Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. Keywords: Clenshaw–Curtis quadrature formula, Gauss quadrature, error term, nodes, quadrature, weight functions Referenced by: §3.5(vii), Other Changes Permalink: http://dlmf.nist.gov/3.5.iv Addition (effective with 1.0.3): The reference to Trefethen (2011) was added at The pn(x) are the monic Legendre polynomials, that is, the polynomials Pn(x) (§18.3) scaled so that the coefficient of the highest power of x in their explicit forms is unity.

Please try the request again. For a = −1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Tabulated weights and abscissae with Mathematica source code, high precision (16 and 256 decimal places) Legendre-Gaussian quadrature weights and abscissas, for n=2 through n=64, with Mathematica source code. This procedure is known as Golub–Welsch algorithm.

Comp. It is accurate for polynomials up to degree 2n–3, where n is the number of integration points (Quarteroni, Sacco & Saleri 2000). MR0331730. Then the weights are given by 3.5.32 wk=β0vk,12, k=1,2,…,n, Symbols: vk: normalized eigenvector and wk: weights Permalink: http://dlmf.nist.gov/3.5.E32 Encodings: TeX, pMML, png See also: info for 3.5(vi) where β0=∫abw(x)dx and vk,1

For an integrand which has 2n continuous derivatives, ∫ a b ω ( x ) f ( x ) d x − ∑ i = 1 n w i f ( For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). Keywords: Christoffel coefficients (or numbers), Gauss quadrature, Gauss–Christoffel quadrature, nodes, quadrature, weight functions Referenced by: §10.74(iii), §18.38(i), §18.41(ii), §3.5(iii), §3.5(iv), §3.5(iv), §6.18(i), §9.17(iii) Permalink: http://dlmf.nist.gov/3.5.v See also: info for 3.5 Let Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. Please try the request again. Other choices lead to other integration rules. Also, the error constant (3.5.20) is given by 3.5.33 γn=β0β1⋯βn.

Comp. The polynomial pn is said to be an orthogonal polynomial of degree n associated to the weight function ω(x). Laurie, Dirk P. (1999), "Accurate recovery of recursion coefficients from Gaussian quadrature formulas", J. The pn(x) are the monic Laguerre polynomials Ln(x) (§18.3).

Appl. The idea underlying the proof is that, because of its sufficiently low degree, h(x) can be divided by p n ( x ) {\displaystyle p_ − 4(x)} to produce a quotient How to Cite Customize Annotate UnAnnotate What's New About the Project 3 Numerical MethodsAreas3.4 Differentiation3.6 Linear Difference Equations §3.5 Quadrature Keywords: integration, quadrature Referenced by: §14.32, §19.36(iv), §29.20(i), §5.21 Permalink: http://dlmf.nist.gov/3.5 See also: info Because p n ( x ) x − x i {\displaystyle {\frac ∫ 0(x)} − 9}}} is a polynomial of degree n-1, we have p n ( x ) x −

In adaptive algorithms the evaluation of the nodes and weights may cause difficulties, unless exact values are known. Gauss–Kronrod rules are extensions of Gauss quadrature rules generated by adding n + 1 points to an n-point rule in such a way that the resulting rule is of order 2n The Golub-Welsch algorithm[edit] The three-term recurrence relation can be written in the matrix form J P ~ = x P ~ − p n ( x ) × e n {\displaystyle On the other hand, p r {\displaystyle p_{r}} is orthogonal to every polynomial of degree less or equal to r − 1.

doi:10.1007/BF01535429. ISBN0-486-61272-4. The system returned: (22) Invalid argument The remote host or network may be down. For the integral of a Gaussian function, see Gaussian integral.

Then the elementary Simpson’s rule is 3.5.6 ∫abf(x)dx=13h(f(a)+4f(12(a+b))+f(b))-190h5f(4)(ξ), Symbols: dx: differential of x and ∫: integral A&S Ref: 25.4.5 (second relation only) Permalink: http://dlmf.nist.gov/3.5.E6 Encodings: TeX, pMML, png See also: info xk wk 0.43831 10175 47540 38348 5 ×10⁻² 0.67009 97891 64937 13603 2 ×10⁻¹ 0.25935 89810 53306 16102 3 ×10⁻¹ 0.11226 41502 86705 74182 8 0.65596 09541 23162 45254 1 ×10⁻¹ 0.13176 01770 39679 90373 2 0.12210 19340 73331 60333 0 0.13521 76490 61934 72513 2 0.19339 52623 74007 11598 3 0.12788 17986 45680 37040 1 0.27677 28387 06102 02439 4 0.11353 29074 90219 42129 0 0.36901 51271 39742 94381 6 0.95205 23978 43586 58511 6 ×10⁻¹ 0.46652 43289 64706 58267 6 0.75389 31416 73959 54339 3 ×10⁻¹ 0.56547 34737 91817 30642 5 0.56078 42449 26537 17992 3 ×10⁻¹ 0.66196 29190 12456 42139 0 0.38768 29537 50182 31110 0 ×10⁻¹ 0.75217 88833 78785 79878 8 The system returned: (22) Invalid argument The remote host or network may be down. the leading coefficient is 1) orthogonal polynomial of degree n and where ( f , g ) = ∫ a b ω ( x ) f ( x ) g (

doi:10.1016/0010-4655(91)90076-W. For computing infinite oscillatory integrals, Longman’s method may be used. The remainder is given by 3.5.19 En(f)=γnf(2n)(ξ)/(2n)!, Symbols: !: factorial (as in n!), γn: coefficients and En(f): error term Referenced by: §3.5(iv) Permalink: http://dlmf.nist.gov/3.5.E19 Encodings: TeX, pMML, png See also: info Lobatto quadrature of function f(x) on interval [−1, 1]: ∫ − 1 1 f ( x ) d x = 2 n ( n − 1 ) [ f ( 1

To generate Gk(h) the quantities G0(h),G0(h/2),…,G0(h/2k) are needed. It is unique up to a constant normalization factor. Math. doi:10.1006/jcph.1996.0258.

Keywords: Gauss quadrature, matrix Referenced by: §3.8(iii) Permalink: http://dlmf.nist.gov/3.5.vi See also: info for 3.5 All the monic orthogonal polynomials {pn} used with Gauss quadrature satisfy a three-term recurrence relation (§18.2(iv)): 3.5.30 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 Your cache administrator is webmaster. Now let h=(b-a)/n, xk=a+kh, and fk=f(xk), k=0,1,…,n.

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 xk wk 0.90426 30962 19965 064 ×10⁻² 0.12095 51319 54570 515 0.53971 26622 25006 295 ×10⁻¹ 0.18636 35425 64071 870 0.13531 18246 39250 775 0.19566 08732 77759 983 0.24705 24162 87159 824 0.17357 71421 82906 921 0.38021 25396 09332 334 0.13569 56729 95484 202 0.52379 23179 71843 201 0.93646 75853 81105 260 ×10⁻¹ 0.66577 52055 16424 597 0.55787 72735 14158 741 ×10⁻¹ 0.79419 04160 11966 217 0.27159 81089 92333 311 ×10⁻¹ 0.89816 10912 19003 538 0.95151 82602 84851 500 ×10⁻² 0.96884 79887 18633 539 0.16381 57633 59826 325 ×10⁻² Symbols: wk: Bibcode:2001JCoAM.127..201L. LCCN64-60036.

Gaussian quadrature From Wikipedia, the free encyclopedia Jump to: navigation, search "Gaussian integration" redirects here. Symbols: Γ(z): gamma function, [a,b]: closed interval, !: factorial (as in n!), γn: coefficients, α: exponent, β: exponent and w: weight Permalink: http://dlmf.nist.gov/3.5.E26 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, Gauss–Laguerre Formula Keywords: Gauss quadrature, Laguerre polynomials See also: info for 3.5(v) 3.5.27 [a,b) =[0,∞), w(x) =xαe-x, γn =n!Γ(n+α+1), α>-1. For detailed comparisons of the Clenshaw–Curtis formula with Gauss quadrature (§3.5(v)), see Trefethen (2008, 2011). §3.5(v) Gauss Quadrature Notes: In this subsection all numerical values of the nodes xk and corresponding

For r = s = 0 {\displaystyle r=s=0} one has ( p 1 , p 0 ) = ( ( x − a 0 , 0 p 0 , p 0 A comparison of several methods, including an extension of the Clenshaw–Curtis formula (§3.5(iv)), is given in Evans and Webster (1999). xk wk 0.25883 27955 92195 54283 3 ×10⁻² 0.43142 75213 32080 78579 0 ×10⁻¹ 0.15209 66234 95602 31720 7 ×10⁻¹ 0.75383 70990 85893 59550 5 ×10⁻¹ 0.38536 55037 21653 27959 8 ×10⁻¹ 0.93053 26745 16630 51372 7 ×10⁻¹ 0.72181 61381 58739 06435 0 ×10⁻¹ 0.10145 67118 49829 75443 7 0.11546 05264 87633 15055 9 0.10320 17620 56072 06905 8 0.16744 28562 75329 68571 8 0.10002 25498 05273 16653 3 0.22698 37872 60202 50336 1 0.93259 79930 02976 78083 7 ×10⁻¹ 0.29275 49609 41545 83299 2 0.84028 95287 19410 56497 1 ×10⁻¹ 0.36327 74298 57858 90453 8 0.73285 58913 00307 40962 8 ×10⁻¹ 0.43695 71400 90768 31848 7 Your cache administrator is webmaster.

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 Golub, Gene H.; Welsch, John H. (1969), "Calculation of Gauss Quadrature Rules", Mathematics of Computation, 23 (106): 221–230, doi:10.1090/S0025-5718-69-99647-1, JSTOR2004418 Gautschi, Walter (1968). "Construction of Gauss–Christoffel Quadrature Formulas". Such a result is exact since the green region has the same area as the red regions.