A Simplified Architecture of the Zhang Neural Network for Toeplitz Linear Systems Solving

Abstract

In this paper, we derive a compact implementation of the discrete time Zhang neural network for computing the solution of an $n\times n$ Toeplitz linear system. In the proposed scheme, the multiplication of the weight matrix by the input vector is performed efficiently in $O(n\log n)$ operations using the fast Fourier transform instead of $O(n^{2})$ in the original Zhang neural network. We have investigated also the use of the proposed neural network to compute the Toeplitz matrices inversion; it consists of dividing the system into n Toeplitz sub-systems requiring n independent subnets similar to the one used for solving the Toeplitz system. In another way, the computation of the first and the last columns of the matrix inverse is sufficient to compute the matrix inverse via the Gohberg–Semencul formulas, in this case only two subnets are needed to define all elements of the matrix inverse.