Notes

For my coursework i have decided to investigate the following integral because I thought that it would be an interesting integral to investigate. To do this I have used several calculations. I first had to work out the midpoint and trapezium rules, for which I chose to use values of n up to 2048, so that my results would have a high level of accuracy. The weighted mean of these was then used to give me the Simpson's rule values for the same values of n, as this is more accurate than both the midpoint and the trapezium rules calculations. When I had these approximations I did not have the exact value of the integral to allow me to calculate absolute or relative errors, so instead I chose to calculate the ratio of differences to analyse error. As Simpson's rule is a fourth order method it should converge to 1/16, which it does start to do with the first order, and is correct to that of five decimal places when n is 128. When n then doubles to 256, the ratio of differences changes to a value below 1/16, and is then only correct to 4 decimal places. As n increases further to 512, the ratio of differences decreases and only agrees with 1/16 to only two decimal places. My ratio of differences for 2048 was the final one that I initially did and this differs drastically to my previous values for the calculation, and is even negative. Because of this I decided to add three more values of n to investigate what happened. The values constantly changed, so I came to the conclusion that this was due to calculations using nearly equal quantities, as the error is very small with only a certain degree of accuracy on Microsoft Excel. Therefore when you divide these numbers by each other the answer given doesn't follow the pattern. I then decided to extrapolate to infinity, but I was unsure which values I was going to use, because as n increases, the values are supposed to get more accurate but any values after n being 512 would have been inappropriate to use. I therefore decided to use the values of the Simpson's rule when n is 64 and 128 to calculate the extrapolated value. This method of extrapolation proved to be very accurate as it agreed with my Simpson's rule value of n being up to 8192.