Since I haven't really had time to post, I figured I would take a quick opportunity to round out the end of the month with a post that illustrates the kind of thing I’ve been doing to
waste my time keep me busy.
Here goes:
If our inner product is defined as and our basis is defined as our Gram-Schmidt process is as follows:
Obviously,
Once we have found a normalized polynomial in our basis we can generate an orthogonal polynomial.
Once we have an orthogonal polynomial we need to normalize it.
Again, now having two normalized, orthogonal polynomials in our basis, we can generate another orthogonal polynomial.
Again we need to normalize the polynomial:
Since we now have three orthonormal polynomials we can complete our basis:
Therefore our new, orthonormal basis is:
Several things to note:
- This is not what I am learning. This is something I learned a year and a half ago. This is something I am doing in order to learn other things.
- This was essentially part “a” of a five part problem.
- The assignment had five problems.
- We were given a week to finish it.
- These are the first couple of Laguerre polynomials. We used them to approximate a function. It was a terrible approximation.
- I’m not bitching. This is actually the class I am doing well in and that I enjoy.
Addendum to anyone who finds this by searching to solve a homework problem: I have no idea if this is right. I would not suggest using it.