Summary
Given an iterative methodM 0, characterized byx (k+1=G 0(x(k)) (k≧0) (x(0) prescribed) for the solution of the operator equationF(x)=0, whereF:X→X is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM p characterized byx (k+1=G p (x(k)) (k≧0) (x(0) prescribed) with order of convergence higher than that ofM o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM 0 is Newton's method.
Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM p, which are referred to as extensions ofM 0. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.
Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.
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Wolfe, M.A. Extended iterative methods for the solution of operator equations. Numer. Math. 31, 153–174 (1978). https://doi.org/10.1007/BF01397473
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DOI: https://doi.org/10.1007/BF01397473