The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions
The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions
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DOI：
10.1016/0096-3003(89)90039-8
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发表时间：
1989-02
影响因子：
4
通讯作者：
J. Boyd
中科院分区：
文献类型：
--
作者：
J. Boyd
当一个函数在其展开区间的末端是奇异的时，它的切比雪夫系数a_n收敛得很差。我们分析了处理(1±x)klog(1±x)形式奇异性的三种数值策略，并在此过程中对切比雪夫展开理论作了一些适度的补充。前两种数值方法是坐标x=Sin[(π/2)y]和x=tanh[Ly⧸(1-y2)12]的改进收敛变化。我们得到了这两个映象和原始未变换的切比雪夫级数在极限n→∞中的渐近切比雪夫系数。对于原函数，一般k的渐近逼近被扩充为关于整数k的切比雪夫系数。数值试验表明，对于k⩾1，正弦映射是很好的，将收敛速度提高到bn=O(1⧸n 4k+1)。尽管对于足够大的n，tanh变换肯定会更好，但我们提供了理论和数值证据来解释为什么正弦映射在实践中通常更好：“足够大的n”通常是巨大的。代替映射，人们可以使用第三种策略：用奇异基函数补充切比雪夫多项式。简单的实验表明，该方法也是成功的。
When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a n converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1±x) k log (1±x), and in the process make some modest additions to the theory of Chebyshev expansions. The first two numerical methods are the convergence-improving changes of coordinate x= sin [(π/2) y] and x= tanh [Ly⧸(1-y 2) 1 2]. We derive the asymptotic Chebyshev coefficients in the limit n→∞ for both mappings and for the original, untransformed Chebyshev series. For the original function, the asymptotic approximation for general k is augmented by the exact Chebyshev coefficients for integer k. Numerical tests show that the sine mapping is excellent for k⩾ 1, increasing the rate of convergence to b n= O (1⧸ n 4k+ 1). Although the tanh transformation is guaranteed to be better for sufficiently large n, we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice:“sufficiently large n” is usually huge. Instead of mapping, one may use a third strategy: supplementing the Chebyshev polynomials with singular basis functions. Simple experiments show that this approach is also successful.