Talk:Newton–Cotes formulas

Original research from Pavel

I looked at this, the English is janky and the analysis is non-standard. This does not mean that it is wrong per se, it is just original research, not peer-reviewed, and probably does not belong here (yet). Please provide references to peer-reviewed consideration of the frequency domain analysis techinque, or I will remove these citations and the paragraph in the text. 2001:BB8:2002:200:BC5E:FF88:CC3A:BD81 (talk) 03:46, 10 February 2016 (UTC)[reply]

Increasing n

In the beginning of the article it is stated that an exponentially increasing error may occur as n increases due to Runge's phenomenon. However, this should not be the case. Runge's phenomenon relates to using an increasingly higher-order polynomial approximation. For a fixed Newton-Cotes formula (e.g., Simpson's 3/8 rule), the order of the approximating polynominal is fixed. As n increases it is the step size h that is decreasing.

Elee1l5 19:00, 1 December 2006 (UTC)[reply]

The n referred to the order of the approximating polynomial. You're thinking of composite rules, which are discussed further down. However, you have a good point that the text is confusing. I tried to improve it. I also moved that part out of the first paragraph; I don't think it's that important because high-order quadrature formulas are not used very often. -- Jitse Niesen (talk) 02:14, 2 December 2006 (UTC)[reply]

After thinking it over, I'm backtracking about it being not so important. Actually, even for order 8 (which is definitely used in practice), Gaussian quadrature has clearly lower error than Newton-Cotes. -- Jitse Niesen (talk) 12:13, 3 December 2006 (UTC)[reply]

Thanks a lot for the improvement - the text is now more clear regarding the meaning of n and the use of composite rules to avoid the catastrophic interpolation effect.

Elee1l5 17:27, 5 December 2006 (UTC)[reply]

I think the way n is used in this article is still misleading. I understood this only after reading this discussion. I think Newton-Cotes is almost always used as the composite rule, so I suggest this rule should be at least shortly mentioned in the very beginning of this article.

One more question to the composite rule: When I use this rule e.g. to integrate a function using 9 points and a polynomial of order 2, do I split it in 4 distinct subintervals and integrate each of them separately or I compute some kind of moving average moving each time by one point only?

--Wassermann7 (talk) 16:06, 25 January 2010 (UTC)[reply]

Coefficients

Closed Newton–Cotes Formulas and Open Newton-cotes Formulas.. seem to be a bit unusual. e.g. Simpson Rule: is should not be 1/3*(1+4+1) better 1/6*... as written on simpson page. or am i understanding something wrong?

Amegon 13:50, 28 December 2008 (UTC)[reply]

It depends on how you define the step size h. In this article, h is

x_{i+1}-x_{i}

while on the Simpson rule page, h is

b-a

which is twice as big. That's why the coefficient differs by a factor 2. I re-stated the definition of h in this article just above the table, but perhaps we use the same definition on both pages. This seems to cause quite a bit of confusion. -- Jitse Niesen (talk) 16:10, 29 December 2008 (UTC)[reply]

In closed Newton-Cotes formula of degree 2 (Simpson's Rule), the coefficient of error term is $-{\frac {{(b-a)}^{5}}{2880}}$ . Shouldn't it be $-{\frac {{(b-a)}^{5}}{90}}$ ?

- Annada (talk) 12:01, 2 June 2013 (UTC)[reply]

This is again a factor two:

{\frac {{(b-a)}^{5}}{2880}}={\frac {1}{90}}{\bigl (}{\frac {b-a}{2}}{\bigr )}^{5}

, which is the formula for the error on the page Simpson's rule. -- Jitse Niesen (talk) 20:11, 2 June 2013 (UTC)[reply]

For the open two-point rule I'm getting an error term ${\frac {3}{4}}h^{3}f(\xi )$ . I acknowledge that both WolframMathWorld and "Abramowitz and Stegun" say it's a 1/4 term but I've just came across another article that claims it's 3/4. Could someone else check this? — Preceding unsigned comment added by 2.219.118.179 (talk) 19:55, 4 December 2018 (UTC)[reply]

It would be useful to see that ″another article″. However it looks like another problem related with the step size (see the reply to the first comment). 1/4 is consistent with the value of h usid in this article.194.230.155.218 (talk) 18:45, 13 February 2021 (UTC)[reply]

Open formula

It's silly that what's called "the open Newton–Cotes formula of degree n" is an approximation using a degree n-2 interpolated polynomial. — Preceding unsigned comment added by 99.245.250.27 (talk) 18:40, 10 December 2012 (UTC)[reply]

I agree; the usage of "degree" through the article is confusing and rather inaccurate.

It seems to be defined as the the number of interpolation points -1 (closed formula) or +1 (open formula); while the value of n in the closed formulas agrees with the common usage, the value used for the open formula does not (it should be -1 as well), and anyway it is rarely referenced to as "degree", as the term is usually reserved to denote the "degree of exactness", the maximum degree of the polynomials which are exactly integrated by the formula.

Besides, I find no reference of such usage in the reference (did I just miss it?)

If there are no objections I will edit it and add a source.194.230.155.218 (talk) 10:56, 13 February 2021 (UTC)[reply]