Abstract:
This paper is devoted to the approximation of quadratic algebraic lattices and grids by integer lattices and rational grids. A General formulation of the problem of approximation of algebraic lattices and corresponding meshes by integer lattices and rational meshes is given. In the case of a simple $p$ of the form $p=4k+3$ or $p=2$, we consider an integer lattice given $m$by a suitable fraction to the number $\sqrt{p}$. The corresponding algebraic lattice and the generalized parallelepipedal grid are written out explicitly. To determine the quality of the corresponding generalized parallelepipedal grid, a quality function is defined, which requires $O(N)$ arithmetic operations for its calculation, where $N$ — is the number of grid points. The Central result is an algorithm for computing a quality function for $O\left(\sqrt{N}\right)$ arithmetic operations. We hypothesize the existence of an algorithm that requires $O\left(\ln{N}\right)$ arithmetic operations. An approach for calculating sums with integral parts of linear functions is outlined.
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