precision arithmetic?

Key findings

• CLOC provides the most accurate approximation of the theoretical order of convergence but requires the exact root, which is impractical in real applications.
• ECLOC consistently produces the narrowest error intervals across all test functions, with min-max error bounds as low as [3.7e-5, 1.5e-3], indicating superior accuracy.
• ACLOC and PCLOC offer viable root-free alternatives, with PCLOC being computationally cheaper but less accurate than ECLOC.
• For all test functions and iterative methods, the computed values of λₙ for all four variants converge to the theoretical order ρ as n increases.
• The number of iterations required to reach 2200-digit precision varied from 6 to 18, depending on the function and method, with ECLOC showing the most consistent performance.
• In all cases, the error in the computed order (|λₖ − ρ|) remained within acceptable bounds, validating the reliability of the proposed variants.