Gauss Quadrature Error Bounds
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The blue line is the polynomial y ( x ) = 7 x 3 − 8 x 2 − 3 x + 3 {\displaystyle y(x)=7x^ ω 2-8x^ ω 1-3x+3} , whose integral in [-1, 1] is 2/3. The trapezoidal rule returns the integral of the orange dashed line, equal to y ( − 1 ) + gaussian quadrature example y ( 1 ) = − 10 {\displaystyle y(-1)+y(1)=-10} . The 2-point Gaussian quadrature rule returns the gaussian quadrature error integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 / 3 gaussian quadrature formula for numerical integration {\displaystyle y({-{\sqrt {\scriptstyle 1/3}}})+y({\sqrt {\scriptstyle 1/3}})=2/3} . Such a result is exact since the green region has the same area as the red regions. In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a
Gaussian Quadrature Pdf
weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i = 1, ..., n. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as gaussian quadrature calculator ∫ − 1 1 f ( x ) d x = ∑ i = 1 n w i f ( x i ) . {\displaystyle \int _{-1}^ − 8f(x)\,dx=\sum _ − 7^ − 6w_ − 5f(x_ − 4).} Gaussian quadrature as above will only produce good results if the function f(x) is well approximated by a polynomial function within the range [−1, 1]. The method is not, for example, suitable for functions with singularities. However, if the integrated function can be written as f ( x ) = ω ( x ) g ( x ) {\displaystyle f(x)=\omega (x)g(x)\,} , where g(x) is approximately polynomial and ω(x) is known, then alternative weights w i ′ {\displaystyle w_ ξ 8'} and points x i ′ {\displaystyle x_ ξ 6'} that depend on the weighting function ω(x) may give better results, where ∫ − 1 1 f ( x ) d x = ∫ − 1 1 ω ( x ) g ( x ) d x ≈ ∑ i = 1 n w i ′ g ( x i ′ ) . {\displaystyle \int _{-1}^ ξ 4f(x)\,dx=\int _{-1}^ ξ 3\omega (x)g(x)\,dx\approx \sum _ ξ 2^ ξ 1w_ ξ 0'g(x_ ξ 9').} Common weighting functions include ω ( x ) = 1 / 1 − x 2 {\displaystyle \omega (x)=1/{\sqrt ξ 2}}\,} (Chebyshev–Gauss) and ω ( x ) = e − x 2 {\displaystyle \omega (x)=e^{-x^ ξ 0}} (Gauss–Hermite). It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points xi are just the roots of
Please note that Internet Explorer version 8.x will not be supported as of January 1, 2016. http://www.sciencedirect.com/science/article/pii/S0377042709002519 Please refer to this blog post for more information. Close http://www.sciencedirect.com/science/article/pii/0377042789903269 ScienceDirectSign inSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Download PDF Opens in a new window. gaussian quadrature Article suggestions will be shown in a dialog on return to ScienceDirect. Help Direct export Export file RIS(for EndNote, Reference Manager, ProCite) BibTeX Text RefWorks Direct Export Content Citation Only Citation and Abstract Advanced search JavaScript is disabled on your browser. Please enable JavaScript to use gauss quadrature error all the features on this page. JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. This page uses JavaScript to progressively load the article content as a user scrolls. Click the View full text link to bypass dynamically loaded article content. View full text Journal of Computational and Applied MathematicsVolume 234, Issue 4, 15 June 2010, Pages 1049–1057Proceedings of the Thirteenth International Congress on Computational and Applied Mathematics (ICCAM-2008), Ghent, Belgium, 7–11 July, 2008Edited By M.J. Goovaerts, S. Vandewalle and M. Van Daele Error bounds of certain Gaussian quadrature formulae ☆Miodrag M. Spalevića, , , Miroslav S. Pranićb, a Department of Mathematics, University of Beograd, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbiab Department of Mathematics and Informatics, University of Banja Luka, Faculty of Science, M. Stojanovića 2, 51000 Banja Luka, Bosnia and HerzegovinaReceived 15 July 2008, Revised 10 April 20
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Download full text in PDF Article Article + other articles in this issue Loading... Export You have selected 1 citation for export. Help Direct export Save to Mendeley Save to RefWorks Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. Journal of Computational and Applied Mathematics Volume 28, December 1989, Pages 145-154 Error bounds for quadrature formulas near Gaussian quadrature Author links open the overlay panel. Numbers correspond to the affiliation list which can be exposed by using the show more link. Opens overlay Helmut Brass, Opens overlay Klaus-Jürgen Förster Institut für Angewandte Mathematik, Technische Universität Braunschweig, Pockelsstraße 14, D-3300 Braunschweig, FRG Received 8 June 1988, Available online 1 April 2002 Show more doi:10.1016/0377-0427(89)90326-9 Get rights and content Under an Elsevier user license Open Archive AbstractLet Rn be the error functional of a quadrature formula Qn on [−1,1] using n nodes. In this paper we consider estimates of the form |Rn[ƒ]|⩽cm∥ƒ(m)∥, ∥ƒ∥≔sup|x|⩽1|ƒ(x)|, with best possible constant cm, i.e., cm = cm(Rn)≔ sup∥ƒ(m)∥⩽1|Rn[ƒ]|. For the error constants c2n−k(RGn) of the Gaussian quadrature formulas QGn we prove results, which are asymptotically sharp, when n increases and k is fixed. For this latter case, comparing with the corresponding error constants c2n−k(R