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
Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Ion 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: on the journal website.
Latex-pdf version of the paper. (soon)
About this paper
Publisher Name
Paper on journal website
Print ISSN
2457-6794
Online ISSN
2501-059X
Google Scholar citations
[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.