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I'm familiar with the QR algorithm for eigen decomposition in symmetric matrices, which takes roughly O(n^3) time. But that O(n^3) only holds if you take a constant number of QR steps per eigenvalue, and I don't know of any convergence guarantees (say in Frobenius distance, or anything else sensible) within that time.

So are there any theoretical runtime guarantees I can make about an eigen decomposition (symmetric matrix or otherwise) for a given allowable numerical error?

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