Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0872.65045
Gutiérrez, José M.
A new semilocal convergence theorem for Newton's method. (English)
[J] J. Comput. Appl. Math. 79, No.1, 131-145 (1997). ISSN 0377-0427

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.
[W.H.Schmidt (Greifswald)]

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
47J25 Methods for solving nonlinear operator equations (general)

Keywords: error estimate; majorizing sequence; Newton method; nonlinear equation; Banach space; semilocal convergence

Cited in: Zbl 0976.65053

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]