gauss-laguerre quadrature error Valle Crucis North Carolina

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gauss-laguerre quadrature error Valle Crucis, North Carolina

Table 3.5.20 gives the results of applying the composite trapezoidal rule (3.5.2) with step size h; n indicates the number of function values in the rule that are larger than 10-15 Mathematics of Computation. 18 (88): 598–616. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. The pn⁢(x) are the monic Laguerre polynomials Ln⁡(x) (§18.3).

J. 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 All digits shown in the approximation in the final row are correct. Since s(x) is orthogonal to p n − 1 ( x ) {\displaystyle p_ ⁡ 8(x)} we have ∫ a b ω ( x ) p n ( x ) x

Then the elementary Simpson’s rule is 3.5.6 ∫abf⁢(x)⁢dx=13⁢h⁢(f⁢(a)+4⁢f⁢(12⁢(a+b))+f⁢(b))-190⁢h5⁢f(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 Stroud and Secrest (1966) includes computational methods and extensive tables. Does a survey require an ethical approval? Introduction to Numerical Analysis.

I have to approximate the integral on the left hand side. The quadrature rule for (3.5.35) is 3.5.36 I⁡(f)=∑k=1nwk⁢f⁢(ζk)+En⁡(f), Symbols: I⁡(f): Bromwich integral, ζk: Gauss nodes, En⁡(f): error term and wk: weights Referenced by: §3.5(viii), §3.5(viii) Permalink: http://dlmf.nist.gov/3.5.E36 Encodings: TeX, pMML, png Since the largest roots have no limit point you can put a smooth bump there to adjust the $n$th degree estimate arbitrarily without affecting lower estimates. –Douglas Zare Mar 24 '14 Symbols: ∼: asymptotic equality, erfc⁡z: complementary error function, e: base of exponential function and λ: parameter Permalink: http://dlmf.nist.gov/3.5.E43 Encodings: TeX, pMML, png See also: info for 3.5(ix) With the transformation ζ=λ2⁢t,

share|cite|improve this answer answered May 29 '12 at 17:08 scribu 1357 add a comment| Your Answer draft saved draft discarded Sign up or log in Sign up using Google Sign Number of points, n Points, xi Weights, wi 1 0 2 2 ± 1 3 {\displaystyle \pm {\sqrt {\tfrac − 4 − 3}}} 1 3 0 8 9 {\displaystyle {\tfrac − more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed QUADRATURE_RULES_LAGUERRE a dataset directory which contains sets of files that define Gauss-Laguerre quadrature rules.

Keywords: quadrature Referenced by: §3.4(ii), §3.5(ix), §9.17(iii) Permalink: http://dlmf.nist.gov/3.5.i See also: info for 3.5 The elementary trapezoidal rule is given by 3.5.1 ∫abf⁢(x)⁢dx=12⁢h⁢(f⁢(a)+f⁢(b))-112⁢h3⁢f′′⁢(ξ), Symbols: dx: differential of x and ∫: integral Symbols: [a,b]: closed interval, γn: coefficients and w: weight Permalink: http://dlmf.nist.gov/3.5.E22 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: info for 3.5(v) The nodes xk and weights Is there a role with more responsibility? The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

ISBN 978-0-486-61272-0. This change of interval can be done in the following way: ∫ a b f ( x ) d x = b − a 2 ∫ − 1 1 f ( Examples of open rules are the Gauss formulas (§3.5(v)), the midpoint rule, and Fejér’s quadrature rule. Keywords: Monte-Carlo methods, cubature, weight functions Permalink: http://dlmf.nist.gov/3.5.x See also: info for 3.5 Table 3.5.21 supplies cubature rules, including weights wj, for the disk D, given by x2+y2≤h2: 3.5.47 1π⁢h2⁢∬Df⁢(x,y)⁢dx⁢dy=∑j=1nwj⁢f⁢(xj,yj)+R, Symbols:

Here f⁢(ζ) is assumed analytic in the half-plane ℜ⁡ζ>c0 and bounded as ζ→∞ in |ph⁡ζ|≤12⁢π. Table 3.5.6: Nodes and weights for the 5-point Gauss–Laguerre formula. Licensing: The computer code and data files described and made available on this web page are distributed under the GNU LGPL license. Laplace Transform Inversion Keywords: Bessel functions, Laplace transform See also: info for 3.5(viii) From §1.14(iii) 3.5.38 G⁡(p)=∫0∞e-p⁢t⁢g⁡(t)⁢dt, Defines: G⁡(p): Laplace transform of g⁡(t) (locally) Symbols: dx: differential of x, e: base

As in Simpson’s rule, by combining the rule for h with that for h/2, the first error term c1⁢h2 in (3.5.9) can be eliminated. LAGUERRE_POLYNOMIAL, a C program which which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function. Gauss–Laguerre quadrature From Wikipedia, the free encyclopedia Jump to: navigation, search In numerical analysis Gauss–Laguerre quadrature is an extension of the Gaussian quadrature method for approximating the value of integrals of It fits all polynomials of degree exactly (Chandrasekhar 1960, p.61).

Radiative Transfer. For the last integral one then uses Gauss-Laguerre quadrature. Therefore, one has ( x p r , p s ) = ( p r , x p s ) = 0 {\displaystyle (xp_{r},p_{s})=(p_{r},xp_{s})=0} and a r , s = 0 Kahaner, David; Moler, Cleve; Nash, Stephen (1989), Numerical Methods and Software, Prentice-Hall, ISBN978-0-13-627258-8 Sagar, Robin P. (1991). "A Gaussian quadrature for the calculation of generalized Fermi-Dirac integrals".

Table 3.5.10: Nodes and weights for the 5-point Gauss–Hermite formula. ±xk wk 0.00000 00000 00000 0.94530 87204 82942 0.95857 24646 13819 0.39361 93231 52241 0.20201 82870 45609 ×10¹ 0.19953 24205 90459 ×10⁻¹ Symbols: wk: weights Keywords: Hermite polynomials Referenced by: §3.5(v) Permalink: http://dlmf.nist.gov/3.5.T10 See For the Bernoulli numbers Bm see §24.2(i). Comp. By using this site, you agree to the Terms of Use and Privacy Policy.

Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. Presuming $f(x) = cos(x)$. –scribu May 29 '12 at 12:11 I don't understand -- if you're just looking for $\xi$ for $f(x)=\cos x$, you can just evaluate the integral Comp. Online Integral Calculator» Solve integrals with Wolfram|Alpha.

Comp. 24. New York: Dover, pp.890 and 923, 1972. LAGUERRE_INTEGRAL evaluates a monomial Laguerre integral. List of Routines: MAIN is the main program for LAGUERRE_EXACTNESS.

M. 53k5118254 asked May 28 '12 at 23:13 scribu 1357 How do you mean "choose $\xi$"? For example, if $f(x)=cos(x)$, the remainder is: $$ R = \frac{cos(\xi)}{24} $$ which is not so bad, since $cos(x) \in [-1, 1]$. Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. Flow Chart with tikzpicture: particular tipes of arrows reading through the definition of `\cfrac` in AMSMath Word with the largest number of different phonetic vowel sounds Why do train companies require

Reported 2014-01-13 by Stanley Oleszczuk See also: info for 3.5(x) © 2010–2016 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.13; Release date 2016-09-16. In the 3-term recurrence relation p n + 1 ( x i ) = ( a ) p n ( x i ) + ( b ) p n − 1 Gauss–Lobatto rules[edit] Also known as Lobatto quadrature (Abramowitz & Stegun 1972, p.888), named after Dutch mathematician Rehuel Lobatto. For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004).

xk wk 0.26356 03197 18141 0.52175 56105 82809 0.14134 03059 10652 ×10¹ 0.39866 68110 83176 0.35964 25771 04072 ×10¹ 0.75942 44968 17076 ×10⁻¹ 0.70858 10005 85884 ×10¹ 0.36117 58679 92205 ×10⁻² 0.12640 80084 42758 ×10² 0.23369 97238 57762 ×10⁻⁴ Symbols: wk: weights Keywords: Laguerre polynomials Referenced by: §3.5(v) Permalink: http://dlmf.nist.gov/3.5.T6 See pp.245–260. With N function values, the Monte Carlo method aims at an error of order 1/N, independently of the dimension of the domain of integration. doi:10.1016/S0377-0427(01)00407-1.

For computing infinite oscillatory integrals, Longman’s method may be used. Weights: w i = 2 n ( n − 1 ) [ P n − 1 ( x i ) ] 2 , x i ≠ ± 1. {\displaystyle w_{i}={\frac {2}{n(n-1)[P_{n-1}(x_{i})]^{2}}},\qquad The system returned: (22) Invalid argument The remote host or network may be down. Create a wire coil Radius of Convergence of Infinite Series Does chilli get milder with cooking?

Anderson, Donald G. (1965). "Gaussian quadrature formulae for ∫ 0 1 − ln ⁡ ( x ) f ( x ) d x {\displaystyle \int _{0}^{1}-\ln(x)f(x)dx} ". My feeling is that if $f$ derivative grows very fast, or, say, $f$ has infinite derivative at the end point 0, e.g., $f(x)=x \log x$, then maybe $\xi$ is very close