Abstract
The classical Kantorovich theorem for Newton’s method assumes that the derivative of the involved operator satisfies a Lipschitz condition ∥;F’(x 0)-1 [F’(x) -’’(y)] ∥≤ L∥x - y∥ In this communication, we analyse the different modifications of this condition, with a special emphasis in the center-Lipschitz condition: ∥F’(x0)-1 [F’(x) - F-(x0)]∥≤ω(∥x - x0∥) being ω a positive increasing real function and x0 the starting point for Newton’s iteration.
In this paper we make a survey of the convergence of Newton’s method in Banach spaces. So, let X, Y be two Banach spaces and let F : X → Y be a Fréchet differentiable operator. Starting from x0 ∈ X, the well-known Newton’s method is defined by the iterates xn+1 = xn - F’(xn)-1F(xn), n = 0, 1, 2, . . . (1) provided that the inverse of the linear operator F’(xn) is defined at each step.
Research of both authors has been supported by a grant of the Universidad de La Rioja (ref. API-99/B14) and two grants of the DGES (refs. PB98-0198 and PB96- 0120-C03-02).
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Gutiérrez, J.M., Hernández, M.A. (2001). Newton’s Method under Different Lipschitz Conditions. In: Vulkov, L., Yalamov, P., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2000. Lecture Notes in Computer Science, vol 1988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45262-1_43
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DOI: https://doi.org/10.1007/3-540-45262-1_43
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