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arXiv · 1806.09725

Determination of Positive Definiteness through Shift-and-Invert Iteration in Weakly Polynomial Complexity

Martin Neuenhofen
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Abstract

We propose a numerical method, based on the shift-and-invert power iteration, that answers whether a symmetric matrix is positive definite ("yes") or not ("no"). Our method uses randomization. But, it returns the correct answer with high probability. A thorough proof for the probability is presented. If the method answers "yes", the result is true with a high constant probability. If it answers "no", it provides proof that the matrix is not positive definite. The method has the following benefits: The cost for a constant probability of success scales logarithmically with the condition number. Further, since essentially consisting of vector iterations, our method is easy to implement.

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Martin Neuenhofen. 2018-06-25. Determination of Positive Definiteness through Shift-and-Invert Iteration in Weakly Polynomial Complexity. https://arxiv.org/abs/1806.09725

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