GS-PIA算法的收敛性证明  

Convergence Proof of GS-PIA Algorithm

作　　者：王志好 李亚娟[1] 邓重阳[1] Wang Zhihao;Li Yajuan;Deng Chongyang(School of Science,Hangzhou Dianzi University,Hangzhou 310018)

机构地区：[1]杭州电子科技大学理学院,杭州310018

出　　处：《计算机辅助设计与图形学学报》2018年第11期2035-2041,共7页Journal of Computer-Aided Design & Computer Graphics

基　　金：国家自然科学基金(61502128;61872121;61761136010);浙江省自然科学基金(LQ17A010009)

摘　　要：非均匀三次B样条曲线插值的GS-PIA算法具有简单、稳定及收敛速度较快等优点.文中详细阐述了GS-PIA算法的几何意义,严格证明了算法的收敛性.首先定义算法配置矩阵的比较矩阵,借助矩阵理论的正则分裂证明比较矩阵对应的迭代矩阵的收敛性;然后利用矩阵的相似性,证明了非均匀三次B样条曲线插值的GS-PIA算法的收敛性.为GS-PIA算法的进一步研究及其在计算机图形学等相关领域的应用打下了理论基础.The GS-PIA algorithm for non-uniform cubic B-spline curve interpolation has the advantages of simplicity,stability,fast convergence and so on.In this paper,we elaborate the detailed geometric meaning of the GS-PIA algorithm and prove the convergence of the algorithm.We first define comparison matrix of configuration matrix of the algorithm.And the convergence of the iterative matrix corresponding to the comparison matrix is proved by the regular splittings of the matrices.Using the similarity of the matrices,we prove that the GS-PIA algorithm for non-uniform cubic B-spline curve interpolation is convergent.It lays out a theoretical foundation for further research of GS-PIA algorithm and applications in computer graphics and related fields.

关 键 词：GS-PIA算法 收敛性证明 迭代速度 曲线插值 非均匀三次B样条曲线 

分 类 号：O245[理学—计算数学]