The principle of minimized iterations in the solution of the matrix eigenvalue problem

Abstract

An interpretation of Dr. Cornelius Lanczos' iteration method, which he has named "minimized iterations", is discussed in this article, expounding the method as applied to the solution of the characteristic matrix equations both in homogeneous and non-homogeneous form. This interpretation leads to a variation of the Lanczos procedure which may frequently be advantageous by virtue of reducing the volume of numerical work in practical applications. Both methods employ essentially the same algorithm, requiring the generation of a series of orthogonal functions through which a simple matrix equation of reduced order is established. The reduced matrix equation may be solved directly in terms of certain polynomial functions obtained in conjunction with the generated orthogonal functions, and the convergence of the solution may be observed as the order of the reduced matrix is successively increased with the order of the original matrix as a limit. The method of minimized iterations is recommended as a rapid means for determining a small number of the larger eigenvalues and modal columns of a large matrix and as a desirable alternative for various series expansions of the Fredholm problem.