On some algorithms for the solution of the complete eigenvalue problem

Abstract

In this article we give the basis and details of algorithms used in the solution of the complete problem of the eigenvalues of a matrix, already briefly explained in a note of the author's [1], and their extension to provide the solution of the partial problem. In § 1 we consider algorithms I and II, which solve the problem for a real non-symmetric non-singular matrix with real eigenvalues, different in modulus. In § 2 we consider algorithms III and IV, which enable us to find the complete spectrum and eigenvectors of the matrices AA′ and A′ A, starting only from the matrices A, and without having to construct A′ A and AA′. In § 3 we consider a modification of these four algorithms which enables us to solve the partial problem. All these algorithms are iterative. They are similar to Jacobian processes in their realization [2] (and also can be carried out using plane rotations), but differ from them as far as their region of convergence is concerned, approaching the triangular step method [3] and the LR algorithm [4]. They can easily be put into effect on a computer, and make it possible to determine both the eigenvalues and the eigenvectors in one operation, thus simplifying the programming. They also allow the greater part of the computations to be made in fixed point without any consequent loss of accuracy. We should expect the algorithms to be stable with respect to rounding errors, since each elementary step in the process reduces to the formation of a linear combination of two columns (rows) of a matrix with coefficients which are less than unity in absolute magnitude. In all the algorithms the sequence of matrices A k and the sequence of left triangular matrices Λ k = ( l ij ( k) ) are constructed recurrently, where Λ k is obtained from the corresponding matrix A k by multiplying it on its right by the orthogonal matrix τ k = ( t ij ( k) ). The orthogonal matrices are constructed using elementary rotation or reflection matrices. In our examples below we shall dwell briefly on the construction of the latter. The reader can find a detailed account in the book [5].