Extended iterative methods for the solution of operator equations

SummaryGiven an iterative methodM0, characterized byx(k+1=G0(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 methodsMp characterized byx(k+1=Gp(x(k)) (k≧0) (x(0) prescribed) with order of convergence higher than that ofMo. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM0 is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsMp, which are referred to as extensions ofM0. 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.