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On the Chebyshev method for approximating the eigenvalues of linear operators

Approaches to the problem of approximating the eigenvalues of linear operators by the Newton method have been done in a series of papers ([1], [3], [4], [5]). There is a special interest in using the Newton method because the operatorial equation to be solved has a special form, as we shall see. We shall study in the following the convergence of the Chebyshev method attached to this problem and we shall apply the results obtained for the approximation of an eigenvalue and of a corresponding eigenvector for a matrix of real or complex numbers. It is known that the convergence order of Newton method is usually 2 and the convergence order of Chebyshev method is usually 3. Taking into account as well the number of operations made at each step, we obtain that the Chebyshev method is more efficient than the Newton method. Let E be a Banach space over K, where K = R or K = C, and T : E → E a linear operator. It is well known that the scalar λ is an eigenvalue of T if the equation