On some interpolatory iterative methods for the second degree polynomial operators (II)

Emil Cătinaş
(original), inverse interpolation, paper, semilocal convergence, solving F(x)=0 iteratively

Abstract

In this paper we apply some iterative methods obtained by inverse interpolation, in order to solve some specific classes of equations: the Ricatti equation, a Fredholm type equation, and the eigenvalue problem for a class of linear operators.

We obtain some semilocal convergence results, showing the r-convergence orders of the iterates.

Authors

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Keywords

inverse interpolation iterative methods; Ricatti equation; Fredholm type equation; eigenvalue problem; semilocal convergence results; r-convergence order.

Cite this paper as:

E. Cătinaş, I. Păvăloiu, On some interpolatory iterative methods for the second degree polynomial operators (II), Rev. Anal. Numér. Théor. Approx., 28 (1999) no. 2, pp. 133-143.

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Scanned paper: on the journal website.

Latex version of the paper (soon).

About this paper

Journal

Revue d’Analyse et de Theorie de l’Approximation

Publisher Name

Editions de l’Academie Roumaine.

Paper on the journal website.

https://ictp.acad.ro/jnaat/journal/article/view/1999-vol28-no2-art4

Print ISSN

1222-9024

Online ISSN

2457-8126

MR

?

ZBL

?

Google Scholar citations

Google Scholar citations

[1] M.P. Anselone and L.B. Rall, The Solution of Characteristic Value-Vector Problems by  Newton Method, Numer. Math. 11 (1968), 38-45.
[2] E. Catinas and I. Pavaloiu, On the Chebyshev method for approximating  the eigenvalue  of linear operators, Rev. Anal. Numer.  Theor. Approx., 25, 1-2 (1996), 43-56.
post
[3] E. Catinas and I. Pavaloiu, On a Chebyshev-type  Method for approximating  the solutions of Polynomial  Operator Equations  of Degree 2.  Proceedings of the International  Conference  on Approximation and  Optimization, Cluj-Napoca, July 29-August 1, 1996, Vol. 1, 219-226.
[4] E. Catinas and I. Pavaloiu, On approximating the eigenvalues and eigenvectors of linear continuous operators, Rev. Anal. Numer. Theor. Approx. 26 1-2 (1997),  19-27.
post
[5] E. Catinas and I. Pavaloiu, On some interpolatory iterative methods for the second degree polynomial operators (I).  Rev. Anal. Numer. Theor. Approx., to appear.
post
[6] F. Chatelin,  Valeurs propres  de matrices. Mason, Paris, 1988.
[7]  L. Collatz, Functionalanalysis   und Numerische  Mathematic. Springer-Verlag, Berlin,  1964.
 [8] J.J. Dongarra,  C.B. Moler and J.H. Wilkinson, Improving the Accurarcy  of the Computed  Eigenvalues and  Eigenvectors.  SIAM J.  Numer. Anal., 20 1 (1983), 23-45.
[9] S. M. Grzegórski, On the Scaled Newton Method for the Symmetric Eigenvalue Problem. Computingh, 45 (1990), pp.277-282.
[10] V. S, Kartîşov and L. Iuhno, O nekotorîh Modifikatiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci. J. Vîcisl. matem. i matem. fiz.,33 9 (1973), pp. 1403-1409 (in Russìan).
[11] M. L. Krasnov, lntegral Equations, Theoretical Introduction, Nauka Moskow, 1975 (in russian).
[12] J. M. Ortega and W. C Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press. New York. 1970
[13] C. Peters and J H. Wilkison. Inverse Iteration, III-Conditioned Equations and Newton’s Method, SIAM Review.21. 3 (1979), pp.339-360.
[14] M. C. Santos. A Note on the Newton Iteration for the Algebraic Eigenvalue Problem, SIAM J. Matrix Anal. Appl., 9 4 (1988), pp.561-569.
[15] R. A. Tapia und L. D. Whitley. The Projected Newton Method has Order 1+√2 for the Symmetric Eigenvalue Problem, SIAM J. Numer. Anal. 25,6 (1988), pp.1376-1382.
[16] S. Ul’m,  On the Iterative Method with Simultaneous Approximation of the Inverse of the Operator, Izv. Acad. Nauk. Estonskoi S.S.R., 16 4 (1967), pp. 40-411.
[17] K. Wu. Y. Saad and A. Stathopoulos, Inexact Newton Preconditioning Techniques for Eigenvalue Problems, Lawrence Berkeley National Laboratory report number 41382 and Minnesota Supercomputing Institute report number UMSI 98-10, 1998.
[18] T. Yamamoto, Error Bounds for Computed Eigenvalues and Eigenvectors, Numer. Math., 34 (1980), pp.189-199.
1999

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