Abstract
We provide a ball comparison between some 4-order methods to solve nonlinear equations involving Banach space valued operators. We only use hypotheses on the first derivative, as compared to the earlier works where they considered conditions reaching up to 5-order derivative, although these derivatives do not appear in the methods. Hence, we expand the applicability of them. Numerical experiments are used to compare the radii of convergence of these methods.
Highlights
Let E1, E2 be Banach spaces and D ⊂ E1 be a nonempty and open set
We are motivated by four iterative methods given as y j = x j − λ 0 ( x j ) −1 λ ( x j )
Upper error bounds on k x j − s∗ k and uniqueness results are not reported with this technique
Summary
Let E1 , E2 be Banach spaces and D ⊂ E1 be a nonempty and open set. Set LB(E1 , E2 ) = { M : E1 → E2 }, bounded and linear operators. The third-order derivative of function λ000 ( x ) is not bounded on D. Cannot be applicable to such problems or their special cases that require the hypotheses on the third or higher-order derivatives of λ. These works do not give a radius of convergence, estimations on k x j − s∗ k, or knowledge about the location of s∗. The novelty of our work is that we provide this information, but requiring only the derivative of order one, for these methods. It is vital to note that the local convergence results are very fruitful, since they give insight into the difficult operational task for choosing the starting points/guesses.
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