使用任意精度浮点算术的自适应求积法与双指数公式的性能评估
Performance Evaluation of an Adaptive Quadrature and a Double Exponential Formula Using Arbitrary-Precision Floating-Point Arithmetic
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中文总结 AI 辅助
该研究借助GNU多精度浮点库实现AQE11D与DE公式，针对Kahaner的21个测试问题评估性能，发现AQE11D在两类容差下均达目标精度但求函数次数多，DE公式抗端点奇点强但积分区间内有奇点时失效。
中文摘要 AI 辅助
借助GNU多精度浮点可靠库提供的任意精度算术，我们实现了AQE11D——即Ninomiya的自适应9点牛顿-科茨规则，该规则扩展了一系列高阶规则——以及Takahasi和Mori的双指数（DE）公式。我们针对Kahaner的21个测试问题对它们进行评估。对于绝对容差10⁻⁵⁰和10⁻¹⁰⁰，AQE11D在全部21个问题上均达到目标精度；但对于如1/√x这类强端点奇点，在10⁻¹⁰⁰容差下它需要约5.4×10⁷次函数求值，大致是DE公式的7×10⁴倍。该公式在两种容差下均在18个问题上收敛，展现了其对抗端点奇点的优势，但也表明就目前而言，它在积分区间内存在奇点的问题上会失效。
英文摘要
Using arbitrary-precision arithmetic provided by the GNU Multiple Precision Floating-Point Reliable Library, we implement AQE11D---that is, Ninomiya's adaptive 9-point Newton--Cotes rule extended with a sequence of higher-order rules---and Takahasi and Moris' double exponential (DE) formula. We evaluate them for Kahaner's 21 test problems. For both absolute tolerances $10^{-50}$ and $10^{-100}$, AQE11D attains target accuracy on all 21 problems; however, for strong endpoint singularity such as $1/\sqrt{x}$, it requires about $5.4\times10^{7}$ function evaluations at $10^{-100}$, roughly $7\times10^{4}$ times as many as the DE formula. The formula converges on 18 problems at both tolerances, demonstrating its strength against endpoint singularities but also its failure, as it stands, on problems with a singularity inside the integration interval.
发表机构
- Faculty of Science and Engineering, Otemon Gakuin University（大东文化大学理工学部）
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