Comp. 19 (91): 477â€“481. Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. SpaleviÄ‡a, , , Miroslav S. Collatz, G.

IntroductionWe study the kernels Kn(z)Kn(z) in the remainder terms Rn(f)Rn(f) of the Gaussian quadrature formula equation(1.1)∫−11f(t)w(t)dt=Gn[f]+Rn(f),Gn[f]=∑Î½=1nÎ»Î½f(Ï„Î½)(n∈N) for analytic functions on elliptical contours with foci at âˆ“1âˆ“1 and the sum of semi-axes BraÃŸ, G. FÃ¶rster On the estimation on linear functionals Analysis, 7 (1987), pp. 237â€“258 [5] H. Furthermore, all these nodes xi will lie in the open interval (a, b) (Stoer & Bulirsch 2002, pp.172â€“175).

First of all, the polynomials defined by the recurrence relation starting with p 0 ( x ) = 1 {\displaystyle p_{0}(x)=1} have leading coefficient one and correct degree. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. Number of points, n Points, xi Weights, wi 1 0 2 2 ± 1 3 {\displaystyle \pm {\sqrt {\tfrac âˆ’ 4 âˆ’ 3}}} 1 3 0 8 9 {\displaystyle {\tfrac âˆ’ JavaScript is disabled on your browser.

You need to reset your browser to accept cookies or to ask you if you want to accept cookies. For the error constants c2nâˆ’k(RGn) of the Gaussian quadrature formulas QGn we prove results, which are asymptotically sharp, when n increases and k is fixed. This procedure is known as Golubâ€“Welsch algorithm. ScienceDirect Â® is a registered trademark of Elsevier B.V.RELX Group Close overlay Close Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered?

BraÃŸ, K.-J. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Gaussian Quadrature by Chris Maes and Anton Antonov, Wolfram Demonstrations Project. MR0228171.

Meinardus, H. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error. By using this site, you agree to the Terms of Use and Privacy Policy. SchmeiÃŸer Error estimates for interpolatory quadrature formulae Numer.

Please enable JavaScript to use all the features on this page. Some of these are tabulated below. doi:10.1090/S0025-5718-1970-0285117-6. Bibcode:1996JCoPh.129..406Y.

Therefore, ( p r + 1 , p s ) = ( x p r , p s ) − a r , s ( p s , p s ) Comp. 24. Why Does this Site Require Cookies? It is similar to Gaussian quadrature with the following differences: The integration points include the end points of the integration interval.

Screen reader users, click the load entire article button to bypass dynamically loaded article content. This site uses cookies to improve performance by remembering that you are logged in when you go from page to page. In this paper we consider estimates of the form |Rn[Æ’]|â©½cmâˆ¥Æ’(m)âˆ¥, âˆ¥Æ’âˆ¥â‰”sup|x|â©½1|Æ’(x)|, with best possible constant cm, i.e., cm = cm(Rn)â‰” supâˆ¥Æ’(m)âˆ¥â©½1|Rn[Æ’]|. Such a result is exact since the green region has the same area as the red regions.

ElsevierAbout ScienceDirectRemote accessShopping cartContact and supportTerms and conditionsPrivacy policyCookies are used by this site. This change of interval can be done in the following way: ∫ a b f ( x ) d x = b − a 2 ∫ − 1 1 f ( Gaussâ€“Lobatto rules[edit] Also known as Lobatto quadrature (Abramowitz & Stegun 1972, p.888), named after Dutch mathematician Rehuel Lobatto. Eng.

Rivlin The Chebyshev Polynomials Wiley, New York (1974) [10] G. So, if q(x) is a polynomial of at most nth degree we have ∫ a b ω ( x ) p n ( x ) x − x i d x Comp. 27 (124). The trapezoidal rule returns the integral of the orange dashed line, equal to y ( − 1 ) + y ( 1 ) = − 10 {\displaystyle y(-1)+y(1)=-10} .

ISBN0-387-98959-5. The weights can be computed from the corresponding eigenvectors: If ϕ ( j ) {\displaystyle \phi ^{(j)}} is a normalized eigenvector (i.e., an eigenvector with euclidean norm equal to one) associated This page uses JavaScript to progressively load the article content as a user scrolls. Math. 5 (1).

On the other hand, p r {\displaystyle p_{r}} is orthogonal to every polynomial of degree less or equal to r âˆ’ 1. Other choices lead to other integration rules. The blue line is the polynomial y ( x ) = 7 x 3 − 8 x 2 − 3 x + 3 {\displaystyle y(x)=7x^ Ï‰ 2-8x^ Ï‰ 1-3x+3} , whose The method is not, for example, suitable for functions with singularities.

Werner (Eds.), Numerical Methods of Approximation Theory, 7, BirkhÃ¤user, Basel (1984), pp. 19â€“31 [3] H. The idea underlying the proof is that, because of its sufficiently low degree, h(x) can be divided by p n ( x ) {\displaystyle p_ âˆ’ 4(x)} to produce a quotient Your cache administrator is webmaster. doi:10.1007/BF01535429.

ElsevierAbout ScienceDirectRemote accessShopping cartContact and supportTerms and conditionsPrivacy policyCookies are used by this site. Goovaerts, S. Error estimates[edit] The error of a Gaussian quadrature rule can be stated as follows (Stoer & Bulirsch 2002, Thm3.6.24). It is accurate for polynomials up to degree 2nâ€“3, where n is the number of integration points (Quarteroni, Sacco & Saleri 2000).

Van Daele Error bounds of certain Gaussian quadrature formulae â˜†Miodrag M. To fix this, set the correct time and date on your computer. Appl. To prove this, note that using Lagrange interpolation one can express r(x) in terms of r ( x i ) {\displaystyle r(x_ Î¾ 6)} as r ( x ) = ∑

Appl. 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 class of orthogonal polynomials. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures.MSCprimary, 41A55; secondary, 65D30, 65D32KeywordsKernel; Remainder term; Gauss quadrature; Analytic function; Elliptic contour; Error bound1.

Setting Your Browser to Accept Cookies There are many reasons why a cookie could not be set correctly. We can write ∏ 1 ≤ j ≤ n j ≠ i ( x − x j ) = ∏ 1 ≤ j ≤ n ( x − x j )