Talk:Gaussian quadrature

message: 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)
replies: text: 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)

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

text: 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)
thread_title: First Comment(s)

message: 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)
thread_title:  Divide by zero? 

message: I am no expert in this but is this also not known as gauss legendres method ? Poticecream (talk) 10:03, 28 October 2011 (UTC)
replies: text: Yes, the case with  given at Gaussian quadrature#Rules for the basic problem is often called Gauss-Legendre quadrature. Qwfp (talk) 12:43, 29 October 2011 (UTC)
thread_title: Gauss-Legendre

message: 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)
thread_title:  n=5 quadrature wrong? 

message: 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)
replies: text: 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)
thread_title:  Error estimate wrong? 

message: Known to whom?  I would say,  always, where  and .  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)
thread_title:  where &omega; is a known function? 

message: This website rosettacode has a worked example in nearly 30 different programming languages.--Billymac00 (talk) 01:12, 22 March 2017 (UTC)
thread_title:  Example 

message: I think the example in the picture at the beginning might need some more explanation. It will, for instance not be clear to the reader, even with some experience, that the integral of the black dashed line just equals y(-w)+y(w) with w = \scriptstyle{-\sqrt{1/3}}. Madyno (talk) 16:51, 15 January 2018 (UTC)
thread_title:  Picture 

message: Can we add the alternative names to the first part of the title similar to here?  (e.g. Gauss-Legendre quadrature, Legendre quadrature) The WP:LEDE and table of contents is a bit confusing:  On first reading, I thought Gauss-Legendre quadrature was a special case of Gauss quadrature rather than an alternative name.  --David Tornheim (talk) 02:11, 22 December 2018 (UTC)
thread_title:  Alternative names 

message: Once the existence of the x_i,\,w_i families is proven, the w(\xi) are easily obtained from the special case Q\left(z\right)=P_{n-1}\left(z\right)\,\prod_{j=1}^{n-1}\left(z-x_{i}\right). This dramatically reduces the length of the proof. Pldx1 (talk) 10:04, 23 December 2018 (UTC)
thread_title: Obtaining the weights

message: Gauss-Legendre quadrature deserves a separate page from the general Gaussian quadrature page. The other cases of Gaussian quadrature rules listed on the
general each have their own pages. The sections on Gauss-Legendre in the
general Gaussian quadrature page do not detail the state-of-the-art
algorithms, which are orders of magnitude more efficient than the Golub-Welsch
algorithm and allow for computation of much larger quadrature rules. There has been much work in this area. We have an applied functional analysis class of students that plan to contribute the rest of the content to the Gauss-Legendre quadrature page. Cpt49 (talk) 17:35, 10 February 2020 (UTC)
thread_title:  Split Gauss-Legendre quadrature page 

message: The article really needs some information on the convergence rate — which is *exponential* for functions that are analytic in a neighborhood of the integration domain.  Trefethen's review article is a good starting place. — Steven G. Johnson (talk) 14:04, 20 June 2023 (UTC)
thread_title:  Convergence rate