Important: The numtheory package has been deprecated.  Use the superseding command NumberTheory[NearestLatticePoint] instead.

In the lattice given by the basis B, the nearestp command returns a vector w that is near the vector alpha. The vector w will be a nearest vector to alpha in the lattice in the following sense:

There is a constant K depending only on n, the dimension of the lattice, such that for every other vector u in the lattice given by B,

The output of the nearestp command is of the form:

which is a list of integers such that

The command with(numtheory,nearestp) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numtheory}\right)\:$

$B≔\left[\left[2^{\frac{1}{3}}\,0\,0\right]\,\left[\exp \left(2\right)\,1\,0\right]\,\left[0\,\mathrm{\pi }\cdot 100\,1\right]\right]$

$B≔\left[\left[2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,0\,0\right]\,\left[{\ⅇ}^2\,1\,0\right]\,\left[0\,100\mathrm{\pi }\,1\right]\right]$

$\mathrm{alp}≔\left[7.01\,8.01\,\mathrm{\gamma }\right]$

$\mathrm{alp}≔\left[7.01\,8.01\,\mathrm{\gamma }\right]$

$\mathrm{nearestp}\left(B\,\mathrm{alp}\right)$

$\left[\mathrm{-36}\,7\,0\right]$

$b≔\left[\left[-\frac{127230625}{746351104}\,1\,0\right]\,\left[\frac{90325}{71691}\,0\,0\right]\,\left[-\frac{726529225}{1119526656}\,\frac{477}{2995}\,1\right]\right]$

$b≔\left[\left[-\frac{127230625}{746351104}\,1\,0\right]\,\left[\frac{90325}{71691}\,0\,0\right]\,\left[-\frac{726529225}{1119526656}\,\frac{477}{2995}\,1\right]\right]$

$\mathrm{blp}≔\left[71.01\,18.01\,{\mathrm{\pi }}^4\right]$

$\mathrm{blp}≔\left[71.01\,18.01\,{\mathrm{\pi }}^4\right]$

$\mathrm{nearestp}\left(b\,\mathrm{blp}\right)$

$\left[2\,107\,97\right]$