Complexity of real root isolation using continued fractions
نویسندگان
چکیده
منابع مشابه
On the complexity of real root isolation using continued fractions
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method’s good performance in practice. We derive an expected complexity bound of ÕB(d + d4τ2), where d is the polynomial degree and τ bounds the coefficient bit size, usin...
متن کاملImproved 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 ÕB(d + dτ + dτ) for isolating the real roots of a polynomial with integer coeffic...
متن کاملUnivariate Polynomial Real Root Isolation: Continued Fractions Revisited
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method’s good performance in practice. We improve the previously known bound by a factor of dτ , where d is the polynomial degree and τ bounds the coefficient bitsize, thu...
متن کاملOn the Complexity of Real Root Isolation
We introduce a new method to isolate the real roots of a square-free polynomial F = ∑i=0 Aix with real coefficients Ai, where |An| ≥ 1 and |Ai| ≤ 2τ for all i. It is assumed that each coefficient of F can be approximated to any specified error bound. The presented method is exact, complete and deterministic. Due to its similarities to the Descartes method, we also consider it practical and easy...
متن کاملRevisited Using Continued Fractions
If the equation of the title has an integer solution with k ≥ 2, then m > 109.3·10 6 . This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m > 1010 7 . Here we achieve m > 1010 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , wit...
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ژورنال
عنوان ژورنال: Theoretical Computer Science
سال: 2008
ISSN: 0304-3975
DOI: 10.1016/j.tcs.2008.09.017