QR因子R的认证及格基约化性的认证
Certification of the QR factor R, and of lattice basis reducedness
AI总结：
本文提出一种基于QR分解R因子误差界认证的算法，以12n^3+O(n^2)次浮点运算严格验证格基是否LLL约化，成本仅为QR分解的六倍，并实验证明其能有效认证最快LLL启发式算法的输出。
摘要：
给定Z^n中n个向量构成的格基，我们提出一种算法，使用12n^3+O(n^2)次浮点运算来检查该基是否为LLL约化的。若基是约化的，算法有望回答“是”。若基不是约化的，或所用精度相对于n及基的数值性质不足，算法将回答“失败”。因此，肯定回答即为严格认证。为实现认证本身，我们提出一种浮点算法，用于计算QR矩阵分解中R因子各项的（已认证）误差界。该算法考虑了所有可能的近似误差和舍入误差。认证的成本12n^3+O(n^2)仅为计算QR分解本身的数值算法成本的六倍，且认证可仅使用矩阵库例程实现。我们报告实验，表明对于维度适当且质量良好的约化基，认证成功，并证实了认证的有效性。该有效性被用于认证现有最快的LLL约化浮点启发式算法的输出，且不拖慢整个过程。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
