Gauss Quadrature Error
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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 ) + y gaussian quadrature example ( 1 ) = − 10 {\displaystyle y(-1)+y(1)=-10} . The 2-point Gaussian quadrature rule returns the integral gaussian quadrature formula for numerical integration of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 / 3 {\displaystyle y({-{\sqrt
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{\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 weighted sum
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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 ∫ − 1 gaussian quadrature matlab 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 a polynomial belonging to a clas
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Mathematics>Numerical Methods>Numerical Integration> Interactive Entries>Interactive Demonstrations> Gaussian Quadrature Seeks to obtain the gauss quadrature method numerical integration example best numerical estimate of an integral by picking optimal abscissas at which to evaluate the function . The fundamental gauss legendre method theorem of Gaussian quadrature states that the optimal abscissas of the -point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomial for the same interval and weighting function. Gaussian quadrature https://en.wikipedia.org/wiki/Gaussian_quadrature is optimal because it fits all polynomials up to degree exactly. Slightly less optimal fits are obtained from Radau quadrature and Laguerre-Gauss quadrature. interval are roots of1 To determine the weights corresponding to the Gaussian abscissas , compute a Lagrange interpolating polynomial for by letting (1) (where Chandrasekhar 1967 uses instead of ), so (2) Then fitting a Lagrange interpolating polynomial through the points http://mathworld.wolfram.com/GaussianQuadrature.html gives (3) for arbitrary points . We are therefore looking for a set of points and weights such that for a weighting function , (4) (5) with weight (6) The weights are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For orthogonal polynomials with , ..., , (7) (Hildebrand 1956, p.322), where is the coefficient of in , then (8) (9) where (10) Using the relationship (11) (Hildebrand 1956, p.323) gives (12) (Note that Press et al. 1992 omit the factor .) In Gaussian quadrature, the weights are all positive. The error is given by (13) (14) where (Hildebrand 1956, pp.320-321). Other curious identities are (15) and (16) (17) (Hildebrand 1956, p.323). In the notation of Szegö (1975), let be an ordered set of points in , and let , ..., be a set of real numbers. If is an arbitrary function on the closed interval , write the Gaussian quadrature as (18) Here are the abscissas and are the Cotes numbers. SEE ALSO: Chebyshev Quadrature, Chebyshev-Gauss Quadrature, Chebyshev-Radau Quadrature, Fundamental Theorem of Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Laguerre-Gauss Quadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Radau Quadrature REFERENCES: Abramowit
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Download full text http://www.sciencedirect.com/science/article/pii/0377042785900366 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 gaussian quadrature 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 Volumes 12–13, May 1985, Pages 425-431 Practical error estimation in numerical integration Author links open the overlay panel. Numbers gauss quadrature error correspond to the affiliation list which can be exposed by using the show more link. Opens overlay Dirk P. Laurie Department of Mathematics and Applied Mathematics, Potchefstroom University for C.H.E., 1900 Vanderbijlpark, Republic of South AfricaSouth Africa Received 17 May 1984, Available online 28 March 2002 Show more doi:10.1016/0377-0427(85)90036-6 Get rights and content Under an Elsevier user license Open Archive AbstractTheoretical error estimates for quadrature rules involve quantities that are usually difficult if not impossible to obtain in practice. Various heuristics to obtain computable error estimates are compared by calculating their performance profiles on the Lyness family of integrands. Two sets of tests are used, corresponding to adaptive and single-rule quadrature. In the single rule case, Gaussian quadrature with error estimate provided by dropping one point from the formula performs best. In the adaptive case, the best heuristic is non-linear extrapolation based on Gaussian quadratu
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