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[Dieter]

(01-15-2018 01:40 AM)sa-penguin Wrote:  Thank you all, for inspiring my continued education. Today's subject: integral I set up a quick Excel spreadsheet, solving \(\int_{1}^{2}e^x dx\) 4096 steps, and I still had a error of 23.2E-9
Simpson rule was next, and 32 steps had the error down to 24.7E-9.

That's the plain vanilla implementation of Simpson's rule.
Here 32 intervals indeed give a result with an error of 2,47 E–8.

But if you got 32 intervals you can also use the data to calculate a Simpson approximation for 16 intervals. This has an error of 3,96 E–7.

And now for the interesting part: use both (!) Simpson approximations, i.e. both the value for 32 as well as the one for 16 intervals, and calculate (16 · S₃₂ – S₁₆)/15. Et voilà: the result has an error of merely 9,2 E–12 (!).

I have implemented this method in a small HP-41 program that can be found on this site.

This method also has another advantage: you can start with n=4 and 2 intervals. Calculate the combined approximation as shown above. Then continue with n=8 and 4 (where only the new f(x) have to be evaluated), and get a new, better approximation. Repeat until the desired accuracy is reached. So you can estimate the error and continue adding more nodes until the result is sufficiently accurate. Take a look at the sequence of Simpson approximations in the linked thread.

Edit:

(01-15-2018 01:40 AM)sa-penguin Wrote:  Now, if I can just wrap my head around Romberg's Method [as explained on Wikipedia] without my brain melting...

The combined Simpson approximations as described above are equivalent to the results in the third column of the Romberg matrix. ;-)

Dieter