Algorithms for Polynomial Real Root Isolation

Abstract

This thesis investigates algorithms for polynomial real root isolation of polynomials with integer and real algebraic number coefficients. A real root isolation algorithm computes isolating intervals (intervals containing exactly one root) for all of the real roots of a polynomial. Improved algorithms are derived and implemented for both integral and algebraic polynomials. Some of the improvements include fast algorithms for polynomial permutation and transformation, improved algorithms for computing the sign of a real algebraic number, algorithms for computing a root bound of a polynomial with real algebraic number coefficients, the use of multiple extensions, and the use of interval arithmetic. Algorithms based on Sturm sequences, Rolle's theorem and the derivative sequence, and Descartes' rule of signs and polynomial transformations are considered. The algorithms are carefully compared both theoretically and empirically. Improved computing time bounds are obtained, strong conjectures are provided for the average computing times, and insight is given into the performance of the various algorithms. This thesis contains the first careful comparison and implementation of these algorithms for polynomials with real algebraic number coefficients. Real root isolation is an essential and time consuming part of G. E. Collins' cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) algorithm. The improved algorithms presented in this thesis significantly reduce the computing time of Collins' algorithm. The different root isolation algorithms are compared using inputs that arise in the CAD algorithm. The CAD algorithm is also used to study the effect of perturbation of the coefficients of a polynomial on the number of real roots of the polynomial. Results from this study have implications for algorithms that use interval arithmetic.