Certification of the QR factor R, and of lattice basis reducedness
release_cfmuh472wje45b7ymthlc2ag2e
by
Gilles Villard
2007
Abstract
Given a lattice basis of n vectors in Z^n, we propose an algorithm using
12n^3+O(n^2) floating point operations for checking whether the basis is
LLL-reduced. If the basis is reduced then the algorithm will hopefully answer
''yes''. If the basis is not reduced, or if the precision used is not
sufficient with respect to n, and to the numerical properties of the basis, the
algorithm will answer ''failed''. Hence a positive answer is a rigorous
certificate. For implementing the certificate itself, we propose a floating
point algorithm for computing (certified) error bounds for the entries of the R
factor of the QR matrix factorization. This algorithm takes into account all
possible approximation and rounding errors. The cost 12n^3+O(n^2) of the
certificate is only six times more than the cost of numerical algorithms for
computing the QR factorization itself, and the certificate may be implemented
using matrix library routines only. We report experiments that show that for a
reduced basis of adequate dimension and quality the certificate succeeds, and
establish the effectiveness of the certificate. This effectiveness is applied
for certifying the output of fastest existing floating point heuristics of LLL
reduction, without slowing down the whole process.
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