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A new semilocal convergence theorem for Newton’s method. (English) Zbl 0872.65045
The well known Newton method for solving a nonlinear equation $F(x) =0$ in a Banach space is considered and a new semilocal convergence theorem is proved under different assumptions from those of the Kantorovich theorem. Here it is assumed that the second Fréchet derivative $F''$ exists and is continuous and bounded and that the condition $$\biggl|F'(x_0)^{-1} \bigl[F''(x)- F''(x_0)\bigr] \biggr|\le k|x-x_0 |$$ is satisfied in a certain neighbourhood of $x_0$. The proof of convergence is similar to that of Huang, a suitable cubic polynomial is checked in the proof. The author shows uniqueness of the solution and estimates the errors. Two examples are added to show situations where the Kantorovich assumptions fail but those of the discussed theorem are fulfilled or vice versa.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI
[1] Argyros, I. K.: Newton-like methods under mild differentiability conditions with error analysis. Bull. austral. Math. soc. 37, 131-147 (1988) · Zbl 0629.65061
[2] Argyros, I. K.: A Newton-like method for solving nonlinear equations in Banach space. Acta math. Hungh. 27, 368-378 (1992) · Zbl 0687.47052
[3] Gutiérrez, J. M.; Hernández, M. A.; Salanova, M. A.: Accessibility of solutions by Newton’s method. Internat. J. Comput. math. 57, 239-247 (1995) · Zbl 0844.47035
[4] Hernández, M. A.: Newton-raphson’s method and convexity. Zb. rad. Prirod.-mat. Fak. ser. Mat. 22, No. 1, 159-166 (1992) · Zbl 0801.65045
[5] Huang, Z.: A note on the Kantorovich theorem for Newton iteration. J. comput. Appl. math. 47, 211-217 (1993) · Zbl 0782.65071
[6] Kantorovich, L. V.: On Newton’s method for functional equations. Dokl. akad. Nauk SSRR 59, 1237-1240 (1948)
[7] Kantorovich, L. V.: The majorant principle and Newton’s method. Dokl. akad. Nauk SSRR 76, 17-20 (1951)
[8] Kantorovich, L. V.; Akilov, G. P.: Functional analysis. (1982) · Zbl 0484.46003
[9] Ostrowski, A. M.: Solution of equations in Euclidean and Banach spaces. (1943)
[10] Rall, L. B.: Quadratic equations in Banach spaces. Rend. circ., mat. Palermo, No. 2, 314-332 (1961)
[11] Rall, L. B.: Computational solution of nonlinear operator equations. (1979) · Zbl 0476.65033
[12] Rheinboldt, W. C.: A unified convergence theory for a class of iterative process. SIAM J. Numer. anal. 5, 42-63 (1968) · Zbl 0155.46701
[13] Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. math. 49, 203-220 (1986) · Zbl 0607.65033