The effect of rounding errors on a certain class of iterative methods

• Autorzy: Ioannis K. Argyros
• Języki publikacji: EN

Abstrakty

EN

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the mth Fréchet derivative (m ≥ 2 an integer) instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.

Słowa kluczowe

EN

Fréchet derivative; Lipschitz conditions; Newton-like method; inexact Newton-like method; Banach space

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 2000
• Tom: 27
• Numer: 3
• Strony: 369-375

Daty

wydano

2000

otrzymano

1999-12-22

Twórcy

autor

Ioannis K. Argyros

• Cameron University, Department of Mathematics, Lawton, OK 73505, U.S.A.

Bibliografia

• [1] I. K. Argyros, On the convergence of some projection methods with perturbations, J. Comput. Appl. Math. 36 (1991), 255-258.
• [2] I. K. Argyros, Concerning the radius of convergence of Newton's method and applications, Korean J. Comput. Appl. Math. 6 (1999), 451-462.
• [3] I. K. Argyros and F. Szidarovszky, The Theory and Application of Iteration Methods, CRC Press, Boca Raton, FL, 1993.
• [4] R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19 (1982), 400-408.
• [5] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982.
• [6] T. J. Ypma, Numerical solution of systems of nonlinear algebraic equations, Ph.D. thesis, Oxford, 1982.
• [7] T. J. Ypma, Affine invariant convergence results for Newton's method, BIT 22 (1982), 108-118.
• [8] T. J. Ypma, The effect of rounding errors on Newton-like methods, IMA J. Numer. Anal. 3 (1983), 109-118.