Talk:Numerical integration

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Mathematics rating:	B Class	High Priority	Field: Applied mathematics
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. No discussion of order. Article could use some pictures and perhaps an example. Section on adaptivity is weak. Perhaps discuss more quadrature rules; definitely, compare them to each other. -- Jitse Niesen (talk) 13:28, 2 August 2007 (UTC)

What happened to the tex of this page? It used to be fine... Loisel 05:10 May 14, 2003 (UTC)

Strange, it worked perfectly before :-( ... Will try to ask an admin Nixdorf 05:48 May 14, 2003 (UTC)

Thanks for fixing it.Loisel 02:22 10 Jun 2003 (UTC)

[edit] Equation reference is broken new version of page

In the section "Conservative (a priori) error estimation" there is a refernce to function (*) this function is present in older versions of the page (i.e. oldid=976581), but not in the current version. The index "n" for the simplified formula for the Riemann sum is therefore not explained to the reader.

130.95.29.24 04:01, 1 December 2006 (UTC)Anders

[edit] Introduction needed.

This subject could use an introduction that is easy for the casual reader to grasp. The true mark of higher intelligence is the ability to take a complex subject and, not only understands it, but re-presents it in terms that are simple and easy to understand. Anyone can paraphrase a textbook, but to truly understand the concept means being able to re-craft it, eliminating what is esoteric and teaching the core principles. Any layman can appreciate that approach and, if they are intrigued by the simple introduction, can pursue the more formal instruction further in the article.

do it yourself, then What's up Dr. Strangelove 06:17, 7 August 2007 (UTC)

[edit] Distinction between numerical integration and numerical solution of diff. eqs.

There needs to be a clear distinction made between numerical integration and numerical solution of differential equations. Which is a subset of which? What determines when one method can be applied but not another? The current explanation does not make this clear. Jdpipe 00:43, 26 March 2007 (UTC)

[edit] Reasons for numerical integration

in this section the author almost implies that exp(-t^2) is impossible , however it is infact sqrt(pi) if it varies between -inf to +inf —The preceding unsigned comment was added by 82.35.33.103 (talk) 13:54, 3 April 2007 (UTC).

That's a very special case. If it's from 3 to 5, or from −½ to 0, then we don't have a formula. -- Jitse Niesen (talk) 14:46, 3 April 2007 (UTC)

At the moment, the page refers to an antiderivitive "which is a simple combination of elementary functions". Elementary functions are defined as being closed under 'simple combination' so the phrase 'simple combination' is redundant.

[edit] Definition of "quadrature"

This continues the discussion by edit summary, which went

"quadrature is integration, numerical quadrature is the most common phrase it is used in, but o/w the 'numerical' would be redundant."

and then

"undo: the first section says otherwise (Davis and Rabinowitz quote) and we have to distinguish from symbolic integration. if you have a conflicting source, we can discuss this on the talk page..."

The Springer Encyclopaedia of Mathematics defines quadrature as "The calculation of an area or an integral (of a function of a single variable)." [1]. The Oxford English Dictionary says "the calculation of the area bounded by, or lying under, a curve; the calculation of a definite integral, esp. by numerical methods." The book by Davis and Rabinowitz is not clear, as far as I can see, but on page 2, they write "The terms mechanical or approximate quadrature are also employed for this type of numerical process [i.e., approximate integration."

Finally, the quote in the article is wrong and should be removed. There is no Greek word quadratos; in fact, the Greeks did not have the letter Q or an equivalent. As the Oxford English Dictionary says, quadrature comes from the Latin word quadratura, which derives via quadrum (a square) from quattuor (four). I don't know what the Greek word for quadrature is, but they used tetragōnos for square (which led to the English word tetragon), from tettares (four). -- Jitse Niesen (talk) 18:16, 16 January 2008 (UTC)

I concur that quadrature means the finding of area, not necessarily by (numerical) integration. But I also agree that its most common use today is in reference to numerical integration (e.g., the MATLAB quad* commands, or quadrature points). As I look at the num. analysis books on my shelves I see books that have "Numerical Quadrature" in their titles (e.g. Stroud) but not just "Quadrature". The "numerical" is not redundant, in my opinion, though it's arguably somewhat pedantic; the text should read "numerical quadrature" (not merely "quadrature").

BTW, I don't think of symbolic integration as quadrature as it's the finding of an antiderivative, not an area. Quadrature harkens back to the days when to find the area of something meant to have a geometric means of constructing a square with the same area as the object ("squaring the circle" being the most notable example of such a problem). JJL (talk) 18:42, 16 January 2008 (UTC)

I'm also very much into the area myself -- I'm currently writing a PhD on adaptive quadrature -- and I agree that "quadrature" is not formally defined as "numerical integration", yet I don't think I have ever seen it used for anything other than numerical integration.

May I suggest wording this as

• "The term quadrature is more or less a synonym for integration, especially as applied to one-dimensional integrals and is generally used to mean numerical integration.

Unless, of course, I'm wrong and it is also used for analytical integration...

Cheers and thanks, pedro gonnet - talk - 17.01.2008 07:14

I am the one who made the original edit. I was taught that quadrature is the calculus of the area under the curve and that numerical quadrature was the same done numerically. Unfortunately, neither of the books I used (Trefethen and Bau as well as Stoer and Bulirsch) explicitly define the term. However both only talk about numerical quadrature using the entire term. I think that JJLJitse Niesen's comment is illuminating, and that he gets it right he also highlights why its rare to talk about quadrature when it is not numerical. I also strongly agree with JJLJitse Niesen that the quote should be purged (if there is no greek Q and the OED disagrees on etymology) and that it does not define numerical quadrature as the replying author did. Pdbailey (talk) 00:40, 18 January 2008 (UTC) (edited by pdbailey to replace JJL with Jitse Niesen, sorry for any confusion. Pdbailey (talk) 04:52, 19 January 2008 (UTC)

[edit] diff eq comparison

It's my opinion that the comparison with differential equations could go at the bottom because it stands on its own, is useful primarily if you know math from more advanced classes than the rest of the article requires. Does anyone object? Pdbailey (talk) 15:02, 19 January 2008 (UTC)

[edit] oddness in the figure

I have always felt there is something "off" about this figure presently on the main page and I just put my finger on it. What does the grey that outlines the trapezoids between the lines represent? If it is the integral, shouldn't it go the the edges (in some way, like a line with zero slope, or the edges should be sampled to allow for this). If it isn't the integral, how is it interesting, and why is it there? Pdbailey (talk) 20:02, 6 April 2008 (UTC)

Uppon closer inspection, I see the the integral boundaries are sampled first, by now I wonder, why is there only one additional point in the first three innovations, but then four in the last one? What exactly is going on in this figure? Is it really the best figure to explain this idea? Pdbailey (talk) 20:05, 6 April 2008 (UTC)