精确的商差算法：误差分析、改进与应用
Accurate Quotient-Difference algorithm: error analysis, improvements and applications
AI总结：
本文提出改进的商差算法，通过误差自由变换提升传统商差算法的稳定性，并引入新条件数以直接推导相对前向舍入误差界，展示了其在继续分数和极点零点检测中的应用。
摘要：
补偿商差（Compqd）算法被提出并应用于某些领域。主要动机基于标准商差（qd）算法在数值上不稳定。Compqd算法通过应用误差自由变换来改进传统qd算法。我们详细研究了qd和Compqd算法的误差分析，并引入新的条件数，以便直接推导相对前向舍入误差界。我们的数值实验表明，Compqd算法比qd算法更准确，将条件数的影响降至计算机舍入单位的二次顺序。展示了新算法在继续分数获得和极点和零点检测中的三个应用。
英文摘要：
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.
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