How Do You Determine Partition Points Using the Composite Trapezoidal Rule?

[Hiche]

I'm not sure if this is the right place to ask..

Anyway.

Assume we have some integral I with 0 and 2 as limits. I = 3∫xe^xdx from 0 to 2. What exactly do we have to do to find the partition points (and what are they?) but using the composite trapezoidal rule? I = 25.1671683 upon computing normally.

Another unrelated question. In Euler's method for approximation, how do we choose our h value? The smaller the h is, the better the approximation, but is there a way to compute it from a given IVP?

 

[HallsofIvy]

We don't, We can use whatever number of partitions we want (the more partitions the better the approximation) and whatever partition points we want. For simplicity we often choose them equally spaced but for some functions we might be guided by the function. For example, the error in using a "nth" order method (Euler's method is 1st order, trapezoidal rule is 2nd order, Simpson's rule is 3rd order) is proportional to the nth derivative. if there are some intervals on which that nth derivative changes sharply, and others where it is close to level, we might choose more partition points in those intervals with sharp increase or decrease, fewer where the derivative doesn't change much. But we choose those points, we don't "calculate" them.

As for Euler's method, as you say, the smaller h is, the better the approximation. But also, of course, since you have to do the same calculations on each interval, the more work you do. There is no way to "compute it from the given IVP" because the IVP does not know how accurate a solution you need or how much work you are willing to do. Those are things you have to decide.

In fact, the accuracy of the trapezoid methods increases as the number of intervals squared, Simpson's method as the number of intervals cubed which is why we would prefer to use Simpson's method- but the number of intervals still depends on how much accuracy you want and how much work you are willing to do.

I = 3∫xe^xdx from 0 to 2. What exactly do we have to do to find the partition points (and what are they?) but using the composite trapezoidal rule? I = 25.1671683 upon computing normally.

I take it that by "computing normally", you mean "finding and evaluating an anti-derivative". It's easy to integrate that "by parts" to get the exact answer $3(e^2+ 1)$ which, evaluated at 2 and 0, is just about that number you give.
Again, the more partitions you use (and I would use equally spaced partition points for simplicity) the more accurate your result but the more work you will have to do.