Displaying similar documents to “A correction to “Cayley's problem”, AM 35 (1990) No. 2, 140-146”

Cayley's problem

Peter Petek (1990)

Aplikace matematiky

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Newton's method for computation of a square root yields a difference equation which can be solved using the hyperbolic cotangent function. For the computation of the third root Newton's sequence presents a harder problem, which already Cayley was trying to solve. In the present paper two mutually inverse functions are defined in order to solve the difference equation, instead of the hyperbolic cotangent and its inverse. Several coefficients in the expansion around the fixed points are...

Computer-Assisted Proofs and Symbolic Computations

Krämer, Walter (2010)

Serdica Journal of Computing

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We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions...

Inexact Newton methods and recurrent functions

Ioannis K. Argyros, Saïd Hilout (2010)

Applicationes Mathematicae

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We provide a semilocal convergence analysis for approximating a solution of an equation in a Banach space setting using an inexact Newton method. By using recurrent functions, we provide under the same or weaker hypotheses: finer error bounds on the distances involved, and an at least as precise information on the location of the solution as in earlier papers. Moreover, if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained. Furthermore,...

Local convergence theorems for Newton's method from data at one point

Ioannis K. Argyros (2002)

Applicationes Mathematicae

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We provide local convergence theorems for the convergence of Newton's method to a solution of an equation in a Banach space utilizing only information at one point. It turns out that for analytic operators the convergence radius for Newton's method is enlarged compared with earlier results. A numerical example is also provided that compares our results favorably with earlier ones.