Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition

Kornilova, Mariya I.; Kovalnogov, Vladislav N.; Fedorov, Ruslan V.; Zamaleev, M. M.; Katsikis, Vasilios N.; Mourtas, Spyridon D.; Simos, Theodore E.

doi:10.3390/math10081208

Cited by 19 publications

(5 citation statements)

References 24 publications

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“…From Figure 1, it can be seen that the relative residual norm of CGS2,SCGS, GCGS CGS, and GCGS2 methods shows irregular convergence behavior, while the relative residual norm of the QMRGCGS2 method tends to show regular convergence behavior. Tat is to say, the QMRGCGS2 method has smoother convergence behavior than the CGS method and GCGS2 method [41][42][43]. Finally, CGS2 and CGS methods with good performance are selected to compare their performance with the proposed QMRGCGS2 method for practical application problems.…”

Section: Applicationmentioning

confidence: 99%

Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

2022

Journal of Mathematics

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The complexity of open linear algebraic equations makes it difficult to obtain analytical solutions, and preprocessing techniques can be applied to coefficient matrices, which has become an effective method to accelerate the convergence of iterative methods. Therefore, it is important to preprocess the structure of open linear algebraic equations to reduce their complexity. Open linear algebraic equations can be divided into symmetric linear equations and asymmetric linear equations. The former is based on 2 × 2. The latter is preprocessed by the improved QMRGCGS method, and the applications of the two methods are analyzed, respectively. The experimental results show that when the step is 500, the pretreatment time of quasi-minimal residual generalized conjugate gradient square 2 method is 34.23 s, that of conjugate gradient square 2 method is 35.14 s, and that of conjugate gradient square method is 45.20 s, providing a new reference method and idea for solving and preprocessing non-closed linear algebraic equations.

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Section: Applicationmentioning

confidence: 99%

Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

2022

Journal of Mathematics

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“…Today their use has expanded to include the resolution of generalized inversion issues, including time-varying Drazin inverse [33], time-varying ML-weighted pseudoinverse [34], time-varying outer inverse [35], time-varying pseudoinverse [36], and core and core-EP inverse [37]. Their use has expanded to include the resolution of linear programming tasks [38], quadratic programming tasks [39,40], systems of nonlinear equations [41,42], and systems of linear equations [43,44]. The creation of a ZNN model typically involves two fundamental steps.…”

Section: Introductionmentioning

confidence: 99%

Towards Higher-Order Zeroing Neural Networks for Calculating Quaternion Matrix Inverse with Application to Robotic Motion Tracking

et al. 2023

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The efficient solution of the time-varying quaternion matrix inverse (TVQ-INV) is a challenging but crucial topic due to the significance of quaternions in many disciplines, including physics, engineering, and computer science. The main goal of this research is to employ the higher-order zeroing neural network (HZNN) strategy to address the TVQ-INV problem. HZNN is a family of zeroing neural network models that correlates to the hyperpower family of iterative methods with adjustable convergence order. Particularly, three novel HZNN models are created in order to solve the TVQ-INV both directly in the quaternion domain and indirectly in the complex and real domains. The noise-handling version of these models is also presented, and the performance of these models under various types of noises is theoretically and numerically tested. The effectiveness and practicality of these models are further supported by their use in robotic motion tracking. According to the principal results, each of these six models can solve the TVQ-INV effectively, and the HZNN strategy offers a faster convergence rate than the conventional zeroing neural network strategy.

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“…To overcome this shortcoming of numerical methods, recursive neural networks (RNNs) were further designed and studied. At present, RNNs were universally applied in practical engineering and application problems [9][10][11][12][13][14][15][16][17][18][19][20][21]. Additionally, RNNs have the characteristics of parallel distributed processing; hence, they have been extensively employed for solving the time-dependent Lyapunov equation (TDLE) [22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation

et al. 2022

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Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order 10−5 when solving CTDLE under complex linear noises, which is much lower than order 10−1 of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to 2∥A∥F/ζ3 quickly and stably, while the residual error of the NTZNN model is divergent.

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Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition

Cited by 19 publications

References 24 publications

Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

Structure Preprocessing Method for the System of Unclosed Linear Algebraic Equations

Towards Higher-Order Zeroing Neural Networks for Calculating Quaternion Matrix Inverse with Application to Robotic Motion Tracking

Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation

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