引用本文:顾传青,黄逸铮,陈之兵.广义逆张量Padé逼近的连分式递推算法[J].控制与决策,2019,34(8):1702-1708
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广义逆张量Padé逼近的连分式递推算法
顾传青1, 黄逸铮1, 陈之兵2
(1. 上海大学理学院,上海200444;2. 深圳大学数学与统计学院,广东深圳518052)
摘要:
张量指数函数已经广泛应用于控制论、图像处理和各个工程领域.鉴于此,在矩阵广义逆的基础上,首次在张量内积空间上定义一种有效的张量广义逆,从而构造张量Padé逼近的一种连分式算法.利用张量t-积成功计算张量的幂,由此递推地给出张量指数函数的幂级数展开式.在前面两个工作的基础上,利用设计的连分式算法逼近张量指数函数,其特点在于,该算法可以编程实现递推计算,而且在计算过程中不必计算张量的乘积,也不必计算张量的逆.给出的两个张量指数函数的数值实验表明,将连分式算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,所提出算法是有效的.如果张量的维数较大,基于张量广义逆的连分式算法仍然具有一定优势.
关键词:  张量  张量t-积  张量广义逆  张量Padé逼近  张量指数函数  张量连分式算法
DOI:10.13195/j.kzyjc.2018.0019
分类号:O231.1
基金项目:国家自然科学基金项目(11371243);上海市重点学科建设项目(S30104).
A continued fractional recurrence algorithm for generalized inverse tensor Padé approximation
GU Chuan-qing1,HUANG Yi-zheng1,CHEN Zhi-bing2
(1. College of Sciences,Shanghai University,Shanghai200444,China;2. School of Mathematics and Statistics,Shenzhen University,Shenzhen518052,China)
Abstract:
The tensor exponential function has been widely used in cybernetics, image processing and various engineering fields. Based on the generalized matrix inverse, an effective tensor generalized inverse is defined for the first time on the scalar inner product space, thus constructing a continued fractional algorithm for the tensor Pad$\acutee$ approximation. On the other hand, we successfully use the tensor t- product to calculate the power of the tensor, and recursively giving the power series expansion of the tensor exponential function. Based on the previous two work, the continuous fractional algorithm designed in this paper is used to approximate the tensor exponential function. Its characteristic is that the algorithm can be programmed to implement recursive calculations, and in the calculation process, it is not necessary to calculate the product of the tensor and to calculate the inverse of the tensor. The numerical experiments of the two tensor exponential functions given in this paper show that comparing the continuous fractional algorithm with the commonly used truncation method, the proposed algorithm is effective without reducing the approximation order. If the dimension of the tensor is relatively large, a continuous fractional algorithm based on the generalized inverse of tensors will also have certain advantages.
Key words:  tensor  tensor t-product  tensor generalized inverse  tensor Padé approximation  tensor exponential functions  continued fraction expression algorithm

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