I have a linear algebra problem that I need help with.
Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large Hermitian matrices (6^n x 6^n, where n>= 3, to be specific). Currently, we are just using MATLAB's eig() function to get them. I am trying to find optimizations for the simulations to cut down on computing time. There are three matrices that we use.
H_constant - generated before the loop. Real and symmetric about the diagonal. Does not change after initial calculation.
H_location - generated during each iteration. Diagonal.
H_final - the addition of H_constant and H_location. Therefore, it is also real and symmetric about the diagonal.
It is H_final that we need the eigenvalues and eigenvectors of. My theory is that we calculate the eigenvalues and eigenvectors of H_constant (which won't change after the initial calculation) once. We use this result with the eigenvalues of H_location (the diagonal), to get the eigenvalues and eigenvectors of H_final1. This would reduce our computation from tens of thousands of eig() calls to 1 eig() call and tens of thousands of very simple operations. I don't remember enough of my linear algebra to prove such a theory.
I hope I was able to explain the problem well enough. I hope someone is able to help me with this problem.