APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE
Autor(a) principal: | |
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Data de Publicação: | 2016 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1137/140993752 http://hdl.handle.net/11449/162043 |
Resumo: | 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. |
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APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATUREapproximate calculation of sumsGaussian type quadrature formula for sumsorthogonal Gram polynomialszeros of Gram polynomialszeros of Legendre polynomialsnatural splinemonosplineWeierstrass-Dochev-Durand-Kerner methoderror analysisThe 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.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Ministerio de Economia y Competitividad of SpainEuropean Community fund FEDERUniv Vigo, Dept Matemat Aplicada 2, EE Aeronaut & Espazo, Campus Lagoas, Orense 32004, SpainUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, BrazilUniv Vigo, Dept Matemat Aplicada 2, EE Ind, Campus Lagoas Marcosende, Vigo 36310, SpainUniv Fed Sao Paulo UNIFESP, ICT, Dept Ciencia Computacao, BR-12231280 Sao Jose Dos Campos, SP, BrazilUniv Estadual Paulista, IBILCE, Dept Matemat Aplicada, BR-15054000 Sao Jose Do Rio Preto, SP, BrazilCNPq: 307183/2013-0FAPESP: 2009/13832-9FAPESP: 2013/23606-1Ministerio de Economia y Competitividad of Spain: MTM2012-38794-C02-01Siam PublicationsUniv VigoUniversidade Estadual Paulista (Unesp)Universidade Federal de São Paulo (UNIFESP)Area, IvanDimitrov, Dimitar K. [UNESP]Godoy, EduardoPaschoa, Vanessa G.2018-11-26T17:06:38Z2018-11-26T17:06:38Z2016-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article2210-2227http://dx.doi.org/10.1137/140993752Siam Journal On Numerical Analysis. Philadelphia: Siam Publications, v. 54, n. 4, p. 2210-2227, 2016.0036-1429http://hdl.handle.net/11449/16204310.1137/140993752WOS:000385274300009Web of Sciencereponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengSiam Journal On Numerical Analysis2,657info:eu-repo/semantics/openAccess2021-10-23T16:01:19Zoai:repositorio.unesp.br:11449/162043Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462021-10-23T16:01:19Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
title |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
spellingShingle |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE Area, Ivan approximate calculation of sums Gaussian type quadrature formula for sums orthogonal Gram polynomials zeros of Gram polynomials zeros of Legendre polynomials natural spline monospline Weierstrass-Dochev-Durand-Kerner method error analysis |
title_short |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
title_full |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
title_fullStr |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
title_full_unstemmed |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
title_sort |
APPROXIMATE CALCULATION OF SUMS II: GAUSSIAN TYPE QUADRATURE |
author |
Area, Ivan |
author_facet |
Area, Ivan Dimitrov, Dimitar K. [UNESP] Godoy, Eduardo Paschoa, Vanessa G. |
author_role |
author |
author2 |
Dimitrov, Dimitar K. [UNESP] Godoy, Eduardo Paschoa, Vanessa G. |
author2_role |
author author author |
dc.contributor.none.fl_str_mv |
Univ Vigo Universidade Estadual Paulista (Unesp) Universidade Federal de São Paulo (UNIFESP) |
dc.contributor.author.fl_str_mv |
Area, Ivan Dimitrov, Dimitar K. [UNESP] Godoy, Eduardo Paschoa, Vanessa G. |
dc.subject.por.fl_str_mv |
approximate calculation of sums Gaussian type quadrature formula for sums orthogonal Gram polynomials zeros of Gram polynomials zeros of Legendre polynomials natural spline monospline Weierstrass-Dochev-Durand-Kerner method error analysis |
topic |
approximate calculation of sums Gaussian type quadrature formula for sums orthogonal Gram polynomials zeros of Gram polynomials zeros of Legendre polynomials natural spline monospline Weierstrass-Dochev-Durand-Kerner method error analysis |
description |
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. |
publishDate |
2016 |
dc.date.none.fl_str_mv |
2016-01-01 2018-11-26T17:06:38Z 2018-11-26T17:06:38Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1137/140993752 Siam Journal On Numerical Analysis. Philadelphia: Siam Publications, v. 54, n. 4, p. 2210-2227, 2016. 0036-1429 http://hdl.handle.net/11449/162043 10.1137/140993752 WOS:000385274300009 |
url |
http://dx.doi.org/10.1137/140993752 http://hdl.handle.net/11449/162043 |
identifier_str_mv |
Siam Journal On Numerical Analysis. Philadelphia: Siam Publications, v. 54, n. 4, p. 2210-2227, 2016. 0036-1429 10.1137/140993752 WOS:000385274300009 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Siam Journal On Numerical Analysis 2,657 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
2210-2227 |
dc.publisher.none.fl_str_mv |
Siam Publications |
publisher.none.fl_str_mv |
Siam Publications |
dc.source.none.fl_str_mv |
Web of Science reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
_version_ |
1799965756896575488 |