A generalized eigenvalue algorithm for tridiagonal matrix pencils based on a nonautonomous discrete integrable system

A generalized eigenvalue algorithm for a certain class of tridiagonal matrix pencils is presented. The algorithm appears as the time evolution equation of a nonautonomous discrete integrable system associated with a polynomial sequence which has some orthogonality on the support set of the zeros of the characteristic polynomial for a tridiagonal matrix pencil. The convergence of the algorithm is discussed by using the solution to the initial value problem for the corresponding discrete integrable system.