5,375 research outputs found
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
A Near-Optimal Algorithm for Computing Real Roots of Sparse Polynomials
Let be an arbitrary polynomial of degree with
non-zero integer coefficients of absolute value less than . In this
paper, we answer the open question whether the real roots of can be
computed with a number of arithmetic operations over the rational numbers that
is polynomial in the input size of the sparse representation of . More
precisely, we give a deterministic, complete, and certified algorithm that
determines isolating intervals for all real roots of with
many exact arithmetic operations over the
rational numbers.
When using approximate but certified arithmetic, the bit complexity of our
algorithm is bounded by , where
means that we ignore logarithmic. Hence, for sufficiently sparse polynomials
(i.e. for a positive constant ), the bit complexity is
. We also prove that the latter bound is optimal up to
logarithmic factors
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
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
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
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