求解线性方程组迭代终止条件的探究
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摘要
迭代算法的一个重要问题是迭代的终止准则,不同的迭代算法给出的终止准则不同,因此求解线性方程组时会遇到此类问题——对某些迭代算法使用一些常用的迭代终止条件可能会提前终止迭代。本文针对这一问题给出了一个可以提高精度的终止迭代的条件。通过数值算例用随机Kaczmarz方法验证了此条件,同时这一方法也适用于其它迭代法的迭代终止,并能得到更高精度的数值解。
One of the important questions for solving linear equations is when to terminate the iteration in the algorithm.Different iterative algorithms give different termination iteration conditions.Therefore,the problem of some iterative algorithms terminating the iteration early by using some common iterative termination conditions will occur in solving linear equations.This paper gives a new condition for terminating the iteration to improve the accuracy.The new iterative termination condition given in this pape is verified by randomized Kaczmarz's method.At the same time,it is also verified that this condition is applicable to different iterative algorithms under normal circumstances,and the accuracy of the obtained numerical solution is better.
One of the important questions for solving linear equations is when to terminate the iteration in the algorithm.Different iterative algorithms give different termination iteration conditions.Therefore,the problem of some iterative algorithms terminating the iteration early by using some common iterative termination conditions will occur in solving linear equations.This paper gives a new condition for terminating the iteration to improve the accuracy.The new iterative termination condition given in this pape is verified by randomized Kaczmarz's method.At the same time,it is also verified that this condition is applicable to different iterative algorithms under normal circumstances,and the accuracy of the obtained numerical solution is better.
引文
[1] 王能超.计算方法:算法设计及其MATLAB实现[M].第2版.武汉:华中科技大学出版社,2016.
[2] HESTENES M R,STIEFEL E.Methods of conjugate gradients for solving linear systems[J].Journal of Research of the National Bureau of Standards,1952,49(6):409-436.
[3] ELBLE J M,SAHINIDIS N V,VOUZIS P.GPU computing with Kaczmarz's and other iterative algorithms for linear systems[J].Parallel Computing,2010,36(5/6):215-231.
[4] STROHMER T,VERSHYNIN R.Comments on the randomized Kaczmarz method[J].Journal of Fourier Analysis & Applications,2009,15(4):437-440.
[5] STROHMER T,VERSHYNIN R.A randomized Kaczmarz algorithm with exponential convergence[J].Journal of Fourier Analysis & Applications,2009,15(2):262.
[6] 向徐.随机数值方法及其在随机Kaczmarz算法中的应用[D].长沙:国防科学技术大学,2015.
[7] 富明慧,李勇息,张文志.求解病态线性方程的一种精细格式及迭代终止准则[J]. 应用力学学报, 2018,35(2): 346-350+454.
[8] 邓兴升,孙虹虹.自适应谱修正LU分解法解算高病态法方程[J].大地测量与地球动力学,2014,34(6):135-139.
[9] 戴蓉,黄成.一种图像降噪的自适应迭代终止策略[J].计算机应用与软件,2016,33(5):204-206.
[10] 徐少平,曾小霞,姜尹楠,等.基于残差图像的迭代终止条件及其在NCSR算法中的应用[J].光电子·激光,2018,29(4):411-422.
[11] 江顺亮,姜尹楠,曾小霞,等.适用于迭代型去模糊算法的自适应迭代终止条件[J].计算机应用研究,2019(3):1-11.
[2] HESTENES M R,STIEFEL E.Methods of conjugate gradients for solving linear systems[J].Journal of Research of the National Bureau of Standards,1952,49(6):409-436.
[3] ELBLE J M,SAHINIDIS N V,VOUZIS P.GPU computing with Kaczmarz's and other iterative algorithms for linear systems[J].Parallel Computing,2010,36(5/6):215-231.
[4] STROHMER T,VERSHYNIN R.Comments on the randomized Kaczmarz method[J].Journal of Fourier Analysis & Applications,2009,15(4):437-440.
[5] STROHMER T,VERSHYNIN R.A randomized Kaczmarz algorithm with exponential convergence[J].Journal of Fourier Analysis & Applications,2009,15(2):262.
[6] 向徐.随机数值方法及其在随机Kaczmarz算法中的应用[D].长沙:国防科学技术大学,2015.
[7] 富明慧,李勇息,张文志.求解病态线性方程的一种精细格式及迭代终止准则[J]. 应用力学学报, 2018,35(2): 346-350+454.
[8] 邓兴升,孙虹虹.自适应谱修正LU分解法解算高病态法方程[J].大地测量与地球动力学,2014,34(6):135-139.
[9] 戴蓉,黄成.一种图像降噪的自适应迭代终止策略[J].计算机应用与软件,2016,33(5):204-206.
[10] 徐少平,曾小霞,姜尹楠,等.基于残差图像的迭代终止条件及其在NCSR算法中的应用[J].光电子·激光,2018,29(4):411-422.
[11] 江顺亮,姜尹楠,曾小霞,等.适用于迭代型去模糊算法的自适应迭代终止条件[J].计算机应用研究,2019(3):1-11.