Bounds for positive roots of polynomials
Journal of Computational and Applied Mathematics
There is no “Uspensky's method.”
SYMSAC '86 Proceedings of the fifth ACM symposium on Symbolic and algebraic computation
Elements of computer algebra with applications
Elements of computer algebra with applications
Fast algorithms for Taylor shifts and certain difference equations
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Bounds for absolute positiveness of multivariate polynomials
Journal of Symbolic Computation
A core library for robust numeric and geometric computation
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Polynomial real root isolation using Descarte's rule of signs
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Some inequalities about univariate polynomials
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Univariate polynomial real root isolation: continued fractions revisited
ESA'06 Proceedings of the 14th conference on Annual European Symposium - Volume 14
Complexity analysis of algorithms in algebraic computation
Complexity analysis of algorithms in algebraic computation
New bounds for the Descartes method
Journal of Symbolic Computation
On the complexity of real root isolation using continued fractions
Theoretical Computer Science
Complexity of real root isolation using continued fractions
Theoretical Computer Science
IEEE Transactions on Signal Processing
Algebraic and numerical algorithms
Algorithms and theory of computation handbook
On the computing time of the continued fractions method
Journal of Symbolic Computation
| Hi-index | 0.01 |
The efficiency of the continued fraction algorithm for isolating the real roots of a univariate polynomial depends upon computing tight lower bounds on the smallest positive root of a polynomial. The known complexity bounds for the algorithm rely on the impractical assumption that it is possible to efficiently compute the floor of the smallest positive root of a polynomial; without this assumption, the worst case bounds are exponential. In this paper, we derive the first polynomial worst case bound on the algorithm: for a square-free integer polynomial of degree n and coefficients of bit-length L, the bit-complexity of the continued fraction algorithm is Õ(n7L2),using a bound by Hong to compute the floor of the smallest positive root of a polynomial; here Õ indicates that we are omitting logarithmic factors.