Certified dense linear system solving
Thom Mulders, Arne Storjohann · Repository for Publications and Research Data (ETH Zurich) · 2000
The following problems related to linear systems are studied: finding a diophantine solution; finding a rational solution; proving no diophantine solution exists; proving no rational solution exists. These problems are reduced, via randomization, to that of computing an expected constant number of rational solutions of square nonsingular systems using adic lifting. The bit complexity of the latter problem is improved by incorporating fast arithmetic and fast matrix multiplication. The resulting randomized algorithm for certified dense linear system solving has substantially better asymptotic complexity than previous algorithms for either rational or diophantine linear system solving.