Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals. (arXiv:0910.1841v1 [math.NA])

High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated by
trapezoidal sums. Whereas analytically each radius r up to the radius of
convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.

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Original source : http://arxiv.org/abs/0910.1841...

Finite Elements for a Beam System With Nonlinear Contact Under Periodic Excitation. (arXiv:0910.2092v1 [math.NA])

Solar arrays are structures which are connected to satellites; during launch,
they are in a folded position and submitted to high vibrations. In order to
save mass, the flexibility of the panels is not negligible and they may strike
each other; this may damage the structure. To prevent this, rubber snubbers are
mounted at well chosen points of the structure; a prestress is applied to the
snubber; but it is quite difficult to check the amount of prestress and the
snubber may act only on one side; they will be modeled as one sided springs
(see figure 2). In this article, some analysis for responses (displacements) in
both time and frequency domains for a clamped-clamped Euler-Bernoulli beam
model with a spring are presented. This spring can be unilateral or bilateral
fixed at a point. The mounting (beam +spring) is fixed on a rigid support which
has a sinusoidal motion of constant frequency. The system is also studied in
the frequency domain by sweeping frequencies between two fixed values, in order
to save the maximum of displacements corresponding to each frequency. Numerical
results are compared with exact solutions in particular cases which already
exist in the literature. On the other hand, a numerical and theoretical
investigation of nonlinear normal mode (NNM) can be a new method to describe
nonlinear behaviors, this work is in progress.

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Original source : http://arxiv.org/abs/0910.2092...

Iterative Methods for the Force-based Quasicontinuum Approximation. (arXiv:0910.2013v1 [math.NA])

Force-based atomistic-continuum hybrid methods are the only known pointwise
consistent methods for coupling a general atomistic model to a finite element
continuum model. For this reason, and due to their algorithmic simplicity,
force-based coupling methods have become a popular approach for
atomistic-continuum hybrid methods as well as other types of multiphysics model
coupling. However, the recently discovered unusual stability properties of the
linearized force-based quasicontinuum approximation, especially its
indefiniteness, present a challenge to the development of efficient and
reliable iterative methods. Using a combination of rigorous analysis and
computational experiments, we present a systematic study of the stability and
rate of convergence of a variety of linear stationary iterative methods and
generalized minimal residual methods (GMRES) for the solution of the linearized
force-based quasicontinuum equations.

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Original source : http://arxiv.org/abs/0910.2013...

Barycentric-Remez algorithms for best polynomial approximation in the chebfun system

Abstract  The Remez algorithm, 75 years old, is a famous method for computing minimax polynomial approximations. Most implementations
of this algorithm date to an era when tractable degrees were in the dozens, whereas today, degrees of hundreds or thousands
are not a problem. We present a 21st-century update of the Remez ideas in the context of the chebfun software system, which
carries out numerical computing with functions rather than numbers. A crucial feature of the new method is its use of chebfun
global rootfinding to locate extrema at each iterative step, based on a recursive algorithm combining ideas of Specht, Good,
Boyd, and Battles. Another important feature is the use of the barycentric interpolation formula to represent the trial polynomials,
which points the way to generalizations for rational approximations. We comment on available software for minimax approximation
and its scientific context, arguing that its greatest importance these days is probably for fundamental studies rather than
applications.

• Content Type Journal Article
• DOI 10.1007/s10543-009-0240-1
• Authors
  
  
  • Ricardo Pachón, University of Oxford Computing Laboratory Parks Rd. Oxford OX1 3QD UK
  • Lloyd N. Trefethen, University of Oxford Computing Laboratory Parks Rd. Oxford OX1 3QD UK
  
  

• Journal BIT Numerical Mathematics
• Online ISSN 1572-9125
• Print ISSN 0006-3835

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Original source : http://www.springerlink.com/content/5n17515j262845...

Generating series for irreducible polynomials over finite fields. (arXiv:0910.1680v1 [math.AG])

We count the number of irreducible polynomials in several variables of a
given degree over a finite field. The results are expressed in terms of a
generating series, an exact formula and an asymptotic approximation. We also
consider the case of the multi-degree and the case of indecomposable
polynomials.

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Original source : http://arxiv.org/abs/0910.1680...