November 11, 2003

MME 523

 

The Hunt for Pi Goes On

 

As you are now experienced Numerical Analysts, its time to go back the  problem of  estimation of  p.  Since p is a transcendental number, we can never write it down exactly, being left with the pursuit of more and more decimal place accuracy.  Our goal this week is to see what numerical techniques can contribute to this.  Once again you may work in pairs and document your efforts in a Word document which we will then convert to html and put on the Internet.

 

Areas to explore:

 

review: previously in this course you had produced a Taylor Series for Arctan(x).  This was in the October 14th assignment.  Armed with a series for Arctan, one could substitute x=1 and get a series for p/4.

 

You can begin by reviewing or recreating how accurate this was.  For example, how many terms in the series are needed to get 4 decimal place accuracy.

 

If you review the error expression for Taylor Series, you will see it depends on how far x is from the “base point”.  In this case, x =1 is relatively far away from 0

 

new:  spend some time on the Internet and see what you can find out about  Machin’s Formula and if you can reproduce/verify/derive it.

 

Having done so,  how can it be used to substantially increase the number of decimal places in the estimate of Pi??

 

curricula:  can this kind of material be used in any of your courses??? please comment pro or con.  This is an interesting mix of history, technology, analysis and arithmetic.

 

resources:  please keep any neat Web sites you might find and include them in your paper