7,434 research outputs found
Scalar Levin-Type Sequence Transformations
Sequence transformations are important tools for the convergence acceleration
of slowly convergent scalar sequences or series and for the summation of
divergent series. Transformations that depend not only on the sequence elements
or partial sums but also on an auxiliary sequence of so-called remainder
estimates are of Levin-type if they are linear in the , and
nonlinear in the . Known Levin-type sequence transformations are
reviewed and put into a common theoretical framework. It is discussed how such
transformations may be constructed by either a model sequence approach or by
iteration of simple transformations. As illustration, two new sequence
transformations are derived. Common properties and results on convergence
acceleration and stability are given. For important special cases, extensions
of the general results are presented. Also, guidelines for the application of
Levin-type sequence transformations are discussed, and a few numerical examples
are given.Comment: 59 pages, LaTeX, invited review for J. Comput. Applied Math.,
abstract shortene
The Effective Prepotential of N=2 Supersymmetric SU(N_c) Gauge Theories
We determine the effective prepotential for N=2 supersymmetric SU(N_c) gauge
theories with an arbitrary number of flavors N_f < 2N_c, from the exact
solution constructed out of spectral curves. The prepotential is the same for
the several models of spectral curves proposed in the literature. It has to all
orders the logarithmic singularities of the one-loop perturbative corrections,
thus confirming the non-renormalization theorems from supersymmetry. In
particular, the renormalized order parameters and their duals have all the
correct monodromy transformations prescribed at weak coupling. We evaluate
explicitly the contributions of one- and two-instanton processes.Comment: 34 pages, Plain TeX, no macros needed, no figure
From "Dirac combs" to Fourier-positivity
Motivated by various problems in physics and applied mathematics, we look for
constraints and properties of real Fourier-positive functions, i.e. with
positive Fourier transforms. Properties of the "Dirac comb" distribution and of
its tensor products in higher dimensions lead to Poisson resummation, allowing
for a useful approximation formula of a Fourier transform in terms of a limited
number of terms. A connection with the Bochner theorem on positive definiteness
of Fourier-positive functions is discussed. As a practical application, we find
simple and rapid analytic algorithms for checking Fourier-positivity in 1- and
(radial) 2-dimensions among a large variety of real positive functions. This
may provide a step towards a classification of positive positive-definite
functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text
Approximation error of the Lagrange reconstructing polynomial
The reconstruction approach [Shu C.W.: {\em SIAM Rev.} {\bf 51} (2009)
82--126] for the numerical approximation of is based on the
construction of a dual function whose sliding averages over the interval
are equal to (assuming
an homogeneous grid of cell-size ). We study the deconvolution
problem [Harten A., Engquist B., Osher S., Chakravarthy S.R.: {\em J. Comp.
Phys.} {\bf 71} (1987) 231--303] which relates the Taylor polynomials of
and , and obtain its explicit solution, by introducing rational numbers
defined by a recurrence relation, or determined by their generating
function, , related with the reconstruction pair of . We
then apply these results to the specific case of Lagrange-interpolation-based
polynomial reconstruction, and determine explicitly the approximation error of
the Lagrange reconstructing polynomial (whose sliding averages are equal to the
Lagrange interpolating polynomial) on an arbitrary stencil defined on a
homogeneous grid.Comment: 31 pages, 1 table; revised version to appear in J. Approx. Theor
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
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