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A two sided iterative method for real roots of \(f(x)=0\). (English) Zbl 0671.65036

Let f(x) be a real-valued continuously differentiable function defined on an interval [a,b]. The iterative method used for enclosing a root r of the nonlinear equation \(f(x)=0\) is based on the interval analysis. It thus generates a sequence \(\{X_ k\}^{\infty}_{k=0}\) of intervals \(X_ k=\{x^ 1_ k,x^ 2_ k]\) such that \(X_{k+1}\subseteq X_ k\), and gives an approximate solution with rigorous error bounds. The algorithm is a modification and in some respects an improvement of some earlier algorithms of the same category (due to R. Krawczyk and J. Herzberger). The authors show that their algorithm generates two monotonic sequences \[ (1)\quad x^ 1_ 0\leq x^ 1_ 1\leq...\leq x^ 1_ k\leq x^ 1_{k+1}\leq...\leq r\leq...\leq x^ 2_{k\quad +1}\leq x^ 2_ k\leq...\leq x^ 2_ 0 \] such that \(\lim_{k\to \infty}x^ 1_ k=\lim_{k\to \infty}x^ 2_ k=r\in [a,b]\). The relation (1) is valid provided that f(x) in an interval \(X_ 0=[x^ 1_ 0,x^ 2_ 0]\subseteq [a,b]\) has the properties: \(f(x^ 1_ 0)\leq 0\), \(f(x^ 2_ 0)\leq 0\); there exist constants \(m_ 0\) and \(M_ 0\) such that \[ 0<m_ 0\leq \frac{f(x)-f(r)}{x-r}=\frac{f(x)}{x-r}\leq M_ 0<\infty \] holds for all \(x\in X_ 0\), \(x\neq r\). For computational purposes the quantities \(m_ k=\min_{x\in X_ k}f'(x)\) and \(M_ k=\max_{x\in X_ k}f'(x)\) are needed at each stage of iteration. In the case \(f(x)\in C^ 2[x^ 1_ 0,x^ 2_ 0]\) the rate of convergence is shown to be quadratic. Two simple examples are presented to illustrate the efficiency of the method.
Reviewer: V.Seppäla

MSC:

65H05 Numerical computation of solutions to single equations
65G30 Interval and finite arithmetic
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