Remarks on Some Newton and Chebyshev-type Methods for Approximation Eigenvalues and Eigenvectors of Matrices

It is well known that the Newton and the Chebyshev methods for nonlinear systems require solving of a linear system at each iteration step. In this note we shall study two modified methods which avoid solving of linear systems by using the Schultz method to approximate inverses of Fréchet derivatives. At the same time we shall use the particularities of nonlinear systems arising from eigenproblems, since the Fréchet derivatives of order higher than two are the null multilinear operators. Some numerical examples will be provided in the end of this note. Denote V = Kn and let A = (aij) ∈ Kn×n, where K = R or C. We recall that the scalar λ ∈ K is an eigenvalue of A if there exists v ∈ V , v 6= 0 such that Av − λv = 0. (1)