求解M-张量方程的两种新型算法

doi:10.12286/jssx.j2020-0756

计算数学 ›› 2022, Vol. 44 ›› Issue (2): 206-216.DOI: 10.12286/jssx.j2020-0756

求解M-张量方程的两种新型算法

邵新慧, 祁猛

东北大学理学院, 沈阳 110819

收稿日期:2020-11-26 出版日期:2022-05-14 发布日期:2022-05-06
基金资助:
中央高校基本业务费(N2005013)资助.

PDF ( 94 ) 引用

邵新慧, 祁猛. 求解M-张量方程的两种新型算法[J]. 计算数学, 2022, 44(2): 206-216.

Shao Xinhui, Qi Meng. TWO NEW ALGORITHMS FOR SOLVING MULTI-LINEAR SYSTEMS WITH M-TENSOR[J]. Mathematica Numerica Sinica, 2022, 44(2): 206-216.

TWO NEW ALGORITHMS FOR SOLVING MULTI-LINEAR SYSTEMS WITH M-TENSOR

Shao Xinhui, Qi Meng

College of Science, Northeastern University, Shenyang 110819, China

Received:2020-11-26 Online:2022-05-14 Published:2022-05-06

多重线性系统在当今的工程计算和数据挖掘等领域有很多实际应用,许多问题可以转化为多重线性系统求解问题.在本文中,我们首先提出了一种新的迭代算法来求解系数张量为M-张量的多重线性系统,在此基础上又提出了一种新的改进算法,并对两种算法的收敛性进行了分析.数值算例的结果表明,本文提出的两种算法是有效的并且改进算法的迭代时间更少.

关键词： M-张量, 多重线性系统, 迭代方法, 新算法, 收敛

In engineering and science fields, some problems can be transformed into multi-linear system problems. We propose a new iteration algorithm to solve multi-linear systems with M-tensor. But this algorithm requires solving polynomial functions. For this reason, we give a simplified algorithm to improve. Then we give the convergence analysis of two algorithms. The results of numerical examples show that algorithms we propose are more effective.

Key words: M-tensor, multi-linear systems, iteration method, new algorithms, convergence

MR(2010)主题分类:

65F10
65H10

分享此文:

[1] Qi L Q, Luo Z Y. Tensor Analysis:Spectral Theory and Special Tensors[M]. Philadelphia:SIAM, 2017.
[2] Ding W, Wei Y. Theory and Computation of Tensors[M]. London:Academic Press, 2016.
[3] Ding W Y, Wei Y M. Solving multi-linear systems with M-tensors[J]. Sci. Comput., 2016, 68(2):689-715.
[4] Luo Z, Qi L, Xiu N. The sparsest solutions to Z-tensor complementarity problems[J]. Optimi. Lett., 2017, 11:471-482.
[5] Li X T, Michael K NG. Solving sparse non-negative tensor equations:algorithms and applications[J]. Front. Math. China, 2015, 10(3):649-680.
[6] Du S, Zhang L, Chen C, Qi L. Tensor absolute value equations[J]. Sci. China Math., 2018, 61(9):1695-1710.
[7] Xie Z J, Jin X Q, Wei Y M. Tensor methods for solving symmetric-tensor systems[J]. Sci. Comput., 2018, 74(1):412-425.
[8] Liang M L, Zheng B, Zhao R J. Alternating iterative methods for solving tensor equations with applications[J]. Numerical Algorithms, 2019, 80:1437-1465.
[9] He H, Ling C, Qi L, Zhou G. A globally and quadratically convergent algorithm for solving multilinear systems with M-tensors[J]. J. Scient. Comput., 2018, 76(3):1718-1741.
[10] Han L. A homotopy method for solving multilinear systems with M-tensors[J]. Appl. Math. Lett., 2017, 69(Supplement C):49-54.
[11] Wang X Z, Che M L, Wei Y M. Neural networks based approach solving multi-linear systems with M-tensors[J]. Neurocomputing on ScienceDirect, 2019, 351:33-42.
[12] Liu D D, Li W, Seak-WengVonga. The tensor splitting with application to solve multi-linear systems[J]. Journal of Computational and Applied Mathematics, 2018, 330:75-94.
[13] Liu D D, Li W, Seak-WengVonga. Comparison results for splitting iterations for solving multilinear systems[J]. Applied Numerical Mathematics, 2018, 134:105-121.
[14] Cui L B, Li M H, Song Y S. Preconditioned tensor splitting iterations method for solving multilinear systems[J]. Applied Mathematics Letters, 2019, 96:89-94.
[15] Kelly J. Pearson. Essentially Positive Tensors[J]. International Journal of Algebra, 2010, 4(9):421-427.
[16] Qi L Q. Eigenvalues of a real supersymmetric tensor[J]. Journal of Symbolic Computation, 2005, 40:1302-1324.
[17] Niki H, Harada K, Morimoto M, Sakakihara M. The survey of preconditioners used for accelerating the rate of convergence in the Gauss-Seidel method[J]. Journal of Computational and Applied Mathematics, 2004, 164-165:587-600.
[18] Kotakemori H, Niki H, Okamotob N. Accelerated iterative method for Z-matrices[J]. Journal of Computational and Applied Mathematics, 1996, 75:87-97.
[19] Ding W Y, Qi L Q, Wei Y M. M-tensors and nonsingular M-tensors[J]. Linear Algebra and its Applications, 2013, 439:3264-3278.
[20] Chang K C, Pearson K, Zhang T. Perron-Frobenius Theorem for nonnegative tensors[J]. Communications in mathematical sciences, 2008, 6(2):507-520.
[21] Golub G H, Van Loan C F. Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013).
[22] Wang X, Che M and Wei Y. Existence and uniqueness of positive solution for H+-tensor equations[J]. Appl. Math. Lett., 2019, 98:191-198.
[23] Wang X, Che M and Wei Y. Neural network approach for solving nonsingular multi-linear tensor systems[J]. J. Comput. Appl. Math., 2020, 368:e112569.
[24] Wang X, Che M and Wei Y. Preconditioned tensor splitting AOR iterative met-hods for H-tensor equations[J]. Numer Linear Algebra Appl., 2020, 27:e2329.
[25] Wang X. A simple proof of Descartes' rule of signs[J]. Amer. Math. Monthly, 2004, 111(6):525- 526.
[26] Albert G E. An inductive proof of Descartes' rule of signs[J]. Amer. Math. Monthly, 1943, 50(3):178-180.

[1]	李步扬. 曲率流的参数化有限元逼近[J]. 计算数学, 2022, 44(2): 145-162.
[2]	杨旭, 赵卫东. 非全局Lipschitz条件下跳适应向后Euler方法的强收敛性分析[J]. 计算数学, 2022, 44(2): 163-177.
[3]	宋珊珊, 李郴良. 求解张量互补问题的一类光滑模系矩阵迭代方法[J]. 计算数学, 2022, 44(2): 178-186.
[4]	杨学敏, 牛晶, 姚春华. 椭圆型界面问题的破裂再生核方法[J]. 计算数学, 2022, 44(2): 217-232.
[5]	罗兴钧, 江伟娟, 张荣. 求解非定常Lavrentiev迭代方程的多尺度配置法[J]. 计算数学, 2022, 44(2): 257-271.
[6]	马玉敏, 蔡邢菊. 求解带线性约束的凸优化的一类自适应不定线性化增广拉格朗日方法[J]. 计算数学, 2022, 44(2): 272-288.
[7]	余妍妍, 代新杰, 肖爱国. 非自治刚性随机微分方程正则EM分裂方法的收敛性和稳定性[J]. 计算数学, 2022, 44(1): 19-33.
[8]	米玲, 薛文娟, 沈春根. 球面上$\ell_1$正则优化的随机临近梯度方法[J]. 计算数学, 2022, 44(1): 34-62.
[9]	邵新慧, 亢重博. 基于分数阶扩散方程的离散线性代数方程组迭代方法研究[J]. 计算数学, 2022, 44(1): 107-118.
[10]	古振东. 非线性弱奇性Volterra积分方程的谱配置法[J]. 计算数学, 2021, 43(4): 426-443.
[11]	唐耀宗, 杨庆之. 非线性特征值问题平移对称幂法的收敛率估计[J]. 计算数学, 2021, 43(4): 529-538.
[12]	魏水艳, 陈小山. 逆奇异值问题的一个二阶收敛算法[J]. 计算数学, 2021, 43(4): 471-483.
[13]	唐跃龙, 华玉春. 椭圆最优控制问题分裂正定混合有限元方法的超收敛性分析[J]. 计算数学, 2021, 43(4): 506-515.
[14]	王艺宏, 李耀堂. 一类特征值反问题(IEP)的基于矩阵方程的Ulm型算法[J]. 计算数学, 2021, 43(4): 444-456.
[15]	包学忠, 胡琳. 随机变延迟微分方程平衡方法的均方收敛性与稳定性[J]. 计算数学, 2021, 43(3): 301-321.

阅读次数

全文

摘要

求解M-张量方程的两种新型算法

TWO NEW ALGORITHMS FOR SOLVING MULTI-LINEAR SYSTEMS WITH M-TENSOR

扫码分享