On the maximum computing time of the bisection method for real root isolation

Abstract

The bisection method for polynomial real root isolation was introduced by Collins and Akritas in 1976. In 1981 Mignotte introduced the polynomials $A_{a,n}(x)=x^n-2{(ax-1)}^2$, a an integer, $a\ge 2$ and $n\ge 3$. First we prove that if a is odd then the computing time of the bisection method when applied to $A_{a,n}$ dominates $n^5{(\log d)}^2$ where d is the maximum norm of $A_{a,n}$. Then we prove that if A is any polynomial of degree n with maximum norm d then the computing time of the bisection method, with a minor improvement regarding homothetic transformations, is dominated by $n^5{(\log d)}^2$. It follows that the maximum computing time of the bisection method is codominant with $n^5{(\log d)}^2$.