Time-Varying Matrix Eigenanalyses via Zhang Neural Networks and look-Ahead Finite Difference Equations

2019/04/23 by Frank Uhlig, Uhlig, Frank, Yunong Zhang +1 · 1 citation
Mathematics · Engineering · Physics and Astronomy · #Numerical methods for differential equations #Robotic Mechanisms and Dynamics #Model Reduction and Neural Networks

paper · pdf · doi:10.48550/arxiv.1904.10566

Abstract

This paper adapts look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying symmetric matrix flows A(t). It is based on the Zhang Neural Network (ZNN) model for time-varying problems and uses the associated error function E(t) = A(t)V(t) - V(t) D(t) or e_i(t) = A(t)v_i(t) -\la_i(t)v_i(t) with the Zhang design stipulation that E(t) = - ηE(t) or e_i(t) = - ηe_i(t) with η> 0 so that E(t) and e(t) decrease exponentially over time. This leads to a discrete-time differential equation of the form P(t_k) z(t_k) = q(t_k) for the eigendata vector z(t_k) of A(t_k). Convergent look-ahead finite difference formulas of varying error orders then allow us to express z(t_k+1) in terms of earlier A and z data. Numerical tests, comparisons and open questions complete the paper.

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