63 research outputs found
Convergence of linear barycentric rational interpolation for analytic functions
Polynomial interpolation to analytic functions can be very accurate, depending on the distribution of the interpolation nodes. However, in equispaced nodes and the like, besides being badly conditioned, these interpolants fail to converge even in exact arithmetic in some cases. Linear barycentric rational interpolation with the weights presented by Floater and Hormann can be viewed as blended polynomial interpolation and often yields better approximation in such cases. This has been proven for differentiable functions and indicated in several experiments for analytic functions. So far, these rational interpolants have been used mainly with a constant parameter usually denoted by d, the degree of the blended polynomials, which leads to small condition numbers but to merely algebraic convergence. With the help of logarithmic potential theory we derive asymptotic convergence results for analytic functions when this parameter varies with the number of nodes. Moreover, we present suggestions on how to choose d in order to observe fast and stable convergence, even in equispaced nodes where stable geometric convergence is provably impossible. We demonstrate our results with several numerical examples
On the Lebesgue constant of the trigonometric Floater-Hormann rational interpolant at equally spaced nodes
It is well known that the classical polynomial interpolation gives bad approximation if the nodes are equispaced. A valid alternative is the family of barycentric rational interpolants introduced by Berrut in [4], analyzed in terms of stability by Berrut and Mittelmann in [5] and their extension done by Floater and Hormann in [8]. In this paper firstly we extend them to the trigonometric case, then as in the Floater-Hormann classical interpolant, we study the growth of the Lebesgue constant on equally spaced points. We show that the growth is logarithmic providing a stable interpolation operato
Improved conditioning of the Floater--Hormann interpolants
The Floater--Hormann family of rational interpolants do not have spurious
poles or unattainable points, are efficient to calculate, and have arbitrarily
high approximation orders. One concern when using them is that the
amplification of rounding errors increases with approximation order, and can
make balancing the interpolation error and rounding error difficult. This
article proposes to modify the Floater--Hormann interpolants by including
additional local polynomial interpolants at the ends of the interval. This
appears to improve the conditioning of the interpolants and allow higher
approximation orders to be used in practice.Comment: 13 pages, 4 figures, 1 tabl
A generalization of Floater--Hormann interpolants
In this paper the interpolating rational functions introduced by Floater and
Hormann are generalized leading to a whole new family of rational functions
depending on , an additional positive integer parameter. For , the original Floater--Hormann interpolants are obtained. When we
prove that the new rational functions share a lot of the nice properties of the
original Floater--Hormann functions. Indeed, for any configuration of nodes,
they have no real poles, interpolate the given data, preserve the polynomials
up to a certain fixed degree, and have a barycentric-type representation.
Moreover, we estimate the associated Lebesgue constants in terms of the minimum
() and maximum () distance between two consecutive nodes. It turns out
that, in contrast to the original Floater-Hormann interpolants, for all we get uniformly bounded Lebesgue constants in the case of equidistant and
quasi-equidistant nodes configurations (i.e., when ). In such cases,
we also estimate the uniform and the pointwise approximation errors for
functions having different degree of smoothness.
Numerical experiments illustrate the theoretical results and show a better
error profile for less smooth functions compared to the original
Floater-Hormann interpolants.Comment: 29 page
A barycentric trigonometric Hermite interpolant via an iterative approach
In this paper we extend and generalise an interative approach for
constructing the Hermite interpolant introduced by Cirillo and Hormann (2018)
for the Floater-Hormann family of interpolants. In particular, we apply that
scheme to produce an effective barycentric rational trigonometric Hermite
interpolant using the basis function of the interpolant introduced by Berrut
(1988). In order to give an easy construction of such an interpolant we compute
the differentation matrix analytically and we conclude with various examples
and a numerical study of the rate of convergence at equidistant nodes
Linear barycentric rational quadrature
Linear interpolation schemes very naturally lead to quadrature rules. Introduced in the eighties, linear barycentric rational interpolation has recently experienced a boost with the presentation of new weights by Floater and Hormann. The corresponding interpolants converge in principle with arbitrary high order of precision. In the present paper we employ them to construct two linear rational quadrature rules. The weights of the first are obtained through the direct numerical integration of the Lagrange fundamental rational functions; the other rule, based on the solution of a simple boundary value problem, yields an approximation of an antiderivative of the integrand. The convergence order in the first case is shown to be one unit larger than that of the interpolation, under some restrictions. We demonstrate the efficiency of both approaches with numerical test
On the constrained mock-Chebyshev least-squares
The algebraic polynomial interpolation on uniformly distributed nodes is
affected by the Runge phenomenon, also when the function to be interpolated is
analytic. Among all techniques that have been proposed to defeat this
phenomenon, there is the mock-Chebyshev interpolation which is an interpolation
made on a subset of the given nodes whose elements mimic as well as possible
the Chebyshev-Lobatto points. In this work we use the simultaneous
approximation theory to combine the previous technique with a polynomial
regression in order to increase the accuracy of the approximation of a given
analytic function. We give indications on how to select the degree of the
simultaneous regression in order to obtain polynomial approximant good in the
uniform norm and provide a sufficient condition to improve, in that norm, the
accuracy of the mock-Chebyshev interpolation with a simultaneous regression.
Numerical results are provided.Comment: 17 pages, 9 figure
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