2,611 research outputs found
Accurate calculation of the solutions to the Thomas-Fermi equations
We obtain highly accurate solutions to the Thomas-Fermi equations for atoms
and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method,
numerical integration, power series with Pad\'e and Hermite-Pad\'e approximants
and Chebyshev polynomials. Both the slope at origin and the location of the
right boundary in the magnetic-field case are given with unprecedented
accuracy
Chebyshev expansion on intervals with branch points with application to the root of Keplerâs equation: A ChebyshevâHermiteâPadĂ© method
AbstractWhen two or more branches of a function merge, the Chebyshev series of u(λ) will converge very poorly with coefficients an of Tn(λ) falling as O(1/nα) for some small positive exponent α. However, as shown in [J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 143 (2002) 189â200], it is possible to obtain approximations that converge exponentially fast in n. If the roots that merge are denoted as u1(λ) and u2(λ), then both branches can be written without approximation as the roots of (uâu1(λ))(uâu2(λ))=u2+ÎČ(λ)u+Îł(λ). By expanding the nonsingular coefficients of the quadratic, ÎČ(λ) and Îł(λ), as Chebyshev series and then applying the usual roots-of-a-quadratic formula, we can approximate both branches simultaneously with error that decreases proportional to exp(âÏN) for some constant Ï>0 where N is the truncation of the Chebyshev series. This is dubbed the âChebyshevâShaferâ or âChebyshevâHermiteâPadĂ©â method because it substitutes Chebyshev series for power series in the generalized PadĂ© approximants known variously as âShaferâ or âHermiteâPadĂ©â approximants. Here we extend these ideas. First, we explore square roots with branches that are both real-valued and complex-valued in the domain of interest, illustrated by meteorological baroclinic instability. Second, we illustrate triply branched functions via roots of the Kepler equation, f(u;λ,Ï”)âĄuâÏ”sin(u)âλ=0. Only one of the merging roots is real-valued and the root depends on two parameters (λ,Ï”) rather than one. Nonetheless, the ChebyshevâHermiteâPadĂ© scheme is successful over the whole two-dimensional parameter plane. We also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain
Error saturation in Gaussian radial basis functions on a finite interval
AbstractRadial basis function (RBF) interpolation is a âmeshlessâ strategy with great promise for adaptive approximation. One restriction is âerror saturationâ which occurs for many types of RBFs including Gaussian RBFs of the form Ï(x;α,h)=exp(âα2(x/h)2): in the limit hâ0 for fixed α, the error does not converge to zero, but rather to ES(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases ES(α).) We show experimentally that the saturation error on the unit interval, xâ[â1,1], is about 0.06exp(â0.47/α2)âfââ â huge compared to the O(2Ï/α2)exp(âÏ2/[4α2]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing αâȘ1, the âflat limitâ, but the condition number of the interpolation matrix explodes as O(exp(Ï2/[4α2])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance ÎŽ, then αoptimum(ÎŽ)=1/â2log(ÎŽ/0.06)
Multiprocessing in Meteorological Models
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94918/1/eost7470.pd
Bound states in weakly deformed waveguides: numerical vs analytical results
We have studied the emergence of bound states in weakly deformed and/or
heterogeneous waveguides, comparing the analytical predictions obtained using a
recently developed perturbative method, with precise numerical results, for
different configurations (a homogeneous asymmetric waveguide, a heterogenous
asymmetric waveguide and a homogeneous broken-strip). In all the examples
considered in this paper we have found excellent agreement between analytical
and numerical results, thus providing a numerical verification of the
analytical approach.Comment: 11 pages, 6 figure
Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases
When solving differential equations by a spectral method, it is often
convenient to shift from Chebyshev polynomials with coefficients
to modified basis functions that incorporate the boundary conditions.
For homogeneous Dirichlet boundary conditions, , popular choices
include the ``Chebyshev difference basis", with coefficients here denoted and the ``quadratic-factor
basis functions" with coefficients
. If is weakly singular at the boundaries, then will
decrease proportionally to for some positive
constant , where the is a logarithm or a constant. We prove that
the Chebyshev difference coefficients decrease more slowly by a factor
of while the quadratic-factor coefficients decrease more slowly
still as . The error for the unconstrained
Chebyshev series, truncated at degree , is
in the interior, but is worse by one power of
in narrow boundary layers near each of the endpoints. Despite having nearly
identical error \emph{norms}, the error in the Chebyshev basis is concentrated
in boundary layers near both endpoints, whereas the error in the
quadratic-factor and difference basis sets is nearly uniform oscillations over
the entire interval in . Meanwhile, for Chebyshev polynomials and the
quadratic-factor basis, the value of the derivatives at the endpoints is
, but only for the difference basis
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