Quadratic-time certificates in linear algebra

Authors:
Erich L. Kaltofen;Michael Nehring;B. David Saunders
Affiliations:
NCSU, Raleigh, NC, USA;NCSU, Raleigh, NC, USA;Univ. Delaware, Newark, DE, USA
Venue:
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Year:
2011

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Abstract

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. In addition, we give certificates of the same space and randomized validation time complexity for the Frobenius form, which includes the characteristic and minimal polynomial. For determinant and rank we have certificates of essentially-quadratic binary space and time complexity via Storjohann's algorithms.