Modified discrete iterations for computing the inverse and pseudoinverse of the time-varying matrix

Abstract The general discretization scheme for transforming continuous-time ZNN models for matrix inversion and pseudoinversion into corresponding discrete-time iterative methods is developed and investigated. The proposed discrete-time ZNN models incorporate scaled Hyperpower iterative methods as well as the Newton iteration in certain cases. The general linear Multi-step method is applied in order to obtain the proposed discretization rule which comprises all previously proposed discretization schemes. Both the Euler difference rule and the Taylor-type difference rules are included in the general scheme. In particular, the iterative scheme based on the 4th order Adams–Bashforth method is proposed and numerically compared with other known iterative schemes. In addition, the ZNN model for computing the time-varying matrix inverse is extended to the singular or rectangular case for the pseudoinverse computation. Convergence properties of the continuous-time ZNN model in the case of the Moore–Penrose inverse and its discretization are also considered.

[1]  Igor Stojanovic,et al.  ZNN models for computing matrix inverse based on hyperpower iterative methods , 2017 .

[2]  Jian Li,et al.  Enhanced discrete-time Zhang neural network for time-variant matrix inversion in the presence of bias noises , 2016, Neurocomputing.

[3]  Long Jin,et al.  Discrete-time Zhang neural network of O(τ3) pattern for time-varying matrix pseudoinversion with application to manipulator motion generation , 2014, Neurocomputing.

[4]  Weiguo Li,et al.  A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix , 2010, Appl. Math. Comput..

[5]  Predrag S. Stanimirovic,et al.  Iterative Method for Computing Moore-penrose Inverse Based on Penrose Equations , 2022 .

[6]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[7]  Y. Wei,et al.  The representation and approximations of outer generalized inverses , 2004 .

[8]  Joan-Josep Climent,et al.  A geometrical approach on generalized inverses by Neumann-type series , 2001 .

[9]  Ying Wang,et al.  Different ZFs Leading to Various ZNN Models Illustrated via Online Solution of Time-Varying Underdetermined Systems of Linear Equations with Robotic Application , 2013, ISNN.

[10]  Shuai Li,et al.  Zeroing neural networks: A survey , 2017, Neurocomputing.

[11]  Shuzhi Sam Ge,et al.  Design and analysis of a general recurrent neural network model for time-varying matrix inversion , 2005, IEEE Transactions on Neural Networks.

[12]  Hebing Wu,et al.  The representation and approximation for the generalized inverse AT, S(2) , 2003, Appl. Math. Comput..

[13]  Xiaoji Liu,et al.  Higher-order convergent iterative method for computing the generalized inverse and its application to Toeplitz matrices , 2013 .

[14]  Yunong Zhang,et al.  Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[15]  Predrag S. Stanimirovic,et al.  Complex ZFs for computing time-varying complex outer inverses , 2018, Neurocomputing.

[16]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[17]  Predrag S. Stanimirovic,et al.  Two improvements of the iterative method for computing Moore-Penrose inverse based on Penrose equations , 2014, J. Comput. Appl. Math..

[18]  Shuai Li,et al.  RNN Models for Dynamic Matrix Inversion: A Control-Theoretical Perspective , 2018, IEEE Transactions on Industrial Informatics.

[19]  Dongsheng Guo,et al.  Novel Discrete-Time Zhang Neural Network for Time-Varying Matrix Inversion , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[20]  Predrag S. Stanimirovic,et al.  A class of numerical algorithms for computing outer inverses , 2014, J. Comput. Appl. Math..

[21]  Long Jin,et al.  Taylor-type 1-step-ahead numerical differentiation rule for first-order derivative approximation and ZNN discretization , 2015, J. Comput. Appl. Math..

[22]  Long Jin,et al.  Three nonlinearly-activated discrete-Time ZNN models for time-varying matrix inversion , 2012, 2012 8th International Conference on Natural Computation.

[23]  Ning Tan,et al.  Zhang neural network solving for time-varying full-rank matrix Moore–Penrose inverse , 2010, Computing.

[24]  Dongsheng Guo,et al.  Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots' kinematic control , 2012, Neurocomputing.

[25]  Are Hjørungnes,et al.  Complex-Valued Matrix Differentiation: Techniques and Key Results , 2007, IEEE Transactions on Signal Processing.

[26]  Binghuang Cai,et al.  From Zhang Neural Network to Newton Iteration for Matrix Inversion , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[27]  Yunong Zhang,et al.  On the Variable Step-Size of Discrete-Time Zhang Neural Network and Newton Iteration for Constant Matrix Inversion , 2008, 2008 Second International Symposium on Intelligent Information Technology Application.

[28]  Yunong Zhang,et al.  From different ZFs to different ZNN models accelerated via Li activation functions to finite-time convergence for time-varying matrix pseudoinversion , 2014, Neurocomputing.

[29]  Liu Weiguo,et al.  A family of iterative methods for computing Moore–Penrose inverse of a matrix , 2013 .

[30]  Gene H. Golub,et al.  The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate , 1972, Milestones in Matrix Computation.