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首页> 外文期刊>Journal of Scientific Computing >Interpolatory Quadrature Rules for Oscillatory Integrals
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Interpolatory Quadrature Rules for Oscillatory Integrals

机译：振荡积分的插值正交规则

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摘要

In this paper we revisit some quadrature methods for highly oscillatory integrals of the form ∫_(-1)~1f(x)e~(iωx)dx,ω> 0. Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some 5-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequency-dependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.
机译：在本文中，我们重新讨论一些形式为∫_（-1）〜1f（x）e〜（iωx）dx，ω> 0的高度振荡积分的正交方法。指数拟合（EF）规则取决于与频率相关的节点，这些节点开始当频率为零时，在高斯-勒格德勒节点处关闭；当频率趋于无穷大时，结束于积分的端点。这使得规则非常适合于大小频率。然而，由于迭代和不良状况，EF节点的计算是昂贵的。通过与Filon类型的规则建立联系可以解决此问题。通过介绍一些5形函数，我们展示了如何构建具有频率相关节点的高斯类型规则，该规则具有随着频率增加而导致的误差衰减的最佳渐近速率，并且对于中小频率也有效。这些与频率相关的节点也可以包含在Filon-Clenshaw-Curtis规则中，以形成一类方法，特别适合在自动软件包中实施。

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