Convergence of second derivative of a family of barycentric Hermite rational interpolants

doi:10.3785/j.issn.1008-9497.2022.03.009

Journal of Zhejiang University (Science Edition)

2022, Vol. 49

Issue (3): 324-328 DOI: 10.3785/j.issn.1008-9497.2022.03.009

Mathematics and Computer Science

Convergence of second derivative of a family of barycentric Hermite rational interpolants

Ning KANG¹(

),Ke JING²

^1.School of Economics，Nanjing University of Finance and Economics，Nanjing 210023，China
^2.School of Applied Mathematics，Nanjing University of Finance and Economics，Nanjing 210023，China

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Abstract

In this paper, we further study a family of barycentric Hermite rational interpolants in a special case $m = 1$ and prove that the second derivatives $r 1 ″ ? x$ of interpolation function converges to corresponding function $f ″ ? x$ at the rate of $O h 2 d - 1$ and $O h 2 d - 3$ at interpolation nodes and non-interpolation nodes, respectively. Finally, numerical examples further verify the effectiveness of the method.

Key words： barycentric rational interpolation Hermite interpolation convergence rates second derivatives

Received: 06 April 2021 Published: 24 May 2022

CLC:

O 241.3

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Cite this article:

Ning KANG, Ke JING. Convergence of second derivative of a family of barycentric Hermite rational interpolants. Journal of Zhejiang University (Science Edition), 2022, 49(3): 324-328.

URL:

https://www.zjujournals.com/sci/EN/Y2022/V49/I3/324

一类重心权Hermite有理插值的二阶导数收敛性

研究了一类特殊情形 $m = 1$ 的重心权Hermite有理插值，证明了该插值函数的二阶导数 $r 1 ″ ? x$ 在插值节点和非插值节点处分别以 $O h 2 d - 1$ 和 $O h 2 d - 3$ 的速度收敛于函数 $f ″ ? x$ 。数值例子进一步验证了方法的有效性。

关键词： 重心权有理插值, Hermite插值, 收敛速度, 二阶导数

实验

函数

f

参数

d

插值

区间

插值节点

x_i

1 / 1 + x 2

3

- 5,5

- 5 + 10 i / n

[1 + t a n h - 9 x + 1] / 2

2

0,1

i / n

Table 1 Functions

f

， parameters

d

， and interpolation nodes

x i

插值节点数 $n$	实验1		实验2
插值节点数 $n$	数值逼近误差 $e 2 ″$	收敛阶	数值逼近误差 $e 2 ″$	收敛阶
10	2.756×10^-1		8.059
20	2.528×10^-2	3.4	8.928×10^-1	3.2
40	1.474×10^-3	4.1	1.282×10^-1	2.8
80	1.496×10^-4	3.3	2.603×10^-2	2.3
160	1.746×10^-4	3.1	6.975×10^-3	1.9
320	2.182×10^-5	3.0	3.486×10^-3	1.0
640	2.728×10^-6	3.0	1.243×10^-3	1.0

Table 2 Approximation errors and convergence orders

Fig.1 Plot of the second derivatives of barycentric Hermite rational interpolation in experiment 1

Fig.2 Plot of the second derivatives of barycentric Hermite rational interpolation in experiment 2


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