Neural networks for computing eigenvalues and eigenvectors

Two new algorithms and associated neuron-like network architectures are proposed for solving the eigenvalue problem in real-time. The first approach is based on the solution of a set of nonlinear algebraic equations by employing optimization techniques. The second approach employs a multilayer neural network with linear artificial neurons and it exploits the continuous-time error back-propagation learning algorithm. The second approach enables us to find all the eigenvalues and the associated eigenvectors simultaneously by training the network to match some desired patterns, while the first approach is suitable to find during one run only one particular eigenvalue (e.g. an extreme eigenvalue) and the corresponding eigenvector in realtime. In order to find all eigenpairs the optimization process must be repeated in this case many times for different initial conditions. The performance and convergence behaviour of the proposed neural network architectures are investigated by extensive computer simulations.