## Abstract

We present a semilocal convergence result for a Newton-type method applied to a polynomial operator equation of degree (2).

The method consists in fact in evaluating the Jacobian at every two steps, and it has the *r*-convergence order at least (3). We apply the method in order to approximate the eigenpairs of matrices.

We perform some numerical examples on some test matrices and compare the method with the Chebyshev method. The norming function we have proposed in a previous paper shows a better convergence of the iterates than the classical norming function for both the methods.

## Authors

Emil **Cătinaş**

(Tiberiu Popoviciu Institute of Numerical Analysis)

Ion **Păvăloiu**

(Tiberiu Popoviciu Institute of Numerical Analysis)

## Keywords

nonlinear equations; abstract polynomial equations of degree 2; r-convergence order.

## Cite this paper as:

E. Cătinaş, I. Păvăloiu, *On a third order iterative method for solving polynomial operator equations*, Rev. Anal. Numér. Théor. Approx., **31** (2002) no. 1, pp. 21-28.

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## About this paper

##### Publisher Name

##### Article on the journal website

##### Print ISSN

1222-9024

##### Online ISSN

2457-8126

##### MR

1222-9024

##### Online ISSN

2457-8126

## Google Scholar citations

*The solution of characteristic value-vector problems by Newton method*, Numer. Math., 11(1968), pp. 38–45.

[2] I.K. Argyros, *Quadratic equations and applications to Chandrasekhar’s and related equations* , Bull. Austral. Math. Soc., 38 (1988), pp. 275–292.

[3] E. Catinas and I. Pavaloiu, *On the Chebyshev method for approximating the eigenvalues of linear operators*, Rev. Anal. Num ́er. Th ́eor. Approx., 25 (1996) nos. 1–2, pp. 43-56.

[4] E. Catinas and I. Pavaloiu, *On a Chebyshev-type method for approximating the solutions of polynomial operator equations of degree 2*, Proceedings of International Conference on Approximation and Optimization, Cluj-Napoca, July 29 – august 1, 1996, vol. 1, pp. 219-226.

[5] E. Catinas and I. Pavaloiu, *On approximating the eigenvalues and eigenvectors of linear continuous operators*, Rev. Anal. Num ́er. Th ́eor. Approx., 26 (1997) nos. 1–2, pp. 19–27.

[6] E. Catinas and I. Pavaloiu, *On some interpolatory iterative methods for the second degree polynomial operators (I),* Rev. Anal. Num ́er. Th ́eor. Approx., 27(1998) no. 1, pp. 33-45.

[7] E. Catinas and I. Pavaloiu, *On some interpolatory iterative methods for the second degree polynomial operators (II)* , Rev. Anal. Num ́er. Th ́eor. Approx., 28 (1999) no. 2, pp. 133-143.

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

[9] A. Diaconu, *On the convergence of an iterative method of Chebyshev type*, Rev. Anal. Numer. Theor. Approx. 24 (1995) nos. 1–2, pp. 91–102.

[10] J.J. Dongarra, C.B. Moler and J.H. Wilkinson, *Improving the accuracy of the computed eigenvalues and eigenvectors* , SIAM J. Numer. Anal., 20 (1983) no. 1, pp. 23–45.

[11] S.M. Grzegorski, *On the scaled Newton method for the symmetric eigenvalue problem*, Computing, 45 (1990), pp. 277–282.

[12] V.S. Kartisov and F.L. Iuhno, *O nekotorih Modifikat ̧ah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci*, J. Vicisl. matem. i matem. fiz., 33 (1973) no. 9, pp. 1403–1409 (in Russian).

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

*Sur les procedes iteratifs `a un order eleve de convergence*, Mathematica (Cluj), 12 (35) (1970) no. 2, pp. 309–324.

*Introduction to the Theory of Approximating the Solutions of Equations*, Ed. Dacia, Cluj-Napoca, Romania, 1986 (in Romanian).

*Remarks on some Newton and Chebyshev-type methods for approximating the eigenvalues and eigenvectors of matrices*, Computer Science Journal of Moldova, 7(1999) no. 1, pp. 3–17.

*Inverse iteration, ill-conditioned equations and New*

*ton’s method*, SIAM Review, 21(1979) no. 3, pp. 339–360.

*A note on the Newton iteration for the algebraic eigenvalue problem*, SIAM J. Matrix Anal. Appl., 9 (1988) no. 4, pp. 561–569.

*The projected Newton method has order 1+√2 for the symmetric eigenvalue problem*, SIAM J. Numer. Anal., 25 (1988) no. 6, pp. 1376–1382.

*Newton’s method in floating point arithmetic and iterative refinement of generalized eigenvalue problems*, SIAM J. Matrix Anal. Appl., 22 (2001) no. 4, pp. 1038–1057.

*Inexact Newton preconditioning techniques for large symmetric eigenvalue problems*, Electronic Transactions on Numerical Analysis, 7 (1998) pp. 202–214.

*Error bounds for computed eigenvalues and eigenvectors*, Numer. Math., 34 (1980), pp. 189–199.