Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and Laszlo Lovasz introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floating-point) approximation to the R-factor. In the present article,  investigate the accuracy of the factor R of the QR factorisation of an LLL-reduced basis. Our main contribution is the first fully rigorous perturbation analysis of the R-factor of LLL-reduced matrices under column-wise perturbations. Our results should be very useful to devise LLL-type algorithms relying on floating-point approximations.
['Xiao-Wen Chang', 'Damien Stehlé', 'Gilles Villard']
I am writing a research paper on a idea PERTURBATION ANALYSIS OF THE QR FACTOR R IN THE CONTEXT OF LLL LATTICE BASIS REDUCTION
[{"from": "human", "value": "I am writing a research paper on a idea PERTURBATION ANALYSIS OF THE QR FACTOR R IN THE CONTEXT OF LLL LATTICE BASIS REDUCTION"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and Laszlo Lovasz introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floating-point) approximation to the R-factor. In the present article,  investigate the accuracy of the factor R of the QR factorisation of an LLL-reduced basis. Our main contribution is the first fully rigorous perturbation analysis of the R-factor of LLL-reduced matrices under column-wise perturbations. Our results should be very useful to devise LLL-type algorithms relying on floating-point approximations."}]