Beyond Worst-Case Analysis for Root Isolation Algorithms

Alperen Ergür, Josué Tonelli-Cueto, Elias Tsigaridas
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients
more » ... bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of Descartes solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is O B (d 2 + dτ ), where d is the degree of the polynomial and τ the bitsize of the coefficients.
doi:10.1145/3476446.3535475 fatcat:dsxkshe4bva5he3ym6dpswqtgu