A Randomized Homotopy for the Hermitian Eigenpair Problem

We describe and analyze a randomized homotopy algorithm for the Hermitian eigenvalue problem. Given an $$n\times n$$n×n Hermitian matrix $$A$$A, the algorithm returns, almost surely, a pair $$(\lambda ,v)$$(λ,v) which approximates, in a very strong sense, an eigenpair of $$A$$A. We prove that the expected cost of this algorithm, where the expectation is both over the random choices of the algorithm and a probability distribution on the input matrix $$A$$A, is $$\mathcal{{O}}(n^6)$$O(n6), that is, cubic on the input size. Our result relies on a cost assumption for some pseudorandom number generators whose rationale is argued by us.