Improving the order and rates of convergence for the Super-Halley method in Banach spaces
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In this study we are concerned with the problem of approximating a locally unique solution of an equation on a Banach space. A semilocal convergence theorem is given for the Super-Halley method in Banach spaces. Earlier results have shown that the order of convergence is four for a certain class of operators , , . These results were not given in affine invariant form, and made use of a real quadratic majorizing polynomial. Here, we provide our results in affine invariant form showing that the order of convergence is at least four. In cases that it is exactly four the rate of convergence is improved. We achieve these results by using a cubic majorizing polynomial. Some numerical examples are given to show that our error bounds are better than earlier ones.
AMS Mathematics Subject Classification65J15 65B05 47H17 49D15
Key word and phrasesSuper-Halley method Banach space majorizing sequence order of convergence rate of convergence
- 3.Argyros, I.K.Convergence results for the Super-Halley method using divided differences, Funct. et. Approx. Comment. Math. XXIII, (1994), 109–122.Google Scholar
- 9.Gutierez, J.M. and Hernandez, M.A.Convex acceleration of Newton’s method, Pub. Sem. G. Galdeano, Serie 11, Seccion 1, 8, Univ. Zaragoza, 1995.Google Scholar