Given a list (or a Vector) v of  real numbers, the PSLQ(v) command outputs a list (or a Vector) u of  integers such that  is minimized.  Thus the PSLQ function finds an integer relation between a vector of linearly dependent real numbers if the input has enough precision.

Given a list (or a Vector) v of  complex numbers, the PSLQ(v) command outputs a list (or a vector) u of  complex integers (Gaussian integers) such that the norm of  is minimized.

This is an implementation of Bailey and Ferguson's PSLQ algorithm. You can also use the Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm to find a linear relation.  For more information, see IntegerRelations[LLL]. Generally speaking, PSLQ is faster.

One application of PSLQ is to find the minimal polynomial of an algebraic number given a decimal approximation of the algebraic number.  The examples below illustrate this.  Generally speaking, if the height of the minimal polynomial is  and its degree is , then you need more than  correct decimal digits for the algebraic number.  If the input has  digits and this is insufficient, the output of PSLQ will typically be a list of  integers each with approximately / digits long.