Numerical Methods · Root finding and nonlinear systems
Let g be continuously differentiable on an open interval containing a fixed point…
Problem
Let \(g\) be continuously differentiable on an open interval containing a fixed point \(r=g(r)\), and assume \(|g'(r)|<1\). Prove that there exist \(\delta>0\) and \(K\in\bigl(|g'(r)|,1\bigr)\) such that if \(x_0\in(r-\delta,r+\delta)\) then the iteration \(x_{n+1}=g(x_n)\) remains in \((r-\delta,r+\delta)\) and converges to \(r\). Further prove that the convergence is at least linear, and that it is of order greater than one if and only if \(g'(r)=0\).
Hint
Continuity of \(g'\) upgrades \(|g'(r)|<1\) to a uniform bound \(K<1\) on a whole neighbourhood.
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