  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.
