On a third order iterative method for solving polynomial operator equations
Keywords:two-step Newton method, Chebyshev method, eigenpair problems
AbstractWe 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.
Anselone, M. P. and Rall, L. B., The solution of characteristic value-vector problems by Newton method, Numer. Math., 11, pp. 38-45, 1968, https://doi.org/10.1007/BF02165469 DOI: https://doi.org/10.1007/BF02165469
Argyros, I. K., Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc., 38, pp. 275-292, 1988, DOI: https://doi.org/10.1017/S0004972700009953
Catinaş, E. and Pavaloiu, I., On the Chebyshev method for approximating the eigen-va-lues of linear operators, Rev. Anal. Numér. Théor. Approx., 25, no. 1-2, pp. 43-56, 1996, https://ictp.acad.ro/chebyshev-method-approximating-eigenvalues-linear-operators/
Catinaş, E. and Pavaloiu, I., On the Chebyshev method for approximating the eigenvalues of linear operators, Proceedings of International Conference on Approximation and Optimization, Cluj-Napoca, July 29-August 1, vol. 1, pp. 219-226, 1996,
Catinaş, E. and Pavaloiu, I., On approximating the eigenvalues and eigenvectors of linear continuous operators, Rev. Anal. Numér. Théor. Approx., 26, no. 1-2, pp. 19-27, 1997, https://ictp.acad.ro/approximating-eigenvalues-eigenvectors-linear-continuous-operators/
Catinaş, E. and Pavaloiu, I., On some interpolatory iterative methods for the second degree polynomial operators (I), Rev. Anal. Numér. Théor. Approx., 27, pp. 33-45, 1998, https://ictp.acad.ro/interpolatory-iterative-methods-second-degree-polynomial-operators-i/
Catinaş, E. and Pavaloiu, I., On some interpolatory iterative methods for the second degree polynomial operators (II), Rev. Anal. Numér. Théor. Approx., 28, no. 2, pp. 133-143, 1999, https://ictp.acad.ro/interpolatory-iterative-methods-second-degree-polynomial-operators-ii/
Collatz, L., Functionalanalysis und Numerische Mathematik, Springer-Verlag, Berlin, 1964, DOI: https://doi.org/10.1007/978-3-642-95028-5
Diaconu, A., On the convergence of an iterative method of Chebyshev type, Rev. Anal. Numér. Théor. Approx., 24, no. 1-2, pp. 91-102, 1995, https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art9/Diaconu
Dongarra, J. J., Moler, C. B. and Wilkinson, J. H., Improving the accuracy of the computed eigenvalues and eigenvectors, SIAM J. Numer. Anal., 20, no. 1, pp. 23-45, 1983, https://doi.org/10.1137/0720002 DOI: https://doi.org/10.1137/0720002
Kartîşov, V. S. and Iuhno, F. L., O nekotorîh Modifikaţiah Metoda Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci, J. Vîcisl. matem. i matem. fiz., 33, no. 9, pp. 1403-1409, 1973 (in Russian).
Ortega, J. M. and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
Pavaloiu, I., Sur les procédés itératifs à un ordre élevé de convergence, Mathematica (Cluj), 12 (35), no. 2, pp. 309-324, 1970,
Pavaloiu, I., Introduction to the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca, Romania, 1986 (in Romanian).
Pavaloiu, I. and Catinaş, E., Remarks on some Newton and Chebyshev-type methods for approximating the eigenvalues and eigenvectors of matrices, Computer Science Journal of Moldova, 7, no. 1, pp. 3-17, 1999,
Peters, G. and Wilkinson, J. H., Inverse iteration, ill-conditioned equations and Newton's method, SIAM Review, 21, no. 3, pp. 339-360, 1979, https://doi.org/10.1137/1021052 DOI: https://doi.org/10.1137/1021052
Santos, M. C., A note on the Newton iteration for the algebraic eigenvalue problem, SIAM J. Matrix Anal. Appl., 9, no. 4, pp. 561-569, 1988, https://doi.org/10.1137/0609046 DOI: https://doi.org/10.1137/0609046
Tapia, R. A. and Whitley, L. D., The projected Newton method has order 1+√2 for the symmetric eigenvalue problem, SIAM J. Numer. Anal., 25, no. 6, pp. 1376-1382, 1988, https://doi.org/10.1137/0725079 DOI: https://doi.org/10.1137/0725079
Tisseur, F., Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems, SIAM J. Matrix Anal. Appl., 22, no. 4, pp. 1038-1057, 2001, https://doi.org/10.1137/S0895479899359837 DOI: https://doi.org/10.1137/S0895479899359837
Wu, K., Saad, Y. and Stathopoulos, A., Inexact Newton preconditioning techniques for large symmetric eigenvalue problems, Electronic Transactions on Numerical Analysis, 7, pp. 202-214, 1998, https://www.academia.edu/12708267/Inexact_Newton
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