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

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.