# On the Chebyshev method for approximating the eigenvalues of linear operators

## Abstract

We study the approximation of an eigenpair (an eigenvalue and a corresponding eigenvector) of a a linear operator T from X to X, X be a Banach space.

The eigenpair is regarded as a solution of a nonlinear system obtained by considering the usual definition plus a norming function and then applying the Chebyshev or the Newton method.

We take into account that the augmented mapping has the third Frechet derivative the null operator, and we give a semilocal convergence result, which ensures the r-order three (cubic r-order) of the iterates.

We consider next the finite dimensional case, and analyse the computation of an eigenpair of a matrix. We also compare in this case the efficiency of the Newton and Chebyshev method, and find out that the later is more efficient.

## Authors

E. Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

I. Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

linear operator; eigenvalue; eigenvector; eigenpair; Newton method; Chebyshev method; semilocal convergence; r-convergence order.

## Cite this paper as:

E. Cătinaş, I. Păvăloiu, On the Chebyshev method for approximating the eigenvalues of linear operators, Rev. Anal. Numér. Théor. Approx., 25 (1996) nos. 1-2, pp. 43-56.

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