24 February 2011

How do graphing calculators do numeical integration?

Here's a question I don't know the answer to, which Sam Shah asked: how does the TI-83 do ∫-zz exp(-x2) dx, or more generally numerical integration? Of course as z goes to ∞ this approaches √π, with very small tails. (The link goes to an old post of mine that unfortunately has broken LaTeX; you can read the alt text for the images. The idea is that in∫z exp(-x2) dx, the integrand can be bounded above by the exponential exp(-z2-2xz); integrating this, the original integral is less than exp(-z2)/2z and this is pretty tight. And yes, I know, I should switch to a platform with LaTeX support.)

So you expect to get something near √π for any reasonably large value of z. But if z is large enough -- say 1000 -- then you get a very small value, in this case on the order of 10-7. Presumably if the range of integration is wide enough, then the integration method used by the calculator doesn't actually pick up the central region where the action is actually happening.


Yaroslav said...

I'm using MathJax for math and it works fairly well. Put the snippet from http://pastebin.com/ETPfQuSF right after "head" tag in your blog template (under "Design"/"Edit HTML"), then anything between $ or $$ will render as formulas

zeebo said...
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zeebo said...

On the TI-84 it uses the Gauss-Kronrod quadrature (source)

It's probably the same for the 83+.