John Abbott

John Abbott

We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when refining isolating intervals of simple roots of polynomials (and other well-behaved functions). We assume the use of arbitrary precision rational arithmetic. Unlike Newton's Iteration our method does not need to evaluate the derivative.

John Abbott, Claudia Fassino, Maria-Laura Torrente

Let $X$ be a set of points whose coordinates are known with limited accuracy; our aim is to give a characterization of the vanishing ideal $I(X)$ independent of the data uncertainty. We present a method to compute a polynomial basis $B$ of $I(X)$ which exhibits structural stability, that is, if $\widetilde X$ is any set of points differing only slightly from $X$, there exists a polynomial set $\widetilde B$ structurally similar to $B$, which is a basis of the perturbed ideal $ I(\widetilde X)$.