Assignment 3, Q3

Theres not much to Gaussian Quadrature, however, without brute forcing how would we know the number of terms to use to get within accuracy of 0.001 as required by the question.

For the sake of convenience, the questions asks for the integral between 0 and 3 of e^x, and they want accuracy within 0.001...

Thoughts?

Again preliminary thoughts: I think one has to look at the derivation and the way the error term is formed there?

In which cases was Quadrature exact? I forget now, but minimum error is zero if you use a high enough polynomial, I think.

I think with Gaussian quadrature the error is never actually zero, however the larger the polynomial is the closer you get to an accurate answer. That said, is there some ratio of the highest polynomial term to the accuracy of the result? How does the accuracy part work?

Ah, I got that a bit back to front. Where you know f explicitly f(-1/sqrt(3)) + f(1/sqrt(3)) gives you exactly the integral on [-1,1] of any cubic polynomial on this interval. That generalises to exactness for polynomials of degree up to 2n-1, given n points.

The exact value is for the polynomial which would've approximated f on that interval, right?
integral(p)~integral(f) (on [-1,1]).

Now from here on I may be off on the wrong tack, but hopefully if I am, it's in an interesting way.

e = I-actual - I-est = I(f) - I(p)
so I(p) = I(f) - e which looks a bit stuck for a moment ... but what if we expand I(p)?

Bear in mind that for those specific points there is one and only one polynomial of (degree n -1).

And that polynomial is a estimate for f, just as I(p) ~ I(f) ...


IOW f is exactly that p minus error (down at function level here, not integral level).
IOW f is exactly the right infinite Taylor series minus error that starts at the point of uncertainty.

Going back up to I(p) = I(f) - e_I, could we push in some Taylor series expressions there?...

I suppose so, but I'll have to go and fiddle around on paper with it now. Brain starts to pop when I'm typing this stuff up. The matter of exactness up to some degree might just simplify this enough to make it work out to some simple error term?

Just a minute ... When you use Matlab to calculate an integral the command is not "integral", is it? It's probably ... quad(x) ... ?

OK from both the Wikipedia articles that seem to apply there's mention of using "two different rules of quadrature" and compare their difference. So maybe one takes different t values?

I must hunt down last year's assignment 3 stuff and see whether there was anything like this. Shame on me if there was, I suppose.

"Quadrature" is an archaic term for Integration. (So "quad" can just be an archaic term for a method Matlab implements as Simpson's 1/3 Rule, with a nicer abbr. than intgt.)

I found a page that might lead in the right direction, but it's probably more a source of confusion. If I had a pdf with solutions to the previous Ass 3 I've lost it well and truly. That would probably be the best place to start.

http://www.maths.lancs.ac.uk/~gilbert/m243b/node12.html#SECTION00031000000000000000

(The page on just plain quadrature is perhaps a more gentle start to this).

Yeah I did some reading on "quadrature" to get the essence of it. Its basically a rudimentary way of calculating an area. I've seen that link before - its not easy to understand.

As far as calculating the quadrature in Matlab goes, I haven't tried. It's not that hard to write a program to do it though.

Yes. I saw which page it was that I had open there as I was about to close it, and realised that's the wrong link. The plain quadrature page has something more like a start.

I spoke to the lecturer briefly about this question. The accuracy formula for gaussian quadrature is way beyond the scope of this course apparently. She says the calculations can take many pages.

Her suggestion was to just try increasing the terms until we get to the required accuracy for now. I think 4 terms would be about right....

Phew! OK, I'll stop that hunt then. I did read somewhere that a proper understanding of the error in Gaussian quadrature requires an understanding of matters beyond whatever course that web page belonged to.

So basically your error is a sort of "next term rule" then?

I've got no idea, but she was quite clear in saying its not something we need to know.

She agrees that this book leaves gaps in the students knowledge.