99,588 research outputs found
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy
We describe an algorithm to count the number of distinct real zeros of a
polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations
where n is the number of polynomials (as well as the dimension of the ambient
space), D is a bound on the polynomials' degree, and kappa(f) is a condition
number for the system. Each iteration uses an exponential number of operations.
The algorithm uses finite-precision arithmetic and a polynomial bound for the
precision required to ensure the returned output is correct is exhibited. This
bound is a major feature of our algorithm since it is in contrast with the
exponential precision required by the existing (symbolic) algorithms for
counting real zeros. The algorithm parallelizes well in the sense that each
iteration can be computed in parallel polynomial time with an exponential
number of processors.Comment: We made minor but necessary improvements in the presentatio
Turbo Packet Combining for Broadband Space-Time BICM Hybrid-ARQ Systems with Co-Channel Interference
In this paper, efficient turbo packet combining for single carrier (SC)
broadband multiple-input--multiple-output (MIMO) hybrid--automatic repeat
request (ARQ) transmission with unknown co-channel interference (CCI) is
studied. We propose a new frequency domain soft minimum mean square error
(MMSE)-based signal level combining technique where received signals and
channel frequency responses (CFR)s corresponding to all retransmissions are
used to decode the data packet. We provide a recursive implementation algorithm
for the introduced scheme, and show that both its computational complexity and
memory requirements are quite insensitive to the ARQ delay, i.e., maximum
number of ARQ rounds. Furthermore, we analyze the asymptotic performance, and
show that under a sum-rank condition on the CCI MIMO ARQ channel, the proposed
packet combining scheme is not interference-limited. Simulation results are
provided to demonstrate the gains offered by the proposed technique.Comment: 12 pages, 7 figures, and 2 table
Short and long-term wind turbine power output prediction
In the wind energy industry, it is of great importance to develop models that
accurately forecast the power output of a wind turbine, as such predictions are
used for wind farm location assessment or power pricing and bidding,
monitoring, and preventive maintenance. As a first step, and following the
guidelines of the existing literature, we use the supervisory control and data
acquisition (SCADA) data to model the wind turbine power curve (WTPC). We
explore various parametric and non-parametric approaches for the modeling of
the WTPC, such as parametric logistic functions, and non-parametric piecewise
linear, polynomial, or cubic spline interpolation functions. We demonstrate
that all aforementioned classes of models are rich enough (with respect to
their relative complexity) to accurately model the WTPC, as their mean squared
error (MSE) is close to the MSE lower bound calculated from the historical
data. We further enhance the accuracy of our proposed model, by incorporating
additional environmental factors that affect the power output, such as the
ambient temperature, and the wind direction. However, all aforementioned
models, when it comes to forecasting, seem to have an intrinsic limitation, due
to their inability to capture the inherent auto-correlation of the data. To
avoid this conundrum, we show that adding a properly scaled ARMA modeling layer
increases short-term prediction performance, while keeping the long-term
prediction capability of the model
Tangent Graeffe Iteration
Graeffe iteration was the choice algorithm for solving univariate polynomials
in the XIX-th and early XX-th century. In this paper, a new variation of
Graeffe iteration is given, suitable to IEEE floating-point arithmetics of
modern digital computers. We prove that under a certain generic assumption the
proposed algorithm converges. We also estimate the error after N iterations and
the running cost. The main ideas from which this algorithm is built are:
classical Graeffe iteration and Newton Diagrams, changes of scale
(renormalization), and replacement of a difference technique by a
differentiation one. The algorithm was implemented successfully and a number of
numerical experiments are displayed
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