Successive Difference Substitution Based on Column Stochastic Matrix and Mechanical Decision for Positive Semi-definite Forms

Employing the concept of termination of sequence of SDS sets to describe the positive semidefinite property of a form, we establish a necessary and sufficient condition for deciding whether a given form on Rn+ is positive semi-definite or not, and show that, for a form which is (strictly) positive definite on Rn+, the corresponding sequence of SDS sets is positively terminating. The above results are exactly constructed as follows: We define the column stochastic mean matrix first, and then prove that if we choose countable infinite matrices from finite n×n column stochastic mean ones at random (repeats allowed), then the product of these infinite matrices will converge to a column stochastic mean matrix with rank 1. Finally, we show the proof for relations between termination of the sequence of SDS sets and positive semi-definite property of a form. The Maple program TSDS3, based upon these results, not only automatically prove the polynomial inequalities, but also output counter examples for those false. This method is verified to be very efficient and better than Pòlya method.