带形状参数的三角β-B曲线的渐进迭代逼近

Progressive-iterative approximation by the triangularβ-B curves with shape parameter

为加快渐进迭代逼近法收敛速度,克服一般B样条曲线不能表示圆或椭圆等曲线的缺陷,基于β-B曲线探讨其(加权)渐进迭代逼近法。根据所选β-B基函数求得曲线(加权)渐进迭代逼近法的迭代矩阵,基于谱半径最小、收敛速度最快的结论,推导出β-B曲线迭代速度最快时的最优形状参数β和加权渐进迭代逼近法的最优权值w;然后分别对其进行收敛性分析;最后给出数值实例分析形状参数取不同值时的迭代速度和迭代误差,所得实验结果证实了取最优形状参数和最优权值时收敛速度最快的结论。

In order to accelerate the convergence speed of the progressive-iterative approximation method and overcome the defect that the general B-spline curves cannot represent curves such as circles or ellipses,this paper discusses its(weighted)progressive-iterative approximation based onβ-B curves.Compared with general cubic uniform polynomial B-spline curve,the former has better continuity.The iterative matrix of the(weighted)progressive-iterative approximation is obtained according to the selectedβ-B basis function.Based on the conclusion that the spectral radius is the smallest and the convergence speed is the fastest,the optimal shape parameters of the(weighted)progressive-iterative approximation and the optimal weight w of the weighted progressive-pterative approximation method are derived.Then this paper does convergence analysis on them respectively.Finally,numerical examples are given to analyze the iteration speed and iteration error when the shape parameters are different.The experimental results show that the convergence speed is the fastest when the shape parameter and weight are optimal.