Recurrent neural network for computation of generalized eigenvalue problem with real diagonalizable matrix pair and its applications

Abstract

We present neural networks to compute the left and right eigenvectors of the real diagonalizable matrix pair with real generalized eigenvalues, corresponding to the largest or the smallest generalized eigenvalue. We establish an explicit representation for the solutions of the neural network and analyze the convergence property. We consider how to use the above neural networks for computation of the singular value problem and the generalized singular value problem. In detail, we use our neural networks to compute the left and right singular vectors of a real matrix, corresponding to the largest or the smallest singular value. The right generalized singular vector of matrix pairs, corresponding to the largest or the smallest generalized singular value, can be computed by the neural networks. Numerical examples are given to illustrate our result is reasonable.