Gauss Legendre Quadrature Error
Contents |
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] gaussian quadrature example is 2/3. The trapezoidal rule returns the integral of the orange dashed line,
Gaussian Quadrature Formula For Numerical Integration
equal to y ( − 1 ) + y ( 1 ) = − 10 {\displaystyle y(-1)+y(1)=-10} . The 2-point gaussian quadrature error Gaussian quadrature rule returns the integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 / 3 {\displaystyle gaussian quadrature pdf 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 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 Calculator
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 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 th
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings gaussian quadrature matlab and policies of this site About Us Learn more about Stack gauss legendre method Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users
Gauss Legendre Quadrature Example
Badges Unanswered Ask Question Page Not Found This question was removed from Mathematics Stack Exchange for reasons of moderation. Please refer to the help center for possible explanations why https://en.wikipedia.org/wiki/Gaussian_quadrature a question might be removed. Here are some similar questions that might be relevant: The error of the midpoint rule for quadrature Remainder term for Gauss-Laguerre quadrature Quadrature Error for Trapezoidal Rule Visualizing Gauss-Legendre Quadrature Gauss Kronrod quadrature rule Will Gauss quadrature numerical integration work with a variable dx Gauss-Legendre Quadrature, computation of the abscissas and weights Why do http://math.stackexchange.com/questions/1340528/gauss-legendre-quadrature-error I get a big error when I compute this integral with Gauss-Legendre Quadrature? Gauss Quadrature Proof. Gaussian Legendre quadrature Try a Google Search Try searching for similar questions Browse our recent questions Browse our popular tags If you feel something is missing that should be here, contact us. about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life / Arts Culture / Recreation Science Other Stack Overflow Server Fault Super User Web Applications Ask Ubuntu Webmasters Game Development TeX - LaTeX Programmers Unix & Linux Ask Different (Apple) WordPress Development Geographic Information Systems Electrical Engineering Android Enthusiasts Information Security Database Administrators Drupal Answers SharePoint User Experience Mathematica Salesforce ExpressionEngine® Answers Cryptography Code Review Magento Signal Processing Raspberry Pi Programming Puzzles & Code Golf more (7) Photography Science Fiction & Fantasy Graphic Design Movies & TV Music: Practice & Theory Seasoned Advice (cooking) Home Improvement Personal Finance & Money Academia more (8) English Language & Usage Skeptics Mi Yodeya (Judaism) Travel Christianity English Language Learners Jap
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Applied http://mathworld.wolfram.com/Legendre-GaussQuadrature.html Mathematics>Numerical Methods>Numerical Integration> Legendre-Gauss Quadrature Legendre-Gauss quadrature is a numerical integration method also http://www.sciencedirect.com/science/article/pii/0771050X81900097 called "the" Gaussian quadrature or Legendre quadrature. A Gaussian quadrature over the interval with weighting function . The abscissas for quadrature order are given by the roots of the Legendre polynomials , which occur symmetrically about 0. The weights are (1) (2) where is the coefficient of in . For Legendre gaussian quadrature polynomials, (3) (Hildebrand 1956, p.323), so (4) (5) Additionally, (6) (7) (Hildebrand 1956, p.324), so (8) (9) Using the recurrence relation (10) (11) (correcting Hildebrand 1956, p.324) gives (12) (13) (Hildebrand 1956, p.324). The weights satisfy (14) which follows from the identity (15) The error term is (16) Beyer (1987) gives a table of abscissas and weights up to , and Chandrasekhar (1960) up gauss legendre quadrature to for even. 21.000000300.8888890.55555640.6521450.347855500.5688890.4786290.236927 The exact abscissas are given in the table below. 2130450 The abscissas for order quadrature are roots of the Legendre polynomial , meaning they are algebraic numbers of degrees 1, 2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, ..., which is equal to for (OEIS A052928). Similarly, the weights for order quadrature can be expressed as the roots of polynomials of degree 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ..., which is equal to for (OEIS A008619). The triangle of polynomials whose roots determine the weights is (17) (18) (19) (20) (21) (22) (23) (24) (OEIS A112734). REFERENCES: Abbott, P. "Tricks of the Trade: Legendre-Gauss Quadrature." Mathematica J. 9, 689-691, 2005. Beyer, W.H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp.462-463, 1987. Chandrasekhar, S. Radiative Transfer. New York: Dover, pp.56-62, 1960. Hildebrand, F.B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp.323-325, 1956. Sloane, N.J.A. Sequences A008619, A052928, and A112734 in "The On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha: Legendre-Gauss Quadrature CITE THIS AS: Weisstein, Eric W. "Legendre-Gauss Quadrature." From MathWorld--A Wolfram Web Resource. http://mathwor
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 7, Issue 1, March 1981, Pages 63-66 Some observations on Gauss-Legendre quadrature error estimates for analytic functions Author links open the overlay panel. Numbers correspond to the affiliation list which can be exposed by using the show more link. Opens overlay Frank G. Lether (∗) aDepartment of Mathematics, University of Georgia, Athens, Georgia 30602, USA Available online 20 April 2006 Show more doi:10.1016/0771-050X(81)90009-7 Get rights and content Under an Elsevier user license Open Archive AbstractThis paper is concerned with estimates for the error when a Gauss-Legendre quadrature rule is used to numerically integrate an analytic function. The error in a standard approximation to the kernel function, which appears in a contour integral representation for the quadrature error, is investigated for this purpose. Applications to meromorphic functions with simple poles are discussed. Download full text in PDF References 1. W. Barret Convergence properties of Gaussian quadrature formulae Comp. J., 3 (1960), pp. 272–277 2. M. Chawla, M. Jain Asymptotic error estimates for Gauss quadrature formula Math. Comp., 22 (1968), pp. 91–97 3. A. Curtis, P. Rabinowitz On the Gaussian integration of Chebyshev polynomials Math. Comp., 26 (1972), pp. 207–211 4. P. Davis, P. Rabinowitz Methods of numerical integration Academic Press, New York (1975) 5. J. Donaldson, D. Elliott A unified approach to quadrature rules with asymptotic estimates of their remainders SIAM J. Numer. Anal. (1972), pp. 573–602 6. J. McNamee Error-bounds for the evaluation of integrals by the