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An acceleration of Newton’s method: Super-Halley method. (English) Zbl 1023.65051
Summary: From a study of the convexity we give an acceleration for Newton’s method and obtain a new third order method. Then we use this method for solving nonlinear equations in Banach spaces, establishing conditions on convergence, existence and uniqueness of solution, as well as error estimates.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Altman, M.: Concerning the method of tangent hyperbolas for operator equations. Bull. acad. Pol sci., serie sci. Math., ast. Et phys. 9, 633-637 (1961) · Zbl 0118.11903
[2] Campbell, R.: LES integrales euleriennes et leurs aplications. (1966) · Zbl 0174.36201
[3] Candela, V.; Marquina, A.: Recurrence relations for rational cubic methods I: The halley method. Comput. 44, 169-184 (1990) · Zbl 0701.65043
[4] Candela, V.; Marquina, A.: Recurrence relations for rational cubic methods II: The Chebyshev method. Comput. 45, 355-367 (1990) · Zbl 0714.65061
[5] Chen, D.; Argyros, I. K.; Qian, Q. S.: A local convergence theorem for the super-halley method in a Banach space. App. math. Lett. 7, No. 5, 49-52 (1994) · Zbl 0811.65043
[6] Ciesielski, Z.: Some propierties of convex functions of higher orders. Annales polonici mathematici 7, 1-7 (1959) · Zbl 0128.05704
[7] Döring, B.: Einige sätze über das verfahren der tangierenden hyperbeln in Banach-räumen. Aplikace mat. 15, 418-464 (1970) · Zbl 0214.41201
[8] Gander, W.: On halley’s iteration method. Amer. math. Monthly 92, 131-134 (1985) · Zbl 0574.65041
[9] Gragg, W. B.; Tapia, R. A.: Optimal error bounds for the Newton--Kantorovich theorem. SIAM J. Numer. anal. 11, 10-13 (1974) · Zbl 0284.65042
[10] Gutiérrez, J. M.; Hernández, M. A.; Salanova, M. A.: Accesibility of solutions by Newton’s method. Int. J. Comput. math. 57, 239-247 (1995) · Zbl 0844.47035
[11] Gutiérrez, J. M.; Hernández, M. A.; Salanova, M. A.: Resolution of quadratic equations in Banach spaces. Numer. funct. Anal. opt. 17, No. 1/2, 113-121 (1996) · Zbl 0849.47038
[12] Gutiérrez, J. M.; Hernández, M. A.: A family of Chebyshev--halley type methods in Banach spaces. Bull. austral. Math. soc. 55, 113-130 (1997) · Zbl 0893.47043
[13] Hernández, M. A.: Newton--raphson’s method and convexity. Zb. rad. Prirod.-mat. Fak. ser. Mat. 22, No. 1, 159-166 (1993)
[14] Hernández, M. A.; Salanova, M. A.: A family of Newton type iterative processes. Int. J. Comput. math. 51, 205-214 (1994) · Zbl 0824.65025
[15] Kantorovich, L. V.; Akilov, G. P.: Functional analysis. (1982) · Zbl 0484.46003
[16] Ostrowski, A. M.: Solution of equations in Euclidean and Banach spaces. (1943)
[17] Traub, J. F.: Iterative methods for solution of equations. (1964) · Zbl 0121.11204
[18] Yamamoto, T.: On the method of tangent hyperbolas in Banach spaces. J. comput. Appl. math. 21, 75-86 (1988) · Zbl 0632.65070