56 research outputs found
Computing Real Roots of Real Polynomials
Computing the roots of a univariate polynomial is a fundamental and
long-studied problem of computational algebra with applications in mathematics,
engineering, computer science, and the natural sciences. For isolating as well
as for approximating all complex roots, the best algorithm known is based on an
almost optimal method for approximate polynomial factorization, introduced by
Pan in 2002. Pan's factorization algorithm goes back to the splitting circle
method from Schoenhage in 1982. The main drawbacks of Pan's method are that it
is quite involved and that all roots have to be computed at the same time. For
the important special case, where only the real roots have to be computed, much
simpler methods are used in practice; however, they considerably lag behind
Pan's method with respect to complexity.
In this paper, we resolve this discrepancy by introducing a hybrid of the
Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than
Pan's method, but achieves a run-time comparable to it. Our algorithm computes
isolating intervals for the real roots of any real square-free polynomial,
given by an oracle that provides arbitrary good approximations of the
polynomial's coefficients. ANEWDSC can also be used to only isolate the roots
in a given interval and to refine the isolating intervals to an arbitrary small
size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio
Computing Real Roots of Real Polynomials ... and now For Real!
Very recent work introduces an asymptotically fast subdivision algorithm,
denoted ANewDsc, for isolating the real roots of a univariate real polynomial.
The method combines Descartes' Rule of Signs to test intervals for the
existence of roots, Newton iteration to speed up convergence against clusters
of roots, and approximate computation to decrease the required precision. It
achieves record bounds on the worst-case complexity for the considered problem,
matching the complexity of Pan's method for computing all complex roots and
improving upon the complexity of other subdivision methods by several
magnitudes.
In the article at hand, we report on an implementation of ANewDsc on top of
the RS root isolator. RS is a highly efficient realization of the classical
Descartes method and currently serves as the default real root solver in Maple.
We describe crucial design changes within ANewDsc and RS that led to a
high-performance implementation without harming the theoretical complexity of
the underlying algorithm.
With an excerpt of our extensive collection of benchmarks, available online
at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in
performance of ANewDsc over other subdivision methods also transfers into
practice. These experiments also show that our new implementation outperforms
both RS and mature competitors by magnitudes for notoriously hard instances
with clustered roots. For all other instances, we avoid almost any overhead by
integrating additional optimizations and heuristics.Comment: Accepted for presentation at the 41st International Symposium on
Symbolic and Algebraic Computation (ISSAC), July 19--22, 2016, Waterloo,
Ontario, Canad
Computing Real Roots of Real Polynomials -- An Efficient Method Based on Descartes' Rule of Signs and Newton Iteration
Computing the real roots of a polynomial is a fundamental problem of computational algebra. We describe a variant of the Descartes method that isolates the real roots of any real square-free polynomial given through coefficient oracles. A coefficient oracle provides arbitrarily good approximations of the coefficients. The bit complexity of the algorithm matches the complexity of the best algorithm known, and the algorithm is simpler than this algorithm. The algorithm derives its speed from the combination of Descartes method with Newton iteration. Our algorithm can also be used to further refine the isolating intervals to an arbitrary small size. The complexity of root refinement is nearly optimal
New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS
Univariate polynomial root-finding has been studied for four millennia and is
still the subject of intensive research. Hundreds of efficient algorithms for
this task have been proposed. Two of them are nearly optimal. The first one,
proposed in 1995, relies on recursive factorization of a polynomial, is quite
involved, and has never been implemented. The second one, proposed in 2016,
relies on subdivision iterations, was implemented in 2018, and promises to be
practically competitive, although user's current choice for univariate
polynomial root-finding is the package MPSolve, proposed in 2000, revised in
2014, and based on Ehrlich's functional iterations. By proposing and
incorporating some novel techniques we significantly accelerate both
subdivision and Ehrlich's iterations. Moreover our acceleration of the known
subdivision root-finders is dramatic in the case of sparse input polynomials.
Our techniques can be of some independent interest for the design and analysis
of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table
Fast real and complex root-finding methods for well-conditioned polynomials
Given a polynomial of degree and a bound on a condition
number of , we present the first root-finding algorithms that return all its
real and complex roots with a number of bit operations quasi-linear in . More precisely, several condition numbers can be defined
depending on the norm chosen on the coefficients of the polynomial. Let . We call the
condition number associated with a perturbation of the the hyperbolic
condition number , and the one associated with a perturbation of the
the elliptic condition number . For each of these condition
numbers, we present algorithms that find the real and the complex roots of
in bit
operations.Our algorithms are well suited for random polynomials since
(resp. ) is bounded by a polynomial in with high
probability if the (resp. the ) are independent, centered Gaussian
variables of variance
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
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