New algorithms for numerical integration and ODE solutions

[Wes Loewer]

(03-13-2014 10:44 AM)Namir Wrote:  I have posted on my web page two new algorithms for numerical integration

Namir,

Thank you for the articles.

I have a question about your Romberg-Simpson Method. (I've seen it mentioned in passing that this could be done, but I've never actually seen it implemented.) Your algorithm uses:

R(n,m) = (1/4^m-1)*(4^m*R(n,m-1)-R(n-1,m-1))

for both the Trapezoid and Simpson methods.

The derivation that I've seen for this recursive formula was based on the fact that the Trapezoid Error is proportional to 1/n^2. I'm not certain about this, but since the Simpson Error is proportional to 1/n^4, it seems like the recursive formula for Simpson's would be different. I could be totally wrong, but using a similar derivation, I'm coming up with:

R(n,m) = (1/16^m-1)*(16^m*R(n,m-1)-R(n-1,m-1))

I tried a couple of examples on a spreadsheet and this second formula seems to converge faster. Try it in your VB code and compare the two.

Please correct me if I'm off on this.

-Wes L