Fast Method for computing 3x3 symmetric matrix spectral decomposition

I am working on a project where I'm basically preforming PCA millions of times on sets of 20-100 points. Currently, we are using some legacy code that is using GNU's GSL linear algebra pack to do SVD on covariance matrix. This works, but is very slow.

I was wondering if there are any simple methods to do eigen decompositions on a 3x3 symmetric matrix, so that I can just put it on the GPU and let it run in parallel.

Since the matrices themselves are so small, I wasn't sure what kind of algorithm to use, because it seems like they were designed for large matrices or data sets. There's also the choice of doing a straight SVD on the data set, but I'm not sure what would be the best option.

I have to admit, I'm not stellar at Linear Algebra, especially when considering algorithm advantages. Any help would be greatly appreciated.

(I'm working in C++ right now)