Argh. I like to see that someone is willing to do their own research. But finding a project that uses those methods, AND is interesting, AND in those fields takes some thought. Here is one idea that would seem to be interesting, involving one or more of those methods:
Consider the time required for some of the basic methods in linear algebra. For example, the multiplication of a pair of nxn matrices requires, in theory, O(n^3) flops. Is that true in practice, on a current processor? The idea is to compute the time required for a matrix multiply. (Use timeit to do the time computations, for MANY values of n, perhaps n=1:1:2000, or farther.) Save the time as measured. Then plot the results. Use modeling tools, such as a linear regression, to determine if the curve really does follow the expected power law. (If you log the data, linear regression can be used here.)
If there are significant deviations in the plot for some values of n, verify that they are reproducible. Are they? Now, postulate some reasons as to why that may have happened in what you see. (e.g., does your CPU use caching? Did you go far enough out that there were virtual memory issues? How would that impact the timing?)
IF all of that is easy enough and you have some spare time, consider if other fundamental tools in the field of linear algebra also have known time estimates. chol or qr for example, or maybe you can find some time estimates for the eigenvalues of a symmetric banded matrix. Or, consider how sparseness might impact the matrix multiplies we talked about before.
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