Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation Xp = Q±A(X−1+B)−1AT

IF 1.5 4区 工程技术 Q3 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS Engineering Computations Pub Date : 2023-11-07 DOI:10.1108/ec-07-2023-0322
Raziyeh Erfanifar, Masoud Hajarian
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Abstract

Purpose In this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory. Design/methodology/approach The authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations. Findings The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices . Originality/value Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.
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基于牛顿法和不动点迭代求解非线性矩阵方程Xp = Q±A(X−1+B)−1AT的高效迭代方案
本文研究了非线性矩阵方程Xp=Q±A(X-1+B)-1AT,该方程在滤波、网络系统、最优控制和控制理论等领域有广泛的应用。本文给出了该非线性矩阵方程解的存在性的一些理论结果。基于牛顿法和不动点迭代,提出了求解非线性矩阵方程的两种无反演迭代方案。在一定条件下,所提出的迭代格式收敛于非线性矩阵方程的解。结果给出了所提方案的效率指标,并且由于所提迭代方案的初始猜测成本较高,作者通过改变初始猜测来降低其成本。因此,与以前的方案相比,所提出的方案具有更高的效率指标。最后,通过数值算例验证了所提方案与现有方案的准确性和有效性。此外,作为应用,利用所提出的格式,作者可以得到最优控制器状态反馈$x(t+1) = A x(t) + C v(t)$。
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来源期刊
Engineering Computations
Engineering Computations 工程技术-工程:综合
CiteScore
3.40
自引率
6.20%
发文量
61
审稿时长
5 months
期刊介绍: The journal presents its readers with broad coverage across all branches of engineering and science of the latest development and application of new solution algorithms, innovative numerical methods and/or solution techniques directed at the utilization of computational methods in engineering analysis, engineering design and practice. For more information visit: http://www.emeraldgrouppublishing.com/ec.htm
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