Error Bounds for Gaussian Quadrature of Analytic Functions
Error Bounds for Gaussian Quadrature of Analytic Functions
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DOI：
10.1137/0720087
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发表时间：
1983-12
影响因子：
2.9
通讯作者：
W. Gautschi;R. Varga
中科院分区：
文献类型：
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作者：
W. Gautschi;R. Varga
对于有限区间上的高斯求积规则，应用于解析函数或亚纯函数，我们从余项的轮廓积分表示得到误差界。就像以前关于这个主题的工作一样，我们同时考虑圆形和椭圆轮廓。然而，与以前的工作不同，我们试图确定误差泛函的核在轮廓上的哪个位置达到其最大模数。当轮廓线为圆时，我们成功地回答了一大类权重分布(包括所有雅可比权重)的问题。在较困难的椭圆等值线情形下，我们可以解决某些特殊的带参数的Jacobi权分布的问题，并给出了更一般的Jacobi权的经验结果。我们进一步指出，误差泛函的核，在积分区间外的任何复点，都可以通过递归过程准确而有效地求出。同样的程序也是有用的。
For Gaussian quadrature rules over a finite interval, applied to analytic or meromorphic functions, we develop error bounds from contour integral representations of the remainder term. As in previous work on the subject, we consider both circular and elliptic contours. In contrast with earlier work, however, we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus. We succeed in answering this question for a large class of weight distributions (including all Jacobi weights) when the contour is a circle. In the more difficult case of elliptic contours, we can settle the question for certain special Jacobi weight distributions with parameters $ \pm \frac{1} {2}$, and we provide empirical results for more general Jacobi weights. We further point out that the kernel of the error functional, at any complex point outside the interval of integration, can be evaluated accurately and efficiently by a recursive procedure. The same procedure is useful also t...