Accurate Quotient-Difference algorithm: error analysis, improvements and applications

TL;DR

This paper introduces the compensated quotient-difference (Compqd) algorithm, which enhances numerical stability and accuracy over the standard qd algorithm through error analysis, condition numbers, and practical applications.

Contribution

The paper develops the Compqd algorithm using error-free transformations, providing detailed error analysis and demonstrating improved accuracy in applications.

Findings

Compqd significantly outperforms qd in numerical accuracy.

Error bounds are derived using new condition numbers.

Applications include continued fractions and pole-zero detection.

Abstract

The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference (qd) algorithm can be numerically unstable. The Compqd algorithm is obtained by applying error-free transformations to improve the traditional qd algorithm. We study in detail the error analysis of the qd and Compqd algorithms and we introduce new condition numbers so that the relative forward rounding error bounds can be derived directly. Our numerical experiments illustrate that the Compqd algorithm is much more accurate than the qd algorithm, relegating the influence of the condition numbers up to second order in the rounding unit of the computer. Three applications of the new algorithm in the obtention of continued fractions and in pole and zero detection are shown.