The IntegerRelations package contains two routines, LLL, and PSLQ, which are used to solve specific computational problems. LLL is the Lenstra, Lenstra, Lovasz lattice basis reduction. PSLQ is Bailey and Ferguson's partial sum of least squares algorithm. The LinearDependency routine is a user-level routine for applying PSLQ or LLL to solve the integer relation problem, defined as follows.

Given decimal approximations for $n$ real or complex numbers $x_1,x_2,...,x_n$, find an integer relation between them, that is, find integers $u_1,u_2,...,u_n$ such that $u_1{x}_1+u_2{x}_2+\mathrm{...}+u_n{x}_n$ is small, if such $u_i$ exist.

Each command in the IntegerRelations package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, IntegerRelations:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands.

To display the help page for a particular IntegerRelations command, see Getting Help with a Command in a Package.

$\mathrm{with}\left(\mathrm{IntegerRelations}\right)$

$\left[\mathrm{LLL}\,\mathrm{LinearDependency}\,\mathrm{PSLQ}\right]$

$\mathrm{Digits}≔20$

$\mathrm{Digits}≔20$

$x≔0.31783724519578224473$

$x≔0.31783724519578224473$

$\mathrm{PSLQ}\left(\left[1\,x\,x^2\,x^3\,x^4\right]\right)$

$\left[1\,0\,\mathrm{-10}\,0\,1\right]$

$\mathrm{solve}\left(y^4-10y^2+1\,y\right)$

$\sqrt{3}-\sqrt{2},-\sqrt{3}+\sqrt{2},\sqrt{3}+\sqrt{2},-\sqrt{3}-\sqrt{2}$

$\mathrm{identify}\left(x\right)$

$\sqrt{3}-\sqrt{2}$