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PERTURBATION ANALYSIS OF THE QR FACTOR R IN THE CONTEXT OF LLL LATTICE BASIS REDUCTION
, 2009
"... ... an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLLreduction. The precise definition involves the Rfactor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floati ..."
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... an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLLreduction. The precise definition involves the Rfactor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floatingpoint) approximation to the Rfactor. In the present article, we investigate the accuracy of the factor R of the QR factorisation of an LLLreduced basis. The results we obtain should be very useful to devise LLLtype algorithms relying on floatingpoint approximations.
Rigorous perturbation bounds for some matrix factorizations
 SIAM J. Matrix Anal. Appl
"... Abstract. This article presents rigorous normwise perturbation bounds for the Cholesky, LU and QR factorizations with normwise or componentwise perturbations in the given matrix. The considered componentwise perturbations have the form of backward rounding errors for the standard factorization algor ..."
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Abstract. This article presents rigorous normwise perturbation bounds for the Cholesky, LU and QR factorizations with normwise or componentwise perturbations in the given matrix. The considered componentwise perturbations have the form of backward rounding errors for the standard factorization algorithms. The used approach is a combination of the classic and refined matrix equation approaches. Each of the new rigorous perturbation bounds is a small constant multiple of the corresponding firstorder perturbation bound obtained by the refined matrix equation approach in the literature and can be estimated efficiently. These new bounds can be much tighter than the existing rigorous bounds obtained by the classic matrix equation approach, while the conditions for the former to hold are almost as moderate as the conditions for the latter to hold. AMS subject classifications. 15A23, 65F35 Key words. Perturbation analysis, normwise perturbation, componentwise perturbation,
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 LLLreduced. 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 suffic ..."
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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 LLLreduced. 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 R factor of the QR factorization. This algorithm takes into account all possible approximation and rounding errors. 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.
MULTIPLICATIVE PERTURBATION ANALYSIS FOR QR FACTORIZATIONS
"... (Communicated by Wenyu Sun) Abstract. This paper is concerned with how the QR factors change when a real matrix A suffers from a left or right multiplicative perturbation, where A is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in th ..."
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(Communicated by Wenyu Sun) Abstract. This paper is concerned with how the QR factors change when a real matrix A suffers from a left or right multiplicative perturbation, where A is assumed to have full column rank. It is proved that for a left multiplicative perturbation the relative changes in the QR factors in norm are no bigger than a small constant multiple of the norm of the difference between the perturbation and the identity matrix. One of common cases for a left multiplicative perturbation case naturally arises from computing the QR factorization of A. The newly established bounds can be used to explain the accuracy in the computed QR factors. For a right multiplicative perturbation, the bounds on the relative changes in the QR factors are still dependent upon the condition number of the scaled Rfactor, however. Some “optimized ” bounds are also obtained by taking into account certain invariant properties in the factors.
LLL reducing with the most significant bits
"... Let B be a basis of a Euclidean lattice, and B ̃ an approximation thereof. We give a sufficient condition on the closeness between B ̃ and B so that an LLLreducing transformation U for B ̃ remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformatio ..."
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Let B be a basis of a Euclidean lattice, and B ̃ an approximation thereof. We give a sufficient condition on the closeness between B ̃ and B so that an LLLreducing transformation U for B ̃ remains valid for B. Further, we analyse an efficient reduction algorithm when B is itself a small deformation of an LLLreduced basis. Applications include speedingup reduction by keeping only the most significant bits of B, reducing a basis that is only approximately known, and efficiently batching LLL reductions for closely related inputs.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2000; 00:1–6 Prepared using nlaauth.cls [Version: 2002/09/18 v1.02] On the Perturbation of the Qfactor of the QR Factorization
"... This paper gives normwise and componentwise perturbation analyses for the Qfactor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Qfactor in the range ..."
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This paper gives normwise and componentwise perturbation analyses for the Qfactor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Qfactor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the firstorder perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent firstorder perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Qfactor are presented. Copyright c ○ 2000 John Wiley & Sons, Ltd. key words: QR factorization, perturbation analysis
Identification Using QR Decompositions?
, 2014
"... Multiorder covariance computation for estimates in stochastic subspace identification using QR decompositions ..."
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Multiorder covariance computation for estimates in stochastic subspace identification using QR decompositions