使用指数变换对具有端点奇异性的函数进行埃尔米特谱逼近
Hermite spectral approximation for functions with endpoint singularities using exponential transforms
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中文总结 AI 辅助
研究使用指数变换对有端点奇异性的函数进行埃尔米特谱逼近，分析无缩放和有缩放时的收敛性，得出最优缩放因子，经数值比较其精度性能良好，还讨论了对求积和求根算法的扩展。
中文摘要 AI 辅助
本文介绍了使用指数变换（包括单指数（SE）、双指数（DE）和误差函数（EF）变换）对具有端点奇异性的函数进行埃尔米特谱逼近，并对这些逼近在无缩放和有缩放情况下进行了全面的收敛性分析。在无缩放情况下，这些方法以某种根指数速率收敛。有缩放时，得出各指数变换的最优缩放因子，表明埃尔米特谱逼近的收敛速率可显著提高。与辛克方法进行了数值比较，结果表明在使用相同项数时埃尔米特方法具有相当或更优的精度性能。还讨论了对求积和求根算法的扩展。
英文摘要
In this paper we introduce Hermite spectral approximation for functions with endpoint singularities using exponential transforms, including single exponential (SE), double exponential (DE) and error function (EF) transforms, and present a comprehensive convergence analysis for these approximations without and with scaling. In the case without scaling, we show that these methods converge at some root-exponential rate. In the case with scaling, we derive optimal scaling factors for each of exponential transforms and show that the convergence rate of Hermite spectral approximation can be significantly improved. Numerical comparisons with sinc method are present and it is shown that Hermite method has comparable or superior accuracy performance when using the same number of terms. Extensions to quadrature and rootfinding algorithm are also discussed.
