On Popovski’s method for nonlinear equations (uncorrected proof)
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There is a vast literature on the solution of nonlinear equations and nonlinear systems, see for example strowski , Traub , Neta  and references there. In general, methods for the solution of polynomial equations are treated differently and will not be discussed here. The methods can be classified as bracketting or fixed point methods. The first class include methods that at every step produce an interval containing a root, whereas the other class produces a point which is hopefully closer to the root than the previous one. Here we develop two third-order fixed point type methods based on Popovski’s family of methods . In the first modified method we traded the second derivative by an additional function evaluation. The informational efficiency and efficiency index (see ) are the same as Popovski’s. In the second modified scheme we replaced the second derivative by a finite difference and thus reducing the order slightly and reducing the number of function evaluations. This method is more efficient than Popovski’s.Two diﬀerent modiﬁcations of Popovski’s method are developed, both are free of second derivatives. In the ﬁrst modiﬁed scheme we traded the second derivative by an additional function evaluation. In the second method we replaced the second derivative by a ﬁnite diﬀerence and thus reducing the order slightly and reducing the number of evaluations per step by one. Therefore the second modiﬁcation is more eﬃcient.
Applied Mathematics and Computation, 201, (2008), 710–715.The article of record as published may be located at http://dx.doi.org/10.1016/j.amc.2008.01.012
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