Numerical Properties of the LLL Algorithm
نویسندگان
چکیده
The LLL algorithm is widely used to solve the integer least squares problems that arise in many engineering applications. As most practitioners did not understand how the LLL algorithm works, they avoided the issue by referring to the method as an integer Gram Schmidt approach (without explaining what they mean by this term). Luk and Tracy were first to describe the behavior of the LLL algorithm, and they presented a new numerical implementation that should be more robust than the original LLL scheme. In this paper, we compare the numerical properties of the two different LLL implementations.
منابع مشابه
A modified LLL algorithm for change of ordering of Grobner basis
In this paper, a modied version of LLL algorithm, which is a an algorithm with output-sensitivecomplexity, is presented to convert a given Grobner basis with respect to a specic order of a polynomialideal I in arbitrary dimensions to a Grobner basis of I with respect to another term order.Also a comparison with the FGLM conversion and Buchberger method is considered.
متن کاملFloating-Point LLL: Theoretical and Practical Aspects
The text-book LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the Gram-Schmidt orthogonalisation by floating-point approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floating-point approximations seems to be natural for LLL even from the theoretical point of view: it is t...
متن کاملPartial LLL Reduction
The Lenstra-Lenstra-Lovasz (LLL) reduction has wide applications in digital communications. It can greatly improve the speed of the sphere decoding (SD) algorithms for solving an integer least squares (ILS) problem and the performance of the Babai integer point, a suboptimal solution to the ILS problem. Recently Ling and Howgrave-Graham proposed the so-called effective LLL (ELLL) reduction. It ...
متن کاملAn improved LLL algorithm
6 The LLL algorithm has received a lot of attention as an effective numerical tool for preconditioning 7 an integer least squares problem. However, the workings of the algorithm are not well understood. In this 8 paper, we present a new way to look at the LLL reduction, which leads to a new implementation method 9 that performs better than the original LLL scheme. 10 © 2007 Published by Elsevie...
متن کاملInteger Relation Detection and Lattice Reduction
By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we shall see, integer relation algorithms have a variety of interesting applications, including the recognition of a numeric constant in terms of the mathematical formula that it satisfies. The prob...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007