Parametric linear systems: the two parameter case

Randomized black box algorithms provide a very efficient means for solving sparse linear systems over arbitrary fields. However, when these probabilistic algorithms fail, it is not revealed whether no solution exists or whether the algorithm simply made unlucky random choices. Here we give an efficient algorithm to compute a certificate of inconsistency for a linear system. The cost of producing such a certificate is shown to be about the same as that of solving the system in the black box model, while the cost of using a certificate to prove inconsistency is much smaller. More specifically, given a matrix A • F ~×~ over a field F and a vector b • F ×1, we show how to compute a vector u • F lxn such that uA = 0 and ub-fi O. This clearly demonstrates that Ax = b has no solution x • F ~×I. Intuitively such certificates are dense in the left nullspace of A. If b is not in the column span of A, then b is orthogonal to al~ most a proper subspace of the left nullspace of A. Thus, with high probability a random element of this left nullspace is a certificate. The cost of this algorithm is measured in terms of black-box evaluations w ~ wA, for w • F l×n. The algorithm requires an expected O(r) such black-box evaluations plus an additional O(rn) operations in F, where r is the rank of A. Additional space for O(n) values from F is needed. This is approximately the cost of solving a consistent system having the same black box matrix, using Wiedemann's algorithm. It should be noted that the black box evaluations used here compute vector-times-matrix products rather than the matrix-times-vector products used in Wiedemann's algorithm. We extend our techniques to provide certificates of inconsistency for sparse Diophantine systems Ax-b, where A • Z ~×~, b • Z ~×1 and where integer solutions x • Z ~×1 are sought. The interesting case here is when rational solutions to the system do exist. In this case, a certificate of inconsistency is a vector u • Z l×n and an integer d such that uA ~ 0 rood d while ub :~ 0 rood d. The integer d is a divisor of the largest determinantal divisor of A. A certificate can be found with about the same cost as that of solving a consistent Diophantine system …