Theoretical analysis, numerical verification and geometrical representation of new three-step DTZD algorithm for time-varying nonlinear equations solving
Introduction
The problem of solving nonlinear equations (e.g., square root and inverse square root [1], [2], [3]) is considered to be one of the fundamental issues encountered in numerous fields of science and engineering. Lots of numerical algorithms have thus been investigated for solving nonlinear equations in the form of [4], [5], [6], [7], [8], [9], [10], [11]. For example, an iterative method was studied by Frontini [4] for computing the roots of nonlinear equations. In [5], Sharma presented a composite third order Newton–Steffensen method to solve nonlinear equations. Besides, Neta et al. developed various eighth order methods for nonlinear equations solving [10]. Note that the approaches based on neural dynamics have been considered as a powerful alternative to problems solving [12], [13], [14], [15]. Thus, being different from the aforementioned numerical algorithms, the neural-dynamics methods have been also developed to solve nonlinear equations [9], [16], [17], [18], [19], [20], [21], [22]. For example, a primal neural network was investigated by Zhang [16] to solve a set of nonlinear equations and inequalities. In [17], [18], Zhang et al. presented a method based on gradient dynamics (GD) for nonlinear equations solving. Besides, Xiao and Lu [21] developed a finite-time convergent neural dynamics to find the roots of nonlinear equations. However, most of the reported methods are designed and developed intrinsically for time-invariant (or say, static) nonlinear equations solving, and the time-derivative information (or say, the change trend) of the time-varying coefficients is not involved. Due to the lack of the consideration of such an important information, these methods may be less effective, when they are exploited directly to solve time-varying nonlinear equations [9], [19], [20], [22].
Since 2008, a special class of neural dynamics, i.e., Zhang dynamics (ZD), has been formally proposed for solving various time-varying problems [19], [23]. As for such a ZD, its state dimension can be multiple or one. It is viewed as a systematic approach to the solution of time-varying problems with scalar situation included as well. It is different from the conventional GD in terms of the problem to be solved, indefinite error function, exponent-type design formula, dynamic equation and the utilization of time-derivative information. Specifical for time-varying nonlinear equations solving, a continuous-time ZD (CTZD) model was developed and investigated in [20]. The related theoretical and simulative results substantiated the efficacy of such a CTZD model. In addition, for the purposes of hardware implementation and numerical algorithm development [24], [25], [26], [27], [28], the discrete-time form of the CTZD model was studied in [9]. Specifically, in [9], by using the Euler-type difference rule, the one-step discrete-time ZD (DTZD) algorithm was developed to solve time-varying nonlinear equations. Both theoretical and numerical results substantiated the efficacy of such a one-step DTZD algorithm, and further indicated that the steady-state residual error (SSRE) of the one-step DTZD algorithm is of order [9], where τ denotes the sampling gap.
On the basis of the previous work [9], [19], [29], a new DTZD algorithm with higher computational precision is developed in this paper for time-varying nonlinear equations solving. Specifically, by exploiting the Taylor-type difference rule [29], the three-step DTZD algorithm is proposed and investigated, of which the SSRE is of order . Evidently, such an algorithm can achieve better computational performance than the one-step DTZD algorithm. In addition, theoretical analysis and results are given to show the excellent computational property of the proposed three-step DTZD algorithm. Comparative numerical results with two illustrative examples are further presented to substantiate the efficacy and superiority of the proposed three-step DTZD algorithm, as compared with the one-step DTZD algorithm. Besides, as for the proposed three-step DTZD algorithm, its geometric representation is given for time-varying nonlinear equations solving.
The rest of this paper is organized into five sections. Section 2 presents the problem formulation and investigates the CTZD model and the one-step DTZD algorithm for time-varying nonlinear equations solving. In Section 3, the three-step DTZD algorithm is presented and investigated, together with the corresponding theoretical analysis and results. Section 4 shows the comparative numerical results synthesized by the one-step and three-step DTZD algorithms. In Section 5, the geometric representation of the three-step DTZD algorithm is provided. Section 6 concludes this paper with final remarks. Before ending this section, it is worth summarizing and listing the main contributions of this paper as follows.
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In this paper, the three-step DTZD algorithm is proposed and investigated for time-varying nonlinear equations solving. To the best of the authors' knowledge, such a DTZD algorithm has not been investigated in existing literature.
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This paper presents theoretical analysis and results, which show the excellent computational property of the proposed three-step DTZD algorithm. In addition, numerical results substantiate the efficacy of such an algorithm for time-varying nonlinear equations solving.
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Both theoretical and numerical results indicate that the SSRE of the proposed three-step DTZD algorithm is of order . This is the first time to provide a numerical algorithm with error pattern to solve time-varying nonlinear equations.
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In this paper, the geometric representation of the proposed three-step DTZD algorithm is given for time-varying nonlinear equations solving. It is an important breakthrough in ZD research concerning time-varying nonlinear equations solving, in view of that such a geometric representation has never been investigated in the previous work [9], [19], [22].
Section snippets
Time-varying nonlinear equations
In this section, the results in the existing work [9], [19] are summarized and provided to lay a basis for further discussion. Specifically, based on the previous work [9], [19], the CTZD model and the one-step DTZD algorithm are presented and investigated for time-varying nonlinear equations solving.
Three-step DTZD algorithm
In this section, by exploiting the Taylor-type difference rule [29], the three-step DTZD algorithm [which is quite different from the presented one-step DTZD algorithm (3)] is proposed and investigated for time-varying nonlinear equations solving. Theoretical analysis and results are also presented to show the computational property of such a three-step DTZNN algorithm.
Numerical experiments and comparisons
In the previous section, the three-step DTZD algorithm (4) and its computational property have been proposed and investigated for solving the time-varying nonlinear equation (1). In this section, numerical experiments and comparative results are illustrated to substantiate the efficacy and superiority of the proposed three-step DTZD algorithm (4), as compared with the presented one-step DTZD algorithm (3). Example 4.1 In this example, the time-varying nonlinear equation is presented as follows:
Geometric representation
In this section, the geometric representation of the proposed three-step DTZD algorithm (4) for time-varying nonlinear equations solving is provided, which is illustrated in Fig. 8.
As shown in Fig. 8(a), at time instant tk, and are obtained, which yields the corresponding gradient . In general, such a gradient is exploited as the search direction to find the theoretical solution, and results in the gradient-descent approach [9], [17], [18], [19], [20], [21], [22]. In
Conclusions
By exploiting the Taylor-type difference rule, this paper has proposed and investigated the three-step DTZD algorithm (4) for time-varying nonlinear equations solving. Note that such an algorithm can achieve better computational performance than the one-step DTZD algorithm (3) presented in the previous work [9]. As for the proposed three-step DTZD algorithm (4), theoretical analysis and results have been given to show its excellent computational property. Comparative numerical results have
Acknowledgments
This work is supported by the Natural Science Foundation of Fujian Province (with number 2016J01307), the National Natural Science Foundation of China (with number 61403149 and 61572534), and the Scientific Research Funds of Huaqiao University (with number 15BS410). Besides, the authors would like to thank the editors and anonymous reviewers for their time and effort spent handling this paper, as well as for providing constructive comments to further improve the presentation and quality of this
Dongsheng Guo received the B.S. degree in Automation from Sun Yat-sen University, Guangzhou, China, in 2010, and the Ph.D. degree in Communication and Information Systems at School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China, in 2015. He was the recipient of the 2012 Google Excellence Scholarship, Shanghai, China, on August 16, 2012, as well as the Academic Award for Excellent Doctoral Student granted by the Ministry of Education of China, Guangzhou, on
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Dongsheng Guo received the B.S. degree in Automation from Sun Yat-sen University, Guangzhou, China, in 2010, and the Ph.D. degree in Communication and Information Systems at School of Information Science and Technology, Sun Yat-sen University, Guangzhou, China, in 2015. He was the recipient of the 2012 Google Excellence Scholarship, Shanghai, China, on August 16, 2012, as well as the Academic Award for Excellent Doctoral Student granted by the Ministry of Education of China, Guangzhou, on January 7, 2013. He is currently an Associate Professor with the College of Information Science and Engineering, Huaqiao University, Xiamen. Before joining Huaqiao University in 2015, he has continued the research work under his supervision since 2008. His main research interests include neural networks, numerical methods, and robotics control.
Zhuoyun Nie received the B.S. and Ph.D. degrees in Automation from Central South University. He was awarded the Excellent Doctoral Student granted by the Ministry of Education of China. He was the visiting student with the National University of Singapore, Singapore, from 2007 to 2009. He is currently a lecturer with the College of Information Science and Engineering, Huaqiao University, Xiamen. His research interests include neural networks, robust control, nonlinear control, and financial forecasting.
Laicheng Yan received the B.S degrees in Electrical Engineering and Automation from Chongqing Communication College, Chongqing, China, in 2004, and the M.S degree in Electrical engineering at School of Electrical Engineering, Chongqing University, Chongqing, China, in 2007. He is currently a lecturer with the College of Information Science and Engineering, Huaqiao University, Xiamen. His research interests include neural networks, machine vision and machine learning.