Talk:Gaussian quadrature

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 Field: Analysis

It seems like the J matrix may have Bn and An switched!


— Preceding unsigned comment added by 171.64.221.179 (talk) 00:46, 1 November 2012 (UTC)

We need an expert here please! the part of Computation of Gaussian quadrature rules is a mess. What is A, what is B? somebody needs to define these; also, we define k as an index variable and instead of it we use i and so on. Wihenao Nov 1, 2009 —Preceding undated comment added 22:06, 1 November 2009 (UTC).

This page isn't linked from Quadrature, although that page does link to Numerical integration. I suspect a cleanup/merge is in order.

Also, somewhere should be mentioned quadrature over a simplex, which is useful for Finite element analysis. BenFrantzDale 04:37, Feb 17, 2005 (UTC)

I guess I'm not seeing what should be merged. Numerical integration is an overview of the topic, and mentions Gaussian quadrature in passing, along with several other techniques. Further detail is in the Gaussian quadrature article. This is as it should be, no? Wile E. Heresiarch 07:54, 18 Feb 2005 (UTC)

The formula (eq. 25.4.45 of Abramowitz and Stegun) referred to for Laguerre quadrature is incorrect. Or rather it is if the formulae for the Laguerre polynomial in chapter 22 of the same book are used. I suspect the authors of the two chapters have adopted different definitions of the Laguerre polynomials. GeordieMcBain 01:24, 9 March 2006 (UTC)

What printing of A&S? In the tenth printing, eqn 25.4.45 is marked with an asterisk, indicating a correction. [1] And what is wrong about the formula? -- Jitse Niesen (talk) 09:32, 10 March 2006 (UTC)

When it says : ... "which make the computed integral exact for all polynomials of degree up to 2n − 1" ... , is it including or excluding all the polynomials of degree 2n-1 ?

132.69.230.37 15:58, 3 September 2007 (UTC)

Including. I think "up to" always means including. Anyway, I reformulated it to say "all polynomials of degree 2n − 1 or less", which is definitely clearer. -- Jitse Niesen (talk) 01:33, 4 September 2007 (UTC)

Divide by zero?[edit]

In the Gauss-Lobatto section, we have Weights: w_i = \frac{2}{n(n-1)[P_{n-1}(x_i)]^2} \quad (x_i \ne \pm 1) What happens when x_i = 0, as it does once for every second legendre? --naught101 (talk) 01:57, 21 January 2011 (UTC)

Gauss-Legendre[edit]

I am no expert in this but is this also not known as gauss legendres method ? Poticecream (talk) 10:03, 28 October 2011 (UTC)

Yes, the case with W(x) = 1 given at Gaussian quadrature#Rules for the basic problem is often called Gauss-Legendre quadrature. Qwfp (talk) 12:43, 29 October 2011 (UTC)

n=5 quadrature wrong?[edit]

I think the abscissas for the n=5 quadrature rule are wrong. I implemented it in my code and it gave incorrect answers, so I checked on mathworld, and their abscissas are slightly different, http://mathworld.wolfram.com/Legendre-GaussQuadrature.html 128.83.68.153 (talk) 18:30, 27 July 2012 (UTC)

Error estimate wrong?[edit]

I can't quite believe the error estimate:

 \int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i)
 = \frac{f^{(2n)}(\xi)}{(2n)!} \, (p_n,p_n)

The normalisation for the orthogonal polynomials is unspecified (monic, normalised, whatever, ...) thus the right hand side can have *any* dependence on $n$ here. Just take $\tilde p_n(x)=g(n) p_n(x)$, which also defines orthogonal polynomial for the given weighting function, and you get a factor of $g(n)^2$ in the estimate. (ezander) 89.182.48.90 (talk) 09:22, 9 July 2013 (UTC)

Ok. I checked it in Bulirsch-Stoer. Must be the monic polynomials. I'll edit the article accordingly. (ezander) 89.182.48.90 (talk) 09:56, 9 July 2013 (UTC)

where ω is a known function?[edit]

Known to whom? I would say, f(x) = ω(x) · g(x) always, where ω = f and g ≡ 1. I suppose it should mean one of the a priori supported functions; the text is unclear. Moreover, I suppose the preferred points of evaluation change as well. --Yecril (talk) 11:05, 14 January 2014 (UTC)