I'm helping a friend of mine to do her homework, but i need help understanding some results (sorry but i took numeric methods class a looooong time ago)
So, the task is to fit a cubic spline over some mountain shaped function f(x), then calculate the area using Simpson's rule. From what i know, Simpson's rule is exact for degree <= 3, so:
Area(Simpson over splines, n = 2) should equal Area (Integrate over splines)
But i'm getting different results when using n = 2 (number of intervals). I'm using the following function, over the interval [0, 1]
$$1.0 - cos((sin(pi * x^{1.5})))$$
By using Simpson' rule with n = 2, i get 0.2502099, but the integral says that the area is 0.2059441 (which i get if i use n = 20, for example). Why this is happening, and how can i fix it?
Here are the (x, y) points used to fit the Spline:
x y
0.000000 0.000000
0.111111 0.006731
0.222222 0.051774
0.333333 0.157254
0.444444 0.304818
0.555556 0.429594
0.666667 0.451576
0.777778 0.328222
0.888889 0.116309
1.000000 0.000000
I'm using Python, and i can post the code if needed! Thanks for your help!
If your cubic spline contains multiple segments, that means it is piecewise continuous. Each segment of the cubic spline is actually a different cubic polynomial and it is probably only C1 continuous (at best C2) at the segment joints. So, when you use Simpson's rule with n=2, all the 3 evaluations of the function value could be done against different cubic polynomials. That is the reason you do not get the same value as when using n=20. You can try to use the 3-point Simpson rule (i.e., n=2) for each segment in the spline then sum up all the result.