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Displaying similar documents to “Newton's formula and the continued fraction expansion of d .”

A new Kantorovich-type theorem for Newton's method

Ioannis Argyros (1999)

Applicationes Mathematicae

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A new Kantorovich-type convergence theorem for Newton's method is established for approximating a locally unique solution of an equation F(x)=0 defined on a Banach space. It is assumed that the operator F is twice Fréchet differentiable, and that F', F'' satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value.

Tong’s spectrum for Rosen continued fractions

Cornelis Kraaikamp, Thomas A. Schmidt, Ionica Smeets (2007)

Journal de Théorie des Nombres de Bordeaux

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In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of k consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of...