365 research outputs found
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
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
Near Optimal Subdivision Algorithms for Real Root Isolation
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 , which is better by
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.Comment: 19 pages, 3 figure
When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial
We introduce a new algorithm denoted DSC2 to isolate the real roots of a
univariate square-free polynomial f with integer coefficients. The algorithm
iteratively subdivides an initial interval which is known to contain all real
roots of f. The main novelty of our approach is that we combine Descartes' Rule
of Signs and Newton iteration. More precisely, instead of using a fixed
subdivision strategy such as bisection in each iteration, a Newton step based
on the number of sign variations for an actual interval is considered, and,
only if the Newton step fails, we fall back to bisection. Following this
approach, our analysis shows that, for most iterations, we can achieve
quadratic convergence towards the real roots. In terms of complexity, our
method induces a recursion tree of almost optimal size O(nlog(n tau)), where n
denotes the degree of the polynomial and tau the bitsize of its coefficients.
The latter bound constitutes an improvement by a factor of tau upon all
existing subdivision methods for the task of isolating the real roots. In
addition, we provide a bit complexity analysis showing that DSC2 needs only
\tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches
the best bound known for this fundamental problem. However, in comparison to
the much more involved algorithms by Pan and Sch\"onhage (for the task of
isolating all complex roots) which achieve the same bit complexity, DSC2
focuses on real root isolation, is very easy to access and easy to implement
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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