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. -
王则柯,高堂安.同伦方法引论[M].重庆:重庆出版社,1990:84-104.[2] 黄象鼎,曾钟钢,马亚南.非线性数值分析的理论与方法[M].武汉:武汉大学出版社,2004:117-158.[3] 陶本藻,张勤.GPS非线性数据处理的同伦最小二乘模型[J].武汉大学学报:信息科学版,2003,28(特刊):115-117.TAO Benzao,ZHANG Qin.Homotopy least squares model of GPS nonlinear data processing[J].Geomatics and Information Science of Wuhan University,2003,28(Special Issue):115-117.[4] 张勤,陶本藻.基于同伦法的非线性最小二乘平差统一模型[J].武汉大学学报:信息科学版,2004,29(8):708-710.ZHANG Qin,TAO Benzao.Uniform model of nonlinear least squares adjustment based on homotopy method[J].Geomatics and Information Science of Wuhan University,2004,29(8):708-710.[5] HANI I S,MOHAMAD A S.Finding special points using matrix-free predictor-corrector methods[J].Applied Mathematics and Computation,2007,185(2):554-563.[6] CHOI S H,HARNEY D A,BOOK N L.A robust path tracking algorithm for homotopy continuation[J].Computers and Chemical Engineering,1996,20(6-7):647-655.[7] RHEINBOLDT W C.Numerical continuation methods:A perspective[J].Applied Mathematics and Computation,2000,124(2):229-224.[8] EURENE L A,KURT G.Numerical continuation methods[M].Heidelberg:Springer-Verlag,1990:25-68.[9] 李庆扬,关治,白峰杉.数值计算原理[M].北京:清华大学出版社,2000:275-281.[10] 武汉大学测绘学院测量平差学科组.误差理论与测量平差[M].武汉:武汉大学出版社,2003:131-136.[11] 赵瑞安,吴芳.非线性最优化理论和方法[M].浙江:浙江科学技术出版社,1992:1-12.[12] 王新洲.非线性模型参数估计理论与应用[M].武汉:武汉大学出版社,2002:51-81. -
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