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Complexity of Real Root Isolation Using Continued Vikram Sharma

Summary: Complexity of Real Root Isolation Using Continued
Vikram Sharma
Sophia Antipolis, France
January 25, 2007
Akritas had proposed an algorithm, which utilizes the continued fraction expansion
of real algebraic numbers, for isolating the real roots of a univariate polynomial. The
efficiency of the algorithm 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 a lower bound
that is at a constant distance from the smallest positive root; without this assumption,
the worst case bounds are exponential. In this paper, we derive a polynomial worst
case bound on the algorithm without relying on the above mentioned assumption. In
particular, we show that for a square-free integer polynomial of degree n and coefficients
of bit-length L, the bit-complexity of the algorithm is O(n8L3), where O indicates that
we are omitting logarithmic factors.
1 Introduction


Source: Akritas, Alkiviadis G. - Department of Computer and Communication Engineering, University of Thessaly


Collections: Computer Technologies and Information Sciences