Approximate calculation of sums ii: gaussian type quadrature

Approximate calculation of sums ii: gaussian type quadrature

Author Area, Ivan Google Scholar
Dimitrov, Dimitar K. Google Scholar
Godoy, Eduardo Google Scholar
Paschoa, Vanessa G. Autor UNIFESP Google Scholar
Abstract The present paper is a continuation of a recent article [SIAM T. Numer. Anal., 52 (2014), pp. 1867-1886], where we proposed an algorithmic approach for approximate calculation of sums of the form Sigma(N)(j=1) f (j). The method is based on a Gaussian type quadrature formula for sums, which allows the calculation of sums with a very large number of terms N to be reduced to sums with a much smaller number of summands n. In this paper we prove that the Weierstrass-Dochev-Durand-Kerner iterative numerical method, with explicitly given initial conditions, converges to the nodes of the quadrature formula. Several methods for computing the nodes of the discrete analogue of the Gaussian quadrature formula are compared. Since, for practical purposes, any approximation of a sum should use only the values of the summands f(j), we implement a simple but efficient procedure to additionally approximate the evaluations at the nodes by local natural splines. Explicit numerical examples are provided. Moreover, the error in different spaces of functions is analyzed rigorously.
Keywords Approximate Calculation Of Sums
Gaussian Type Quadrature Formula For Sums
Orthogonal Gram Polynomials
Zeros Of Gram Polynomials
Zeros Of Legendre Polynomials
Natural Spline
Weierstrass-Dochev-Durand-Kerner Method
Error AnalysisFloating-Point Summation
Inverse Iteration
Language English
Sponsor Brazilian foundation CNPq [307183/2013-0]
Brazilian foundation FAPESP [2009/13832-9, 2013/23606-1]
Ministerio de Economia y Competitividad of Spain [MTM2012-38794-C02-01]
European Community fund FEDER
Grant number CNPq: 307183/2013-0
FAPESP: 2009/13832-9
FAPESP: 2013/23606-1
Ministerio de Economia y Competitividad of Spain: MTM2012-38794-C02-01
Date 2016
Published in Siam Journal On Numerical Analysis. Philadelphia, v. 54, n. 4, p. 2210-2227, 2016.
ISSN 0036-1429 (Sherpa/Romeo, impact factor)
Publisher Elsevier Science Inc
Extent 2210-2227
Access rights Closed access
Type Article
Web of Science ID WOS:000385274300009

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