TITLE: Algorithm for solving systems of linear Diophantine inequalities
QUESTION [8 upvotes]: So, I posted on StackOverflow looking for a reasonably fast algorithm to solve systems of linear Diophantine inequalities and was pointed to this article by Cheng-Zhi Gao and Yu-Lin Dong. The problem is, they give the algorithm on pages 350-351, but part of step (3) appears to be missing.
My question for MathOverflow is, therefore, whether anyone knows either of another such algorithm or has an idea as to what the missing part of Gao and Dong's algorithm is.

REPLY [6 votes]: GAP provides a function NullspaceIntMat which solves systems
of linear diophantine equations. The documentation says:
25.1-2 SolutionIntMat

* SolutionIntMat( mat, vec ) ───────────────────────────────────── operation

If  mat  is  a  matrix  with integral entries and vec a vector with integral
entries,  this  function  returns  a vector x with integer entries that is a
solution  of  the  equation x * mat = vec. It returns fail if no such vector
exists.

────────────────────────────────  Example  ─────────────────────────────────
  gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
  gap> SolutionMat(mat,[95,115,182]);
  [ 47/4, -17/2, 67/4, 0, 0 ]
  gap> SolutionIntMat(mat,[95,115,182]);
  [ 2285, -5854, 4888, -1299, 0 ]
────────────────────────────────────────────────────────────────────────────

The source code can be found in the file lib/matint.gi included in the GAP distribution.
There is also a function SolutionNullspaceIntMat which additionally
computes a basis of the integral nullspace of the given matrix.