New zeroing neural dynamics models for diagonalization of symmetric matrix stream
- PDF / 8,865,435 Bytes
- 18 Pages / 439.642 x 666.49 pts Page_size
- 54 Downloads / 172 Views
New zeroing neural dynamics models for diagonalization of symmetric matrix stream Yunong Zhang1,2 · Huanchang Huang1,2 · Min Yang1,2 · Yihong Ling1,2 · Jian Li1,2 · Binbin Qiu1,2 Received: 25 September 2018 / Accepted: 1 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, the problem of diagonalizing a symmetric matrix stream (or say, timevarying matrix) is investigated. To fulfill our goal of diagonalization, two error functions are constructed. By making the error functions converge to zero with zeroing neural dynamics (ZND) design formulas, a continuous ZND model is established and its effectiveness is then substantiated by simulative results. Furthermore, a Zhang et al. discretization (ZeaD) formula with high precision is developed to discretize the continuous ZND model. Thus, a new 5-point discrete ZND (DZND) model is further proposed for diagonalization of matrix stream. Theoretical analyses prove the stability and convergence of the 5-point DZND model. In addition, simulative experiments are carried out, of which the results substantiate not only the efficacy of the proposed 5-point DZND model but also its higher computational precision as compared with the conventional Euler-type and 4-point DZND models for diagonalization of symmetric matrix stream. Keywords Matrix diagonalization · Symmetric matrix stream · Zeroing neural dynamics (ZND) · Discrete models · Simulative results
1 Introduction Matrix diagonalization, also called matrix eigen decomposition, is the operation of taking a square matrix and transforming it into a diagonal matrix, which is closely Binbin Qiu
[email protected] 1
School of Information Science and Technology, Sun Yat-sen University, Guangzhou 510006, People’s Republic of China
2
Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, Guangzhou 510006, People’s Republic of China
Numerical Algorithms
related to the original one [1–3]. Matrix diagonalization is equivalent to finding the matrix’s eigenvalues and eigenvectors, in which the former turn out to be the precise diagonal elements of the diagonalized matrix and the latter make up the new set of axes corresponding to the diagonal matrix. In other words, for matrix A and its diagonalized matrix Y , the interesting relationship between Y and the eigenvectors of A follows a mathematical description [1–3] that matrix A can be decomposed as A = XY X−1 , where the invertible matrix X consists of the eigenvectors of A, and diagonal matrix Y is constructed from the corresponding eigenvalues of A. As known, the eigenvalues and eigenvectors of a matrix are the keys to solve differential equations or study the properties of mechanical systems [4–6]. Therefore, diagonalized matrices play an extremely important role in many areas of science and engineering [1–3, 7, 8] because of their excellent properties for matrix operations and convenience for eigenvalue problems. For example, Lee et al. [7] employed matrix diagonalization to decompose a
Data Loading...
