Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Certification of the QR factor R, and of lattice basis reducedness   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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The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
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The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int