高阶收敛的加速最小二乘渐进迭代逼近
Accelerated Least Squares Progressive-Iterative Approximation of Higher Order of Convergence
DOI: 10.12677/aam.2026.156283, PDF,    科研立项经费支持
作者: 曹一菲, 彭兴璇*：辽宁师范大学数学学院，辽宁 大连
关键词: 渐进迭代逼近；最小二乘拟合；近似逆；B样条；Progressive-Iterative Approximation； Least Squares Fitting； Approximate Inverse； B-Spline
摘要: 最小二乘渐进迭代逼近(progressive iterative approximation for least square fitting, LSPIA)是一种应用于数据拟合问题的迭代算法。针对经典LSPIA算法具有收敛速度较慢的问题，本文引入一种求解非奇异矩阵逆的具有8阶收敛性的迭代算法，与经典LSPIA算法相结合，提出了一类加速LSPIA算法。首先对数据点进行参数化；然后利用加速LSPIA算法构造拟合曲线序列，并证明该序列的极限曲线收敛至给定数据点的最小二乘拟合结果。数值实例进一步表明，相较于经典LSPIA算法，本文方法能显著提升收敛速度，在相同终止误差条件下，可减少迭代次数和运行时间。
Abstract: The progressive iterative approximation for least square fitting (LSPIA) is an iterative algorithm, which is commonly applied to data fitting problems. To address the issue of slow convergence in the classical LSPIA method, this paper integrates an iterative algorithm for computing the inverse of non-singular matrices, which possesses eighth-order convergence. Combined with the classical LSPIA approach, a class of accelerated LSPIA algorithms is proposed. Firstly, the data points are parameterized. Then, an accelerated LSPIA algorithm is employed to generate a sequence of fitting curves. This sequence converges to the least squares fitting result for the given data points. Experimental results demonstrate that the proposed algorithm achieves a significantly accelerated convergence rate compared to the classical LSPIA method, and it can reduce the number of iterations and running time under the same termination error condition.
文章引用：曹一菲, 彭兴璇. 高阶收敛的加速最小二乘渐进迭代逼近[J]. 应用数学进展, 2026, 15(6): 248-257. https://doi.org/10.12677/aam.2026.156283

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