Abstract
The local convergence of a generalized (p+1)-step iterative method of order 2p+1 is established in order to estimate the locally unique solutions of nonlinear equations in the Banach spaces. In earlier studies, convergence analysis for the given iterative method was carried out while assuming the existence of certain higher-order derivatives. In contrast to this approach, the convergence analysis is carried out in the present study by considering the hypothesis only on the first-order Fréchet derivatives. This study further provides an estimate of convergence radius and bounds of the error for the considered method. Such estimates were not provided in earlier studies. In view of this, the applicability of the given method clearly seems to be extended over the wider class of functions or problems. Moreover, the numerical applications are presented to verify the theoretical deductions.
Highlights
Problems in applied mathematics are frequently formulated as the systems of nonlinear equations
The most common approach to estimate the order of convergence of an iterative method includes the Taylor’s series expansions, which inherently involve higher-order derivatives (F(i), i = 1, 2, . . .), and some assumptions on F(i). Such assumptions limit the applicability of techniques, since most require the computation of the first-order derivative only
To verify the theoretical deductions, we provide here the real parameters or functions, as well as the estimated radius of convergence, for each of the following numerical examples, in particular by taking a
Summary
Problems in applied mathematics are frequently formulated as the systems of nonlinear equations. It is worth noticing that the set assumptions on F(i) It is essential to enlarge the convergence domain by avoiding these additional hypotheses In this sense, the convergence analysis of iterative techniques should include a measure of closeness of the initial estimate to the solution. In view of the above facts, we shall study the local convergence analysis of a generalized (p + 1)-step iterative method of order 2p + 1, developed in [4], and which is expressed as follows: yn(0) = xn − aF (xn)−1F(xn), y(n1) =y(n0) − Bn F (xn)−1F(xn), y(n2) =y(n1) − g(xn, yn)F (xn)−1F(y(n1)), y(np−1) = y(np−2) − g(xn, yn)F (xn)−1F(y(np−2)), xn+1 = y(np) = y(np−1) − g(xn, yn)F (xn)−1F(y(np−1)),.
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