1,943 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
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
Exact Symbolic-Numeric Computation of Planar Algebraic Curves
We present a novel certified and complete algorithm to compute arrangements
of real planar algebraic curves. It provides a geometric-topological analysis
of the decomposition of the plane induced by a finite number of algebraic
curves in terms of a cylindrical algebraic decomposition. From a high-level
perspective, the overall method splits into two main subroutines, namely an
algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional
bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic
curve.
Compared to existing approaches based on elimination techniques, we
considerably improve the corresponding lifting steps in both subroutines. As a
result, generic position of the input system is never assumed, and thus our
algorithm never demands for any change of coordinates. In addition, we
significantly limit the types of involved exact operations, that is, we only
use resultant and gcd computations as purely symbolic operations. The latter
results are achieved by combining techniques from different fields such as
(modular) symbolic computation, numerical analysis and algebraic geometry.
We have implemented our algorithms as prototypical contributions to the
C++-project CGAL. They exploit graphics hardware to expedite the symbolic
computations. We have also compared our implementation with the current
reference implementations, that is, LGP and Maple's Isolate for polynomial
system solving, and CGAL's bivariate algebraic kernel for analyses and
arrangement computations of algebraic curves. For various series of challenging
instances, our exhaustive experiments show that the new implementations
outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011.
arXiv admin note: substantial text overlap with arXiv:1010.1386 and
arXiv:1103.469
Experimental Tests of Asymptotic Freedom
Measurements which probe the energy dependence of , the coupling
strength of the strong interaction, are reviewed. Jet counting in
annihilation, combining results obtained in the centre of mass energy range
from 22 to 133 GeV, provides direct evidence for an asymptotically free
coupling, without the need to determine explicit values of . Recent
results from jet production in and in collisions,
obtained in single experiments spanning large ranges of momentum transfer,
, are in good agreement with the running of as predicted by
QCD. Mass spectra of hadronic decays of -leptons are analysed to probe
the running in the very low energy domain, . An update of the world summary of measurements of
consistently proves the energy dependence of and results in a
combined average of .Comment: 11 pages, 8 Figures, LaTeX. To appear in Proc. of QCD Euroconference
96 at Montpellier, France, July 199
Measurement of Rb in e+e- Collisions at 182 - 209 GeV
Measurements of Rb, the ratio of the bbbar cross-section to the qqbar cross-
section in e+e- collisions, are presented. The data were collected by the OPAL
experiment at LEP at centre-of-mass energies between 182 GeV and 209 GeV.
Lepton, lifetime and event shape information is used to tag events containing b
quarks with high efficiency. The data are compatible with the Standard Model
expectation. The mean ratio of the eight measurements reported here to the
Standard Model prediction is 1.055+-0.031+-0.037, where the first error is
statistical and the second systematic.Comment: 21 pages, 5 figures, Submitted to Phys. Letts
Measurement of the partial widths of the Z into up- and down-type quarks
Using the entire OPAL LEP1 on-peak Z hadronic decay sample, Z -> qbarq gamma
decays were selected by tagging hadronic final states with isolated photon
candidates in the electromagnetic calorimeter. Combining the measured rates of
Z -> qbarq gamma decays with the total rate of hadronic Z decays permits the
simultaneous determination of the widths of the Z into up- and down-type
quarks. The values obtained, with total errors, were Gamma u = 300 ^{+19}_{-18}
MeV and Gamma d = 381 ^{+12}_{-12} MeV. The results are in good agreement with
the Standard Model expectation.Comment: 22 pages, 5 figures, Submitted to Phys. Letts.
Genuine Correlations of Like-Sign Particles in Hadronic Z0 Decays
Correlations among hadrons with the same electric charge produced in Z0
decays are studied using the high statistics data collected from 1991 through
1995 with the OPAL detector at LEP. Normalized factorial cumulants up to fourth
order are used to measure genuine particle correlations as a function of the
size of phase space domains in rapidity, azimuthal angle and transverse
momentum. Both all-charge and like-sign particle combinations show strong
positive genuine correlations. One-dimensional cumulants initially increase
rapidly with decreasing size of the phase space cells but saturate quickly. In
contrast, cumulants in two- and three-dimensional domains continue to increase.
The strong rise of the cumulants for all-charge multiplets is increasingly
driven by that of like-sign multiplets. This points to the likely influence of
Bose-Einstein correlations. Some of the recently proposed algorithms to
simulate Bose-Einstein effects, implemented in the Monte Carlo model PYTHIA,
are found to reproduce reasonably well the measured second- and higher-order
correlations between particles with the same charge as well as those in
all-charge particle multiplets.Comment: 26 pages, 6 figures, Submitted to Phys. Lett.
WW Production Cross Section and W Branching Fractions in e+e- Collisions at 189 GeV
From a data sample of 183 pb^-1 recorded at a center-of-mass energy of roots
= 189 GeV with the OPAL detector at LEP, 3068 W-pair candidate events are
selected. Assuming Standard Model W boson decay branching fractions, the W-pair
production cross section is measured to be sigmaWW = 16.30 +- 0.34(stat.) +-
0.18(syst.) pb. When combined with previous OPAL measurements, the W boson
branching fraction to hadrons is determined to be 68.32 +- 0.61(stat.) +-
0.28(syst.) % assuming lepton universality. These results are consistent with
Standard Model expectations.Comment: 22 pages, 5 figures, submitted to Phys. Lett.
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