1,717 research outputs found
Reconstructing Rational Functions with
We present the open-source library for the
reconstruction of multivariate rational functions over finite fields. We
discuss the involved algorithms and their implementation. As an application, we
use in the context of integration-by-parts reductions and
compare runtime and memory consumption to a fully algebraic approach with the
program .Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio
Reconstruction Algorithms for Sums of Affine Powers
In this paper we study sums of powers of affine functions in (mostly) one
variable. Although quite simple, this model is a generalization of two
well-studied models: Waring decomposition and sparsest shift. For these three
models there are natural extensions to several variables, but this paper is
mostly focused on univariate polynomials. We present structural results which
compare the expressive power of the three models; and we propose algorithms
that find the smallest decomposition of f in the first model (sums of affine
powers) for an input polynomial f given in dense representation. We also begin
a study of the multivariate case. This work could be extended in several
directions. In particular, just as for Sparsest Shift and Waring decomposition,
one could consider extensions to "supersparse" polynomials and attempt a fuller
study of the multi-variate case. We also point out that the basic univariate
problem studied in the present paper is far from completely solved: our
algorithms all rely on some assumptions for the exponents in an optimal
decomposition, and some algorithms also rely on a distinctness assumption for
the shifts. It would be very interesting to weaken these assumptions, or even
to remove them entirely. Another related and poorly understood issue is that of
the bit size of the constants appearing in an optimal decomposition: is it
always polynomially related to the bit size of the input polynomial given in
dense representation?Comment: This version improves on several algorithmic result
Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree
In this paper we study the problem of deterministic factorization of sparse
polynomials. We show that if is a
polynomial with monomials, with individual degrees of its variables bounded
by , then can be deterministically factored in time . Prior to our work, the only efficient factoring algorithms known for
this class of polynomials were randomized, and other than for the cases of
and , only exponential time deterministic factoring algorithms were
known.
A crucial ingredient in our proof is a quasi-polynomial sparsity bound for
factors of sparse polynomials of bounded individual degree. In particular we
show if is an -sparse polynomial in variables, with individual
degrees of its variables bounded by , then the sparsity of each factor of
is bounded by . This is the first nontrivial bound on
factor sparsity for . Our sparsity bound uses techniques from convex
geometry, such as the theory of Newton polytopes and an approximate version of
the classical Carath\'eodory's Theorem.
Our work addresses and partially answers a question of von zur Gathen and
Kaltofen (JCSS 1985) who asked whether a quasi-polynomial bound holds for the
sparsity of factors of sparse polynomials
Parallel Integer Polynomial Multiplication
We propose a new algorithm for multiplying dense polynomials with integer
coefficients in a parallel fashion, targeting multi-core processor
architectures. Complexity estimates and experimental comparisons demonstrate
the advantages of this new approach
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
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