1,245 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
State-of-the-art in aerodynamic shape optimisation methods
Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners
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
On the Complexity of Computing with Planar Algebraic Curves
In this paper, we give improved bounds for the computational complexity of
computing with planar algebraic curves. More specifically, for arbitrary
coprime polynomials , and an arbitrary polynomial , each of total degree less than and with integer
coefficients of absolute value less than , we show that each of the
following problems can be solved in a deterministic way with a number of bit
operations bounded by , where we ignore polylogarithmic
factors in and :
(1) The computation of isolating regions in for all complex
solutions of the system ,
(2) the computation of a separating form for the solutions of ,
(3) the computation of the sign of at all real valued solutions of , and
(4) the computation of the topology of the planar algebraic curve
defined as the real valued vanishing set of the polynomial .
Our bound improves upon the best currently known bounds for the first three
problems by a factor of or more and closes the gap to the
state-of-the-art randomized complexity for the last problem.Comment: 41 pages, 1 figur
Faster all-pairs shortest paths via circuit complexity
We present a new randomized method for computing the min-plus product
(a.k.a., tropical product) of two matrices, yielding a faster
algorithm for solving the all-pairs shortest path problem (APSP) in dense
-node directed graphs with arbitrary edge weights. On the real RAM, where
additions and comparisons of reals are unit cost (but all other operations have
typical logarithmic cost), the algorithm runs in time
and is correct with high probability.
On the word RAM, the algorithm runs in time for edge weights in . Prior algorithms used either time for
various , or time for various
and .
The new algorithm applies a tool from circuit complexity, namely the
Razborov-Smolensky polynomials for approximately representing
circuits, to efficiently reduce a matrix product over the algebra to
a relatively small number of rectangular matrix products over ,
each of which are computable using a particularly efficient method due to
Coppersmith. We also give a deterministic version of the algorithm running in
time for some , which utilizes the
Yao-Beigel-Tarui translation of circuits into "nice" depth-two
circuits.Comment: 24 pages. Updated version now has slightly faster running time. To
appear in ACM Symposium on Theory of Computing (STOC), 201
Multipoint passive monitoring in packet networks
Traffic monitoring is essential to manage large networks and validate Service Level Agreements. Passive monitoring is particularly valuable to promptly identify transient fault episodes and react in a timely manner. This paper proposes a novel, non-invasive and flexible method to passively monitor large backbone networks. By using only packet counters, commonly available on existing hardware, we can accurately measure packet losses, in different segments of the network, affecting only specific flows. We can monitor not only end-to-end flows, but any generic flow with packets following several different paths in the network (multipoint flows). We also sketch a possible extension of the method to measure average one-way delay for multipoint flows, provided that the measurement points are synchronized. Through various experiments we show that the method is effective and enables easy zooming in on the cause packet losses. Moreover, the method can scale to very large networks with a very low overhead on the data plane and the management plane
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