Quadrature Error Estimate
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] is 2/3. The trapezoidal rule returns the integral of gaussian quadrature example the orange dashed line, equal to y ( − 1 ) + y ( 1 gaussian quadrature formula for numerical integration ) = − 10 {\displaystyle y(-1)+y(1)=-10} . The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to gaussian quadrature pdf y ( − 1 / 3 ) + y ( 1 / 3 ) = 2 / 3 {\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
Gaussian Quadrature Calculator
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 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 gaussian quadrature matlab 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 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
loginOther institution loginHelpJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Download gaussian quadrature error full text in PDF Article Article + other articles
Gauss Legendre Quadrature Example
in this issue Loading... Export You have selected 1 citation for export. Help Direct
Gauss Hermite Quadrature
export Save to Mendeley Save to RefWorks Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export https://en.wikipedia.org/wiki/Gaussian_quadrature 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 Volumes 12–13, May 1985, Pages 425-431 Practical error estimation in numerical integration Author links http://www.sciencedirect.com/science/article/pii/0377042785900366 open the overlay panel. Numbers 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 adapt
tour help Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow http://scicomp.stackexchange.com/questions/14141/how-do-error-estimates-scale-for-multidimensional-cubature the company Business Learn more about hiring developers or posting ads with us Computational Science http://mathoverflow.net/questions/137825/practical-error-estimates-for-adaptive-newton-cotes-quadrature beta Questions Tags Users Badges Unanswered Ask Question _ Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to gaussian quadrature the top How do error estimates scale for multidimensional cubature? up vote 3 down vote favorite Suppose I have a quadrature method with a theoretical error estimate that scales with the number of points as $e(N)$: $$\int_a^b f(x)\mathrm{d}x = \sum_{i=1}^N w_i f(x_i) + \mathcal{O}\bigl(e(N)\bigr)$$ For example, $e(N) = N^{-4}$ for Simpson's rule, or $e(N) = N^{-2}$ for the trapezoidal rule. In this question I only care about how the error estimate scales with quadrature error estimate $N$, not its dependence on properties of the function or any leading coefficients. (And yes, I know it's just an estimate.) If I use this quadrature method to perform a $d$-dimensional integral by repeated 1D integration, $$\int_{a_0}^{b_0} \int_{a_1(x_0)}^{b_1(x_0)} \cdots \int_{a_d(x_0,\ldots,x_{d-1})}^{b_d(x_0,\ldots,x_{d-1})} f(\vec{x})\mathrm{d}^d\vec{x}$$ how does the error estimate for the multidimensional integral scale with $N$, the number of function evaluations along each dimension? Is it just $e(N)$? Or some function of $e(N)$? (like $[e(N)]^d$) Is it impossible to give a general relation without knowing the specific quadrature method in use? I've done some Google searching and checking in Numerical Recipes but I can't find a straightforward answer to this. I would have thought it would be common. quadrature error-estimation share|improve this question asked Jul 15 '14 at 13:44 David Z 1,48811530 add a comment| 1 Answer 1 active oldest votes up vote 5 down vote accepted Using product quadrature rules for multi-dimensional integrals suffers from the so-called curse of dimensionality. An $O(N^{-2})$ accurate rule using N evaluations in one-dimension is generally $O(N^{-2})$ accurate when applied as a product rule in multiple dimensions, but there will be $M = N^d$ evaluations required. So the accuracy is $O(M^{-2/d}).$ The required number of evaluations becomes intractable beyond 10 dimensions. Monte Carlo and Quasi-Monte Carlo methods seem to be preferred approache
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us MathOverflow Questions Tags Users Badges Unanswered Ask Question _ MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Practical error-estimates for (adaptive) Newton-Cotes Quadrature up vote 3 down vote favorite 1 I am looking for practical error estimates for Newton-Cotes Quadrature rules. Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective methods. Despite an extensive (but perhaps misdrirected) search I found almost nothing dealing with error estimation when automatic computation is concerned. Example: The error of the Simpson-Rule is given by: $Ch^5 f^{(4)}(\xi) $ where $C$ is a constant and $\xi \in [a,b]$. Obviously $\xi$ usually is not known. One could certaily choose the maximum of $f^{(4)}$ to get an upper bound but this can be tricky to determine on the run. This is why I am wondering: what are the best practices for estimating Integration-Error in numerical algorithms ? (above all I am interested in the ones concerning newton-cotes-quadrature) algorithms na.numerical-analysis integration approximation-algorithms share|cite|improve this question edited Jul 27 '13 at 6:04 asked Jul 26 '13 at 13:37 Boldwing 18410 2 This was a standard topic for numerical computation manuals of the 1970s and 1980s. Maybe you aren't looking back far enough. One method that I recall was to use several values of $h$ and compar