## Diagonalize Large Hermitian Matrices Efficiently?

I am running a program that has to diagonalize large, complex Hermitian matrices (the largest they get is about 1000x1000). To diagonalize the matrix once isn't too bad, but I need to diagonalize thousands to millions of different Hermitian matrices each time I run a simulation. If I only need the eigenvalues of each matrix, is there an efficient method that does not require me to diagonalize each matrix to get said eigenvalues? I'm currently using a version of the QR algorithm to diagonalize the matrices but I was hoping there was a faster method. Is there a way to emulate the matrices with a series or polynomial of some sort? Any help is greatly appreciated (if anyone knows of any papers written on this topic that would be perfect). Thanks!
 There are methods using matrix multiplication and fixed points to pick out the largest eigenvalue if you only need that one. Is that the case here?
 That sounds helpful! I only ever need one eigenvalue from the computation (though not necessarily the largest); if you know of any resources on such methods could you point me to them? Hopefully I can adapt them to what I need. Thanks!

## Diagonalize Large Hermitian Matrices Efficiently?

Check out the power iteration method. The matrix must have a strictly largest ev, and this method will pick it out.

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