An iterative method for estimation the roots of real-valued functions
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In this paper we study the recursive sequence x_n+1=x_n+f(x_n)/2 for each continuous real-valued function f on an interval [a,b], where x_0 is an arbitrary point in [a,b]. First, we present some results for real-valued continuous function f on [a,b] which have a unique fixed point c∈ (a,b) and show that the sequence {x_n} converges to c provided that f satisfies some conditions. By assuming that c is a root of f instead of being its fixed point, we extend these results. We define two other sequences by x^+_0=x^-_0=x_0∈ [a,b] and x^+_n+1=x^+_n+f(x^+_n)/2 and x^-_n+1= x^-_n-f(x^-_n)/2 for each n≥ 0. We show that for each real-valued continuous function f on [a,b] with f(a)>0>f(b) which has a unique root c∈ (a,b), the sequence {x^+_n} converges to c provided that f^'≥ -2 on (a,b). Accordingly we show that for each real-valued continuous function f on [a,b] with f(a)<0<f(b) which has a unique root c∈ (a,b), the sequence {x^-_n} converges to c provided that f^'≤ 2 on (a,b). By an example we also show that there exists some continuous real-valued function f:[a,b]→ [a,b] such that the sequence {x_n} does not converge for some x_0∈ [a,b].
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