An interval iteration for multiple roots of transcendental equations (Q1907875)

Let \(f: X^{(0)}\subseteq {\mathbb{R}}\to {\mathbb{R}}\) be a function which has a multiple root \(x^*\) in the compact interval \(X^{(0)}\). Let the function \(\phi\) be defined on \(X^{(0)}\), and let \(\phi\) generate an iterative method via \(z_{k+ 1}= \phi(z_k)\) which converges to \(x^*\) locally with convergence order \(p\). Using interval arithmetic this iterative method is enhanced to a new one which converges globally on \(X^{(0)}\) to \(x^*\) without loosing the order of convergence. Four numerical examples illustrate the efficiency of the method.