Improved complexity bounds for real root isolation using Continued Fractions

We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real roots of univariate polynomials. This allows us to derive a worst case bound of OB(d6 + d4 2 + d3 2 ) for isolating the real roots of a polynomial with integer coefficients using the classic variant of CF, where d is the degree of the polynomial and the maximum bitsize of its coefficients. This improves the previous bound by Sharma [30] by a factor of d3 and matches the bound derived by Mehlhorn and Ray [21] for another variant of CF; it also matches the worst case bound of the subdivisionbased solvers. We present a new variant of CF, we call it iCF, that isolates the real roots of a polynomial with integer coefficients in OB(d5 + d4 ), thus improving the current known bound for the problem by a factor of d. If the polynomial has only real roots, then ...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Elias P. Tsigaridas
• Keywords: CORR 2010, Education, Integer Coefficients, Polynomial, Real Root