Integration (math) in C++
I'm looking for a library to find the integral of a given set of random data (rather than a function) in C++ (or C, but preferably C++). There is another question asking about integration in C but the answers discuss more how to integrate a function (I think...). I understand that this can be done simply by calculating the area under the line segment between each pair of points from start to finish, but I'd rather not reinvent the wheel if this has already been done. I apologize in advance if this is a duplicate; I searched pretty extensively to no avail. My math isn't as strong as I'd like it so it's entirely possible I'm using the wrong terminology.
Thanks in advance for any help!
Chris
Edit: In case anybody is interested, I feel like an idiot. Even adding in a bunch of OO abstraction to make my other code easier to use, that was maybe 30 lines of code. This is what 3 years away from any sort of math will do to you...thanks for all of the help!
Answer
This is trivial. If the points are (x0, y0), (x1, y1), ..., (xN, yN), and the points are ordered so that x0 <= x1 <= ... <= xN, then the integral is
- y0 * (x1 - x0) + y1 * (x2 - x1) + ...
using no interpolation (summing areas of rectangles), and
- (y0 + y1)/2 * (x1 - x0) + (y1 + y2)/2 * (x2 - x1) + ...
using linear interpolation (summing areas of trapezia).
The problem is especially simple if your data is y0, y1, ..., yN and the corresponding x values are assumed to be 0, 1, ..., N. Then you get
- y0 + y1 + ...
using no interpolation (summing areas of rectangles), and
- (y0 + y1)/2 + (y1 + y2)/2 + ...
using linear interpolation (summing areas of trapezia).
Of course, using some simple algebra, the trapezia formulae can be simplified. For instance, in the last case, you get
- y0/2 + y1 + y2 + ...