Archive for March 9, 2018

A Problem involving Simpson’s Rule

Posted in Cute Problems, mathematics with tags , on March 9, 2018 by telescoper

Since I’m teaching a course on Computational Physics here in Maynooth and have just been doing methods of numerical integration (i.e. quadrature) I thought I’d add this little item to the Cute Problems folder. You might answer it by writing a short bit of code, but it’s easy enough to do with a calculator and a piece of paper if you prefer.

Use the above expression, displayed using my high-tech mathematical visualization software, to obtain an approximate value for π/4 (= 0.78539816339…) by estimating the integral on the left hand side using Simpson’s Rule at ordinates x =0, 0.25, 0.5, 0.75 and 1.

Comment on the accuracy of your result. Solutions and comments through the box please.

HINT 1: Note that the calculation just involves two applications of the usual three-point Simpson’s Rule with weights (1/3, 4/3, 1/3). Alternatively you could do it in one go using weights (1/3, 4/3, 2/3, 4/3, 1/3).

HINT 2: If you’ve written a bit of code to do this, you could try increasing the number of ordinates and see how the result changes…

P.S. Incidentally I learn that, in Germany, Simpson’s Rule is sometimes called called Kepler’s rule, or Keplersche Fassregel after Johannes Kepler, who used something very similar about a century before Simpson…

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