Reciprocal Zhang Dynamics (RZD) Handling TVUDLMVE (Time-Varying Under-Determined Linear Matrix-Vector Equation)

Due to the wide application of time-varying underdetermined linear matrix-vector equation (TVUDLMVE), solving it has been a great important topic in both scientific and engineering fields. As results, a lot of algorithms have been developed for solving the TVUDLMVE. Some algorithms originally proposed for solving static problems have also been applied to solving time-varying problems. For example, traditional methods, such as Gaussian elimination method, Newton method, and quasiNewton method, can also be used to solve the TVUDLMVE, but they introduce large lag errors. With the rapid development of neural dynamics, some dynamics methods in the form of recurrent neural dynamics, such as gradient dynamics (GD), has been presented to solve the TVUDLMVE. However, those methods do not eliminate large lag error either. Afterwards, Zhang et al proposed Zhang dynamics (ZD) and the gradient Zhang dynamics (GZD) successively, which are different from both traditional methods and the GD, in solving time-varying problems. These two new kinds of dynamics completely (or almost completely) eliminate lag errors theoretically. Subsequently, Zhang et al proposed the reciprocal Zhang dynamics (RZD) to avoid computing the pseudoinverse of a coefficient matrix. In this paper, we present a continuous RZD model for solving the TVUDLMVE. Two different time-discretization algorithms are obtained by discretizing the continuous GD model and the continuous RZD model with Euler forward discretization formula. Besides, we conduct some numerical experiments to verify the effectiveness of our time-discretization algorithms, and compare the convergence rate and precision of GD and RZD algorithms. Finally, some discussions of experimental results are presented.