Certification of the QR factor R and of lattice basis reducedness

Given a lattice basis of n vectors in Zn, we propose an algorithm using 12n3 + O(n2) 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 12n3 + O(n2) 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 red...

• Type: Conference
• Year: 2007
• Where: ISSAC
• Authors: Gilles Villard
• Keywords: Algorithms, ISSAC 2007, Lattice Basis, QR Matrix Factorization