Continued fraction real root isolation using the Hong root bound. (January 2016) - Collins, George E.
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 A a, n ( x ) = x n − 2 ( a x 2 − ( a + 2 ) x + 1 ) 2, with n ≥ 5 and a ≥ 1 . These polynomials have two roots, r 1 and r 2, in the interval ( 0, 1 ), with | r 1 − r 2 | < 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 n 5 ( 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 n 6 ( 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 n 5 ( 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 n 5 ( ln ⁡ a ) 2 .