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首页> 外文期刊>Applied mathematics and computation >Shafer (Hermite-Pade) approximants for functions with exponentially small imaginary part with application to equatorial waves with critical latitude
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Shafer (Hermite-Pade) approximants for functions with exponentially small imaginary part with application to equatorial waves with critical latitude

机译：Shafer（Hermite-Pade）近似函数用于虚部呈指数减小的函数，并适用于临界纬度的赤道波

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摘要

Quadratic Shafer approximants and their generalization to higher degree polynomials called Hermite-Pade, approximants have been successfully used in quantum mechanics for calculation of exponentially small escape rates. In this paper we test quadratic Shafer approximants in representing growth rates typical for equatorial atmosphere. One of the characteristic features of the equatorial Kelvin wave - the dominant mode in equatorial dynamics - is its exceptionally small linear growth rate. For example, Kelvin wave evolving on the zonal basic state with small linear shear ( has growth rate O(exp(-1/c(2))) in contrast to T(E) similar to O(exp(-1/epsilon)) common to similar problem,, in quantum mechanics. It is interesting to know how well Hermite-Pade approximants handle this more computationally expensive problem. First we apply the quadratic Shafer approximants to calculate the imaginary part of the Gauss-Stieltjes function defined as G(s)(epsilon) equivalent to lim(delta-->0) integral(0)(infinity) exp(-t(2))/t-[1+idelta]/c dt on its branch cut. The imaginary part of Gs(c) can be shown to be T(G(S))(epsilon) = pi exp (-1/epsilon(2)). which is of the same order of magnitude as Kelvin-in-shear growth rate. Next, We use this technique to sum the divergent Rayleigh Schrodinger perturbation series of the Hermite-with-Pole equation u(yy) + (epsilon 1/1+epsilony - lambda - y(2))u = 0. which is a simple model for equatorial waves in shear. The Hermite-with-Pole equation has been previously studied numerically and analytically in [J,P. Bold, A. Nataroc, Stud. Appl. Math. 101 (1998) 43]. We compare Shafer approximant, against numerical integration and find that for,mall epsilon, Shafer approximants are more efficient, primarily because with rational coefficients, one does not need multiple precision at the main stage of the calculation. Although the higher-order approximants are usually more accurate, the overall improvement of accuracy is not monotonic due to the appearance of nearly coincident zeros in the approximants. (C) 2002 Elsevier Science Inc. All rights reserved. [References: 7]
机译：二次Shafer近似值及其推广到称为Hermite-Pade的更高阶多项式的近似值已成功地用于量子力学中，以计算出较小的逸出率。在本文中，我们测试了二次Shafer近似值，以表示赤道大气典型的增长率。赤道开尔文波（赤道动力学的主导模式）的特征之一是其极小的线性增长率。例如，与类似于O（exp（-1 / epsilon）的T（E）相比，开尔文波以较小的线性剪切（具有O（exp（-1 / c（2））的增长率）在带状基本状态上演化。 ）是量子力学中类似问题的共同点。很有趣的是知道Hermite-Pade近似值如何处理这个在计算上更昂贵的问题。首先，我们应用二次Shafer近似值来计算定义为G的Gauss-Stieltjes函数的虚部（s）（ε）等效于lim（delta-> 0）积分（0）（无穷大）exp（-t（2））/ t- [1 + idelta / c dt。 Gs（c）的T可以显示为T（G（S））（epsilon）= pi exp（-1 / epsilon（2）），其大小与切尔文切变增长率相同。接下来，我们使用此技术对Hermite-with-Pole方程u（yy）+（epsilon 1/1 + epsilony-lambda-y（2））u = 0的发散的Rayleigh Schrodinger摄动序列求和。赤道波在剪切过程中的模型。以前在[J，P。 Bold，A。Nataroc，饰钉。应用数学。 101（1998）43]。我们将Shafer近似值与数值积分进行比较，发现对于商场epsilon，Shafer近似值更有效，主要是因为有了有理系数，在计算的主要阶段不需要多重精度。尽管高阶近似值通常更准确，但由于近似值中几乎重合的零点出现，因此精度的整体提高并不是单调的。 （C）2002 Elsevier Science Inc.保留所有权利。 [参考：7]

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