Derivation of interval Newton's method

Question

Saw this formula in my text on interval analysis $$X^{(k+1)} = X^{(k)} \cap N(X^{(k)}) \\ k = 0,1,2,...$$ Where $$N(X^{(k)}) = m(X^{(k)}) - \frac{F(m(X^{(k)}))}{F'(X^{(k)})})$$ X is an interval say [a,b] . While it is similar to the ordinary Newton's method, I can't figure out how to deal with the intersection. Can someone help prove this or rather point me to a material where the formula was derived

Answer 1

The intersection of two intervals $[a_1,b_1]$, $[a_2,b_2]$ is empty if $b_1<a_2$ or $b_2<a_1$, else it is $$[\max(a_1,a_2),\min(b_1,b_2)].$$

If $x^*$ is a root in $X=[a,b]$, then $0=f(x^*)=f(m)+f'(\xi)(x^*-m)$ for some $\xi\in[a,b]$. Thus $x^*\in N(X)$, but other points in the derivative might lead to points outside the interval. Thus the intersection to keep the complexity of the resulting sets down.