Error Analysis Gaussian Quadrature
Contents |
The blue line is the polynomial y ( x ) = 7 x 3 − 8 x 2 − 3 x + 3 {\displaystyle y(x)=7x^ ω 6-8x^ ω 5-3x+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 gaussian quadrature calculator returns the integral of the black dashed curve, equal to y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 gaussian quadrature weights / 3 {\displaystyle y(-{\sqrt ω 2})+y({\sqrt ω 1})=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 gaussian quadrature 2d 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 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
Gaussian Quadrature Python
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 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)
Gaussian quadrature formulaeAuthorsAuthors and affiliationsBjörn von SydowArticleReceived: 09 November 1976DOI: 10.1007/BF01389313Cite this article as: von Sydow,
Gaussian Quadrature C++
B. Numer. Math. (1977) 29: 59. doi:10.1007/BF01389313SummaryWe derive both gaussian quadrature matlab strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a two point gaussian quadrature example general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex https://en.wikipedia.org/wiki/Gaussian_quadrature analysis.Subject ClassificationsAMS (MOS): 41A5541A25CR: 5.16References1.Achieser, N.I.: Vorlesungen über Approximationstheorie, 2. Aufl. Berlin; Akademie-Verlag 19672.Chawla, M.M.: Asymptotic estimates for the error in the Gauss-Legendre quadrature formula. Comput. J.11, 339–340 (1968)3.Davis, P.J.: Errors of numerical approximation. Arch. Rational Mech. Anal.2, 303–313 (1953)4.Duren, P.J.: Theory ofH p spaces. New York: http://link.springer.com/article/10.1007/BF01389313 Academic Press 19705.Freud, G.: Error estimates for Gauss-Jacobi quadrature formulae. In: Topics in numerical analysis (J. Miller, ed.), pp. 113–121. London: Academic Press, 19736.Karlsson, J., von Sydow, B.: The convergence of Padé approximants to series of Stieltjes. Ark. Mat.14, 43–53 (1976)7.Markoff, A.: Differenzenrechnung. Leipzig: Teubner 18968.Meinardus, G.: Approximation of functions: Theory and numerical methods. Berlin-Heidelberg-New York: Springer 19679.Stenger, F.: Bounds on the error of Gauss-type quadratures. Numer. Math.8, 150–160 (1966)10.Szegö, G.: Orthogonal polynomials, 3rd ed. Providence: American Mathematical Society 1967Copyright information© Springer-Verlag 1977Authors and AffiliationsBjörn von Sydow11.Department of MathematicsUniversity of LuleåLuleåSweden About this article Print ISSN 0029-599X Online ISSN 0945-3245 Publisher Name Springer-Verlag About this journal Reprints and Permissions Article actions Log in to check your access to this article Buy (PDF)EUR41,59 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about instit
be down. Please try the request again. Your cache administrator is webmaster. Generated Mon, 10 Oct 2016 12:22:29 GMT by s_ac15 (squid/3.5.20)