A variant of Newton’s method with accelerated third-order convergence. (English) Zbl 0973.65037

Weerakoon, S.; Fernando, T. G. I.

Appl. Math. Lett. 13, No. 8, 87-93 (2000).

Authors’ summary: In the given method, we suggest an improvement to the iteration of Newton’s method. Derivation of Newton’s method involves an indefinite integral of the derivative of the function, and the relevant area is approximated by a rectangle. In the proposed scheme, we approximate this indefinite integral by a trapezoid instead of a rectangle, thereby reducing the error in the approximation. It is shown that the order of convergence of the new method is three, and computed results support this theory. Even though we have shown that the order of convergence is three, in several cases, computational order of convergence is even higher. For most of the functions we tested, of convergence in Newton’s method was less than two and for our method, it was always close to three.

Reviewer: B.Döring (Düsseldorf)

MSC:

65H05

Numerical computation of solutions to single equations

Keywords:

third-order convergence; nonlinear equations; iterative methods; Newton’s method

References:

[1]	Dennis, J. E.; Schnable, R. B., Numerical Methods for Unconstrained Optimisation and Nonlinear Equations (1983), Prentice Hall
[2]	Burden, R. L.; Faires, D. J., Numerical Analysis. PWS (1993), Kent Publishing: Kent Publishing Boston
[3]	Fernando, T. G.I.; Weerakoon, S., Imporved Newton’s method for finding roots of a nonlinear equation, (Proceedings of the \(53^{rd}\) Annual Sessions of Sri Lanka Association for the Advancement of Science (SLAA), E1-22 (1997)), 309 · Zbl 0973.65037