Accurate Quotient-Difference algorithm: error analysis, improvements and applications
For numerical analysts and practitioners using the qd algorithm, this work provides a more accurate alternative, though it is an incremental improvement over existing error-free transformation techniques.
The paper proposes a compensated quotient-difference (Compqd) algorithm to address numerical instability in the standard qd algorithm, achieving accuracy improvements that relegate condition number influence to second order in the rounding unit. Applications in continued fractions and pole/zero detection are demonstrated.
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.