Nonlinear Least Squares Adjustment Based on Improved Homotopy Algorithm
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摘要: 为了寻求一种更有效的非线性最小二乘平差算法,根据同伦思想提出了一种改进的同伦算法.该算法直接从非线性方程入手,将非线性最小二乘平差准则转化为同伦最小二乘平差准则;根据最优化问题的极值条件,将同伦最小二乘平差准则转化为求解非线性方程组的不动点同伦问题;在Li-Yorke算法的基础上,对切向量及步长求解进行改进,并用于求解微分方程初值问题,进而跟踪同伦曲线.对改进同伦算法的收敛性进行了分析,并采用Matlab语言编程进行了试验.结果表明,较之牛顿迭代法和Li-Yorke算法,改进同伦算法是一种结果稳定、精度较高、速度较快和收敛域扩大的整体收敛方法.
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关键词:
- 同伦算法 /
- 非线性最小二乘平差 /
- Li-Yorke算法
Abstract: Based on the homotopy idea,an improved homotopy algorithm was proposed in order to search a more efficient algorithm for nonlinear least squares adjustment.This algorithm directly transforms the rule of nonlinear least squares adjustment into the rule of homotopy least squares adjustment,and the rule of homotopy least squares adjustment is changed into a fixed-point homotopy problem on the basis of the extreme conditions of an optimization problem.The solutions of tangent vector and step size are improved based on the Li-Yorke algorithm.Partial differential equations are solved with the improved homotopy algorithm to follow the homotopy curve.The convergence of the improved homotopy algorithm was investigated and tested with the Matlab program language.The research results show that the improved homotopy algorithm is a global convergent algorithm with a stable result,great accuracy,fast speed and wide convergence,compared with the Newton iterative algorithm and the Li-Yorke algorithm. -
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