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计算张量指数函数的广义逆张量ε-算法

顾传青 唐鹏飞 陈之兵

顾传青, 唐鹏飞, 陈之兵. 计算张量指数函数的广义逆张量ε-算法. 自动化学报, 2020, 46(4): 744-751. doi: 10.16383/j.aas.c180002
引用本文: 顾传青, 唐鹏飞, 陈之兵. 计算张量指数函数的广义逆张量ε-算法. 自动化学报, 2020, 46(4): 744-751. doi: 10.16383/j.aas.c180002
GU Chuan-Qing, TANG Peng-Fei, CHEN Zhi-Bing. Generalized Inverse Tensor ε-algorithom for Computing Tensor Exponential Function. ACTA AUTOMATICA SINICA, 2020, 46(4): 744-751. doi: 10.16383/j.aas.c180002
Citation: GU Chuan-Qing, TANG Peng-Fei, CHEN Zhi-Bing. Generalized Inverse Tensor ε-algorithom for Computing Tensor Exponential Function. ACTA AUTOMATICA SINICA, 2020, 46(4): 744-751. doi: 10.16383/j.aas.c180002

计算张量指数函数的广义逆张量ε-算法

doi: 10.16383/j.aas.c180002
基金项目: 

国家自然科学基金 11371243

上海市重点学科建设资助项目 S30104

详细信息
    作者简介:

    唐鹏飞  上海大学数学系硕士研究生.主要研究方向为张量Padé逼近及其在控制论中应用.E-mail: soccer.fly@163.com

    陈之兵  深圳大学数学与统计学院教授.主要研究方向为矩阵计算, 数值逼近及其应用. E-mail: chenzb@szu.edu.cn

    通讯作者:

    顾传青   上海大学数学系教授.主要研究方向为数值代数, 张量与矩阵计算, 有理逼近及其在控制论应用.本文通信作者. E-mail: cqgu@shu.edu.cn

  • 本文责任编委 黎铭

Generalized Inverse Tensor ε-algorithom for Computing Tensor Exponential Function

Funds: 

National Natural Science Foundation of China 11371243

Key Disciplines of Shanghai Municipality S30104

More Information
    Author Bio:

    TANG Peng-Fei    Master student in the Department of Mathematics, Shanghai University. His research interest covers tensor Padé approximation and its application in control theory

    CHEN Zhi-Bing Professor at the School of Mathematics and Statistics, Shenzhen University. His research interest covers matrix calculation, numerical approximation and its application

    Corresponding author: GU Chuan-Qing    Professor in the Department of Mathematics, Shanghai University. His research interest covers numerical algebra, tensor and matrix calculation, rational approximation and its application in control theory. Corresponding author of this paper
  • Recommended by Associate Editor LI Ming
  • 摘要: 张量指数函数已经广泛应用于工程领域.本文得到了一种有效的张量广义逆, 并以此为基础构造了广义逆张量Padé逼近的一种$\varepsilon$-算法.该算法可以编程实施递推的计算, 其特点是, 在计算过程中, 不必计算张量的乘积, 也不必计算张量的逆.给出的计算张量指数函数的数值实验显示, 将本文的方法与目前通常使用的截断法进行比较, 在不降低逼近阶的条件下, $\varepsilon$-算法能很好地降低计算复杂度, 尤其是在张量的维数比较大的时候.
    Recommended by Associate Editor LI Ming
    1)  本文责任编委 黎铭
  • 图  1  不同维数下两种算法运行时间直方图(s)

    Fig.  1  The time-consuming comparison histogram of two algorithms in different dimensions (s)

    表  1  $[\frac{4}{4}]$型GITPA-算法数值实验

    Table  1  The numerical experiment of $[\frac{4}{4}]$ type GITPA-algorithm

    $x$ $(1, 2, 1)$ $(2, 2, 1)$ $(1, 2, 2)$ $(2, 2, 2)$ $ RES $
    0.2 $E$值 0.08766299 0.87955329 0.12044671 -0.08766299 $5.69\times10^{-13}$
    $A$值 0.08766327 0.87955283 0.12044717 -0.08766327
    0.4 $E$值 0.15420167 0.78130960 0.21869040 -0.15420167 $3.74\times10^{-10}$
    $A$值 0.15420895 0.78129804 0.21870196 -0.15420895
    0.6 $E$值 0.20408121 0.70078192 0.29921808 -0.20408121 $1.40\times10^{-8}$
    $A$值 0.20412606 0.70071136 0.29928864 -0.20412606
    0.8 $E$值 0.24081224 0.63444735 0.36555265 -0.24081224 $1.63\times10^{-7}$
    $A$值 0.24096630 0.63420702 0.36579298 -0.24096630
    1 $E$值 0.26715410 0.57953894 0.42046106 -0.26715410 $1.01\times10^{-6}$
    $A$值 0.26753925 0.57894247 0.42105753 -0.26753925
    下载: 导出CSV

    表  2  算法1和算法2的数值实验比较

    Table  2  The numerical experiment comparison of Algorithm 1 and Algorithm 2

    $[\frac{j+2k}{2k}]$ 张量$\varepsilon$-算法 $n_{\rm{max}}$ $\sum^{n_{\rm{max}}}_{n=0}\frac{1}{n!}\mathcal{A}^nt^n $
    $a_{121}$ $a_{221}$ $a_{122}$ $a_{222}$ $a_{121}$ $a_{221}$ $a_{122}$ $a_{222}$
    $[\frac{2}{2}]$ 0.4235 0.3513 0.6487 -0.4235 1 1.0000 -0.3333 1.3333 -1.0000
    $[\frac{4}{4}]$ 0.3049 0.4141 0.5859 -0.3049 2 -0.3333 1.0556 -0.0556 0.3333
    $[\frac{6}{6}]$ 0.3098 0.4068 0.5932 -0.3098 3 0.7222 -0.0062 1.0062 -0.7222
    - - - - 4 0.1049 0.6116 0.3884 -0.1049
    - - - - 5 0.3931 0.3234 0.6766 -0.3931
    - - - - 6 0.2810 0.4355 0.5645 -0.2810
    - - - - 7 0.3184 0.3981 0.6019 -0.3184
    - - - - 8 0.3075 0.4090 0.5910 -0.3075
    - - - - 9 0.3103 0.4062 0.5938 -0.3103
    - - - - 10 0.3097 0.4069 0.5931 -0.3097
    - - - - 11 0.3098 0.4067 0.5933 -0.3098
    - - - - 12 0.3098 0.4068 0.5932 -0.3098
    下载: 导出CSV

    表  3  算法1和算法2的计算复杂度分析

    Table  3  The analysis of computational complexity of Algorithm 1 and Algorithm 2

    计算一次 $\varepsilon$-算法: $[\frac{6}{6}]$型 截断法:取13项
    复杂度 计算 计算
    $t$-积运算 $l^3n^2$ 5 11
    数乘运算 $l^2n$ 21 12
    范数运算 $l^2n$ 21 0
    总和 $5l^3n^2+42l^2n$ $11l^3n^2+12l^2n$
    下载: 导出CSV

    表  4  不同维数下两种算法的运行时间表(s)

    Table  4  The consuming time of two algorithms in different dimensions (s)

    张量维数 张量$\varepsilon$-算法运行时间 截断法运行时间
    $ 3~\times ~3\times ~3$ 1.755262 1.103832
    $10\times 10\times 10$ 2.272663 2.164860
    $20\times 20\times 20$ 3.831094 4.805505
    $30\times 30\times 30$ 6.419004 10.545785
    $40\times 40\times 40$ 15.814063 30.012744
    下载: 导出CSV
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  • 收稿日期:  2018-01-03
  • 录用日期:  2018-04-06
  • 刊出日期:  2020-04-24

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