Journal of Computer Applications ›› 2015, Vol. 35 ›› Issue (8): 2266-2273.

### Polynomial interpolation algorithm framework based on osculating polynomial approximation

1. 1. School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang Sichuan 621010, China;
2. School of Information Engineering, Southwest University of Science and Technology, Mianyang Sichuan 621010, China;
3. College of Computer Science, Sichuan University, Chengdu Sichuan 610045, China

### 基于密切多项式近似的多项式插值算法框架

1. 1. 西南科技大学 计算机科学与技术学院, 四川 绵阳 621010;
2. 西南科技大学 信息工程学院, 四川 绵阳 621010;
3. 四川大学 计算机学院, 成都 610045
• 通讯作者: 吴亚东(1979-),男,河南周口人,教授,博士,CCF会员,主要研究方向:图形图像处理、信息可视化、人机交互,wyd028@163.com
• 作者简介:赵小乐(1987-),男,四川南部人,硕士研究生,CCF会员,主要研究方向:数字图像处理; 张红英(1976-),女,四川德阳人,教授,博士,主要研究方向:数字图像处理; 赵静(1991-),女,四川隆昌人,硕士研究生,主要研究方向:数字图像处理。
• 基金资助:

国家自然科学基金资助项目(61303127);国家科技支撑计划项目(2013BAH32F01);四川省科技厅科技支撑计划项目(2014SZ0223);四川省教育厅重点项目(13ZA0169);中国科学院"西部之光"人才培养计划项目(13ZS0106);西南科技大学创新基金资助项目(15ycx053)。

Abstract:

Polynomial interpolation technique is a common approximation method in approximation theory, which is widely used in numerical analysis, signal processing, and so on. Traditional polynomial interpolation algorithms are mainly developed by combining numerical analysis with experimental results, lacking of unified theoretical description and regular solution. A uniform theoretical framework for polynomial interpolation algorithm based on osculating polynomial approximation theory was proposed here. Existing interpolation algorithms could be analyzed and new algorithms could be developed under this framework, which consists of the number of sample points, osculating order for sample points and derivative approximation rules. The presentation of existing mainstream interpolation algorithms was analyzed in proposed framework, and the general process for developing new algorithms was shown by using a four-point and two-order osculating polynomial interpolation. Theoretical analysis and numerical experiments show that almost all mainstream polynomial interpolation algorithms belong to osculating polynomial interpolation, and their effects are strongly related to the number of sampling points, order of osculating, and derivative approximation rules.

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