计算柯西主值积分问题中的舍入误差
Roundoff errors in the problem of computing Cauchy principal value integrals
AI总结：
研究使用标准自适应求积快速、准确且可靠计算柯西主值积分时舍入误差的影响，并提出一种易于实现的方法。
摘要：
我们研究了使用标准自适应求积快速、准确且可靠计算柯西主值积分 $\mathrm{P}\!\int_{a}^{b} f(x)(x-\tau)^{-1} dx$ $(a < \tau < b)$ 的可能性。为了适当控制自适应求积的误差容限并获得近似误差的可靠估计，我们研究了舍入误差对计算结果的潜在影响。数值实验证实，所提出的方法可以成功与其他计算此类积分的算法竞争。此外，该方法在任何配备可靠自适应积分子程序的系统上都非常容易实现。
英文摘要：
We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-τ)^{-1} dx$ $(a < τ< b)$ using standard adaptive quadratures. In order to properly control the error tolerance for the adaptive quadrature and to obtain a~reliable estimation of the approximation error, we research the possible influence of round-off errors on the computed result. As the numerical experiments confirm, the proposed method can successfully compete with other algorithms for computing such type integrals. Moreover, the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine.
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