The Bernstein Basis and Real Root Isolation (original) (raw)

Near Optimal Subdivision Algorithms for Real Root Isolation

Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '15, 2015

The problem of isolating real roots of a square-free polynomial inside a given interval I0 is a fundamental problem. Subdivision based algorithms are a standard approach to solve this problem. Given an interval I, such algorithms rely on two predicates: an exclusion predicate, which if true means I has no roots, and an inclusion predicate, which if true, reports an isolated root in I. If neither predicate holds, then we subdivide the interval and proceed recursively, starting from I0. Example algorithms are Sturm's method (predicates based on Sturm sequences), the Descartes method (using Descartes's rule of signs), and Eval (using interval-arithmetic). For the canonical problem of isolating all real roots of a degree n polynomial with integer coefficients of bit-length L, the subdivision tree size of (almost all) these algorithms is bounded by O(n(L + log n)). This is known to be optimal for subdivision based algorithms. We describe a subroutine that improves the running time of any subdivision algorithm for real root isolation. The subroutine first detects clusters of roots using a result of Ostrowski, and then uses Newton iteration to converge to them. Near a cluster, we switch to subdivision, and proceed recursively. The subroutine has the advantage that it is independent of the predicates used to terminate the subdivision. This gives us an alternative and simpler approach to recent developments of Sagraloff (2012) and Sagraloff-Mehlhorn (2013), assuming exact arithmetic. The subdivision tree size of our algorithm using predicates based on Descartes's rule of signs is bounded by O(n log n), which is better by O(n log L) compared to known results. Our analysis differs in two key aspects. First, we use the general technique of continuous amortization from Burr-Krahmer-Yap (2009), and second, we use the geometry of clusters of roots instead of the Davenport-Mahler bound. The analysis naturally extends to other predicates.

Empirical study of an evaluation-based subdivision algorithm for complex root isolation

2012

We provide an empirical study of subdivision algorithms for isolating the simple roots of a polynomial in any desired box region B0 of the complex plane. One such class of algorithms is based on Newton-like interval methods (Moore, Krawczyk, Hansen-Sengupta). Another class of subdivision algorithms is based on function evaluation. Here, Yakoubsohn discussed a method that is purely based on an exclusion predicate. Recently, Sagraloff and Yap introduced another algorithm of this type, called Ceval. We describe the first implementation of Ceval in Core Library. We compare its performance to the above mentioned algorithms, and also to the well-known MPSolve software from Bini and Florentino. Our results suggest that certified evaluation-based methods such as Ceval are encouraging and deserve further exploration.

Cache Complexity and Multicore Implementation for Univariate Real Root Isolation

Journal of Physics: Conference Series, 2012

We present parallel algorithms with optimal cache complexity for the kernel routine of many real root isolation algorithms, namely the Taylor shift by 1. We then report on multicore implementation for isolating the real roots of univariate polynomials. For processing some wellknown benchmark examples with sufficiently large size, our software tool reaches linear speedup on an 8-core machine. In addition, we show that our software is able to fully utilize the many cores and the memory space of a 32-core machine to tackle large problems that are out of reach for a desktop implementation.

A note on the complexity of univariate root isolation

2006

This paper presents the average-case bit complexity of subdivision-based univariate solvers, namely those named after Sturm, Descartes, and Bernstein. By real solving we mean real root isolation. We prove bounds of O B (N 5 ) for all methods, where N bounds the polynomial degree and the coefficient bitsize, whereas their worst-case complexity is in O B (N 6 ). In the case of the Sturm solver, our bound depends on the number of real roots. Our work is a step towards understanding the practical complexity of real root isolation. This enables a better juxtaposition against numerical solvers, the latter having worst-case complexity in O B (N 4 ). Our approach extends to complex root isolation, where we offer a simple proof leading to bounds on the worst and average-case complexities of O B (N 7 ) and O B (N 6 ) respectively, where the latter is output-sensitive.

Parallel Univariate Real Root Isolation on Multicore Processors

2011

We present parallel algorithms with optimal cache complexity for the kernel routine of many real root isolation algorithms, namely, Taylor shift, targeting multicore processors. We then report an efficient multithreaded implementation for isolating the real roots of univariate polynomials based on the parallel Taylor shift algorithms. For processing some wellknown benchmark examples with sufficiently large size, our software tool reaches linear speedup on a 8-core machine. In addition, we show that our software is able to fully utilize the many cores and the memory space of a 32-core machine to tackle large problems that are out of reach for a desktop implementation.

An improved algorithm for the isolation of polynomial real zeros

NASA. Langley Res. Center Proc. of the 1977 …, 1977

THe Collins-Loos algorithm for computing isolating intervals for the zeros of an integer polynomial requires the evaluation of polynomials at rational points. This implies the use of arbitrary precision integer arithmetic. It is shown how careful use of single precision, ...

Real Root Isolation of Regular Chains

Computer Mathematics, 2014

We present an algorithm RealRootIsolate for isolating the real roots of a system of multivariate polynomials given by a zerodimensional squarefree regular chain. The output of the algorithm is guaranteed in the sense that all real roots are obtained and are described by boxes of arbitrary precision. Real roots are encoded with a hybrid representation, combining a symbolic object, namely a regular chain, and a numerical approximation given by intervals. Our isolation algorithm is a generalization, for regular chains, of the algorithm proposed by Collins and Akritas. We have implemented RealRootIsolate as a command of the module SemiAlgebraicSetTools of the RegularChains library in Maple. Benchmarks are reported. * Recent investigations of A. Akritas seem to prove that Uspensky only had an incomplete knowledge of Vincent's paper, from [30, pages 363-368].

Continued Fraction Expansion of Real Roots of Polynomial Systems

Proceedings of the 2009 …, 2009

We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the univariate continued fraction algorithm or alternatively as a fully analog of Bernstein subdivision in the monomial basis. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditionning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. An extension of Vincent's theorem to multivariate polynomials is proved and used for the termination of the algorithm. New complexity bounds are provided for a simplified version of the algorithm. Examples computed with a preliminary C++ implementation illustrate the approach.

The Root Separation of Polynomials and Some Applications

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1995

PETKOVIC, M.; MIGNOTTE, M.; TRAJKOVIC, M. : Root Separation of Polynomials 55 1 I ZAMM . Z. angew. Math. Mech. 75 (1995) 7, 551 -561 PETKOVI~, M.; MIGNOTTE, M.; TRAJKOVIC, M.