Continued fraction real root isolation using the Hong root bound

Abstract

An investigation of the codominance maximum computing time of the continued fractions method (CF) for isolation of the real roots of a squarefree integral polynomial when applied to the two-parameter family of polynomials Aa,n(x)=xn−2(ax2−(a+2)x+1)2, with n≥5 and a≥1. These polynomials have two roots, r1 and r2, in the interval (0,1), with |r1−r2|<a−n. It is proved that for these polynomials the maximum time required by CF to isolate those two close roots would be codominant with n5(ln⁡a)2 even if an “ideal” root bound were available and either the Horner method or the Budan method is used for translations. It is proved that if a power-of-two Hong root bound is used by CF to determine translation amounts then the time required to isolate the two close roots is dominated by n6(ln⁡a) if a multiplication-free Budan translation method is used. Computations reveal that the Hong root bound is surprisingly effective when applied to the transformed polynomials that arise, engendering a minimum efficiency conjecture. It is proved that if the conjecture is true then the time to isolate the two close roots is dominated by n5(ln⁡a)2. There is also evidence for a maximum efficiency conjecture. The two conjectures together, if true, make it likely that this time is codominant with n5(ln⁡a)2.