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

TL;DR

This paper introduces a randomized shift-and-invert iteration method to efficiently determine if a symmetric matrix is positive definite, with high probability and logarithmic complexity relative to the condition number.

Contribution

It presents a novel probabilistic algorithm for positive definiteness testing that is simple to implement and scales logarithmically with the matrix's condition number.

Findings

High probability correctness for positive definiteness detection

Logarithmic complexity scaling with condition number

Easy to implement vector iteration-based method

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.