Abstract
We describe and analyze a randomized homotopy algorithm for the Hermitian eigenvalue problem. Given an \(n\times n\) Hermitian matrix \(A\), the algorithm returns, almost surely, a pair \((\lambda ,v)\) which approximates, in a very strong sense, an eigenpair of \(A\). 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 \(A\), is \(\mathcal{{O}}(n^6)\), 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.
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Acknowledgments
Diego Armentano was partially supported by Agencia Nacional de Investigación e Innovación (ANII). Felipe Cucker was partially funded by a GRF grant from the Research Grants Council of the Hong Kong SAR (project number CityU 100810).
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Communicated by Teresa Krick and James Renegar.
Dedicated to Mike Shub on his 70th birthday, for years of friendship.
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Armentano, D., Cucker, F. A Randomized Homotopy for the Hermitian Eigenpair Problem. Found Comput Math 15, 281–312 (2015). https://doi.org/10.1007/s10208-014-9217-9
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DOI: https://doi.org/10.1007/s10208-014-9217-9