Quadratic-time certificates in linear algebra
Erich Kaltofen, Michael Nehring, B. David Saunders · 2011
We present certificates for the positive semidefiniteness of an n by n matrix A, whose entries are integers of binary length log ||A||, that can be verified in O(n(2+µ) (log ||A||)(1+µ) binary operations for any µ > 0. The question arises in Hilbert/Artin-based rational sum-of-squares certificates (proofs) for polynomial inequalities with rational coefficients. We allow certificates that are validated by Monte Carlo randomized algorithms, as in Rusins Freivalds's famous 1979 quadratic time certification for the matrix product. Our certificates occupy O(n(3+µ) (log ||A||)(1+µ) bits, from which the verfication algorithm randomly samples a quadratic amount.