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

Emil Cătinaş; I Păvăloiu

Authors

Emil Cătinaş Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania
I Păvăloiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

Abstract views: 280

Abstract

In this note we consider the chord (secant) method and the Steffensen method for solving polynomial operators of degree 2 on Banach spaces, \(F:X\rightarrow Y\).

The convergence conditions in this case are simplified, as the divided difference of order 3 is the null trilinear operator.

As particular cases, we study the eigenproblem for quadratic matrices and integral equations of Volterra type.

We obtain semilocal convergence results, which show the r-convergence orders of the iterates.

Downloads

Download data is not yet available.

References

M. P. Anselone and '.L. Rall, The solution of characteristic value-vector problems by Newton, method, Numer. Math. ll (1968), pp. 38-45, https://doi.org/10.1007/bf02165469

I.K. Argyros, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc. 38 (1988), pp. 275-292, https://doi.org/10.1017/s0004972700009953

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

E. Cătinaş and I. Păvăloiu, On a Chebyshev-type Method for Approximating the Solutions of Polynomial Operator Equations of Degree 2, Proceedings on International conference on Approximation and Optimization, Cluj-Napoca, July 29 - August 1, 1996, Vol. 1, pp.219-226.

F. Chatelin, Valeurs propres de matrices, Mason, Paris-Milan-Barcelone-Mexico, 1988.

P. G. Ciarlet, Introduction à I'analyse numtérique matricielle et à l’optimisation, Mason, Paris-Milan Barcelone Mexico, 1990.

L. Collatz, Functionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964, https://doi.org/10.1007/978-3-642-53372-3

A. Diaconu, On the convergence of an iterative method of Chebyshev type, Rev. Anal. Numér. Théorie Approximation 24, 1-2 (1995), pp. 91-102,

A. Diaconu and I. Păvăloiu, Sur quelques méthodes itératives pour la résolution des équations opérationnelles, Rev. Anal. Numér. Theorie Approximation, 1. 1 (1972), pp.45-61,

V.S. Kartîşov and F. L. Iuhno, O nekotorîh Modifikaţiah Metoda Niutona dlea Resenia Nelineinoi spektralnoi Yadaci, J. Vîcisl. matem. i matem. fiz. 33, 0 (1973), pp. 1403-1409.

I. Lazăr, On a Newton-type method, Rev. Anal. Numér. Théorie Aproximation 23, 2 (1994), pp. 167-174,

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York. 1970.

I. Păvăloiu, Sur les procédés itératifs à un ordre élevé de convergence, Mathematica (Cluj) 12, (35), 2 (1970) pp. 309-324.

I. Păvăloiu, Introduction in the Approximation Theory for the solutions ol Equations, Ed. Dacia, Cluj-Napoca, I986 (in Romanian).

I. Păvăloiu, Observations concerning some approximation methods for the soltuions of operator equations, Rev. Anal. Numér. Théorie Approximation 23, 2 (1994), pp.185-196,

I. Păvăloiu, Approximation of the root of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32, 1-2 January-June, 1995, pp. 69-82, https://doi.org/10.1007/bf02576543

R. A. Tapia and L. D. Whitley, The projected Newton method has order 1+√2 for the suymmetric eigenvalue problem, SIAM J. Numer. Anal. 25, 6(l988), pp. 1316-1382, https://doi.org/10.1137/0725079

F.J. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Inc., Englewood Clilfs, N. J., 1964.

[l9] S' Ul'm, On the iterative method with simultuneous approximntion of the inverse of the operator, Izv. Acad. Nauk. Estonskoi S.S.R. 16,4 (1967), pp. 403-411.

T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math.34, (1980),pp. 189-199, https://doi.org/10.1007/bf01396059