21,473 research outputs found
Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation
In this paper, a linear univariate representation for the roots of a
zero-dimensional polynomial equation system is presented, where the roots of
the equation system are represented as linear combinations of roots of several
univariate polynomial equations. The main advantage of this representation is
that the precision of the roots can be easily controlled. In fact, based on the
linear univariate representation, we can give the exact precisions needed for
roots of the univariate equations in order to obtain the roots of the equation
system to a given precision. As a consequence, a root isolation algorithm for a
zero-dimensional polynomial equation system can be easily derived from its
linear univariate representation.Comment: 19 pages,2 figures; MM-Preprint of KLMM, Vol. 29, 92-111, Aug. 201
A Generic Position Based Method for Real Root Isolation of Zero-Dimensional Polynomial Systems
We improve the local generic position method for isolating the real roots of
a zero-dimensional bivariate polynomial system with two polynomials and extend
the method to general zero-dimensional polynomial systems. The method mainly
involves resultant computation and real root isolation of univariate polynomial
equations. The roots of the system have a linear univariate representation. The
complexity of the method is for the bivariate case, where
, resp., is an upper bound on the degree, resp., the
maximal coefficient bitsize of the input polynomials. The algorithm is
certified with probability 1 in the multivariate case. The implementation shows
that the method is efficient, especially for bivariate polynomial systems.Comment: 24 pages, 5 figure
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
Clustering Complex Zeros of Triangular Systems of Polynomials
This paper gives the first algorithm for finding a set of natural
-clusters of complex zeros of a triangular system of polynomials
within a given polybox in , for any given . Our
algorithm is based on a recent near-optimal algorithm of Becker et al (2016)
for clustering the complex roots of a univariate polynomial where the
coefficients are represented by number oracles.
Our algorithm is numeric, certified and based on subdivision. We implemented
it and compared it with two well-known homotopy solvers on various triangular
systems. Our solver always gives correct answers, is often faster than the
homotopy solver that often gives correct answers, and sometimes faster than the
one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update
On the Complexity of Real Root Isolation
We introduce a new approach to isolate the real roots of a square-free
polynomial with real coefficients. It is assumed that
each coefficient of can be approximated to any specified error bound. The
presented method is exact, complete and deterministic. Due to its similarities
to the Descartes method, we also consider it practical and easy to implement.
Compared to previous approaches, our new method achieves a significantly better
bit complexity. It is further shown that the hardness of isolating the real
roots of is exclusively determined by the geometry of the roots and not by
the complexity or the size of the coefficients. For the special case where
has integer coefficients of maximal bitsize , our bound on the bit
complexity writes as which improves the best bounds
known for existing practical algorithms by a factor of . The crucial
idea underlying the new approach is to run an approximate version of the
Descartes method, where, in each subdivision step, we only consider
approximations of the intermediate results to a certain precision. We give an
upper bound on the maximal precision that is needed for isolating the roots of
. For integer polynomials, this bound is by a factor lower than that of
the precision needed when using exact arithmetic explaining the improved bound
on the bit complexity
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