Simpson revival

[Namir]

So my version 1 of my Basic code is the way to go since it does not update the older area value in W.

To be honest, I am a bit disappointed by your algorithm, since the general expectation for improving a calculated value in an iteration should be used in the next iteration and cause enhancing the result and/or reduce the number of iteration. An example is Ostrowski's root solving algorithm where he first uses Newton's method to refine the guess for the root and then (in the same iteration) refines that guess again. The result is that Ostrokswki's method matches the efficiency of Halley's root finding method.

In version 1, the (error of the refined area/area of calculated result) is about 100! It seems that applying the refinement on two good estimates for the integral works well to enhance the final result only.

Namir