scholarly journals On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Arnak Poghosyan
Keyword(s):  
Trigonometric Polynomial ◽  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Convergence Acceleration ◽  
Fast Convergence ◽  
Exact Constants

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice.

scholarly journals Optimal Rational Approximations by the Modified Fourier Basis

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Arnak V. Poghosyan ◽  
Tigran K. Bakaryan
Keyword(s):  
Numerical Experiments ◽  
Pointwise Convergence ◽  
Convergence Acceleration ◽  
Rational Approximations ◽  
Unknown Parameters ◽  
Fourier Basis ◽  
Optimal Values ◽  
Exact Constants ◽  
Selection Of

We consider convergence acceleration of the modified Fourier expansions by rational trigonometric corrections which lead to modified-trigonometric-rational approximations. The rational corrections contain some unknown parameters and determination of their optimal values for improved pointwise convergence is the main goal of this paper. The goal was accomplished by deriving the exact constants of the asymptotic errors of the approximations with further elimination of the corresponding main terms by appropriate selection of those parameters. Numerical experiments outline the convergence improvement of the optimal rational approximations compared to the expansions by the modified Fourier basis.

FOURIER FORMULAE FOR EQUIDISTANT HERMITE TRIGONOMETRIC INTERPOLATION

2011 ◽  
Vol 04 (01) ◽  
pp. 127-144 ◽  
Author(s):  
Arnak Poghosyan
Keyword(s):  
Convergence Acceleration ◽  
Lanczos Method ◽  
Trigonometric Interpolation ◽  
Quadrature Formulae ◽  
Interpolation Polynomials ◽  
Exact Constants ◽  
Interpolation Nodes

A sequence of Hermite trigonometric interpolation polynomials with equidistant interpolation nodes and uniform multiplicities is investigated. We derive relatively compact formula that gives the interpolants as functions of the coefficients in the DFTs of the derivative values. The coefficients can be calculated by the FFT algorithm. Corresponding quadrature formulae are derived and explored. Convergence acceleration based on the Krylov-Lanczos method for accelerating both the convergence of interpolation and quadrature is considered. Exact constants of the asymptotic errors are obtained and some numerical illustrations are presented.

scholarly journals A new approach for constructing mock-Chebyshev grids

2021 ◽  
Author(s):  
Ali IBRAHIMOGLU
Keyword(s):  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Numerical Results ◽  
Uniform Grid ◽  
New Approach ◽  
Chebyshev Nodes ◽  
Chebyshev Points ◽  
Runge Phenomenon ◽  
Equidistant Nodes

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

scholarly journals Polynomial approximation on spheres - generalizing de la Vallée-Poussin

2011 ◽  
Vol 11 (4) ◽  
pp. 540-552 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Trigonometric Polynomial ◽  
Polynomial Approximation ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Fast Convergence ◽  
Piecewise Polynomial ◽  
Smooth Functions ◽  
Higher Dimensional ◽  
Trigonometric Polynomial Approximation ◽  
De La Vallée Poussin

AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}. In all cases the approximation converges uniformly for all continuous functions, and has arbitrarily fast convergence for smooth functions.

scholarly journals On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

2021 ◽  
pp. 1-44
Author(s):  
Arnak V. Poghosyan ◽  
Lusine D. Poghosyan ◽  
Rafayel H. Barkhudaryan
Keyword(s):  
Numerical Experiments ◽  
Finite Interval ◽  
Convergence Acceleration ◽  
Vandermonde Matrix ◽  
Asymptotic Estimates ◽  
Convergence Properties ◽  
Explicit Formulas ◽  
Fair Comparison

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

scholarly journals On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Arnak Poghosyan ◽  
Lusine Poghosyan
Keyword(s):  
Pointwise Convergence ◽  
Convergence Acceleration ◽  
Trigonometric Interpolation ◽  
Special Choice ◽  
Unknown Parameters ◽  
Exact Constants

We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors.

scholarly journals The Markov-WZ Method

10.37236/1806 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Mohamud Mohammed ◽  
Doron Zeilberger
Keyword(s):  
Infinite Series ◽  
Ad Hoc ◽  
Convergence Acceleration ◽  
Fast Convergence ◽  
New Class ◽  
Wz Theory ◽  
Hypergeometric Type ◽  
Wz Method

Andrei Markov's 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger's algorithms, and even if he did, he wouldn't have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type.

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