## 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.

Scanned paper. (soon)

Latex-pdf version of the paper. (soon)

post (soon)

## About this paper

##### Publisher Name

##### Paper on journal website

##### Print ISSN

2457-6794

##### Online ISSN

2501-059X

##### MR

?

##### ZBL

?

## Google Scholar citations

## References

## Paper in html format

## References

[1] M.P. Anselone, L.B. Rall, *The solution of characteristic value-vector problems by Newton method*, Numer. Math. 11 (1968), pp. 38–45.

CrossRef

[2] P.G. Ciarlet, *Introduction a l’analyse numerique matricielle et a l’optimisation*, Mason, Paris, 1990.

[3] F. Chatelin, *Valeurs propres de matrices*, Mason, Paris, 1988.

[4] L. Collatz, *Functionalanalysis und Numerische Mathematik*, Springer-Verlag, Berlin, 1964.

[5] V.S. Kartısov, F.L. Iuhno, *O nekotorıh Modifikatiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci*, J. Vıcisl. matem. i matem. fiz. (33) (1973) 9, pp. 1403–1409.

[6] I. Pavaloiu, *Sur les procedes iteratifs a un ordre eleve de convergence*, Mathematica (Cluj) 12 (35) (1970) 2, pp. 309–324.

paper

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