Numerical simulation of parametrized differential equations is crucial importance in the study real-world phenomena applied science and engineering. Computational methods for real-time many-query such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During last few decades, model order reduction has proved successful providing l...

Lenstra, Lenstra, and Lovász in [7] proved several inequalities showing that the vectors in an LLL-reduced basis are short, and near orthogonal. Here we present generalizations, from which with k = 1, and k = n we can recover their inequalities: Theorem 1. Let b1, . . . , bn ∈ R be an LLL-reduced basis of the lattice L, and d1, . . . , dk arbitrary linearly independent vectors in L. Then ‖b1 ‖ ...