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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2457-6794

Online ISSN

2501-059X

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References

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[7] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Inc., Englewood Cliffs, N. J., 1964.

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