Algorithm for solving systems of linear Diophantine inequalities

Mathematica implements an algorithm: see the manual here:

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(Added in response to a comment query.)

The paper

Hochbaum, Dorit S., and Anu Pathria. "Can a System of Linear Diophantine Equations be Solved in Strongly Polynomial Time?." (1994). (PDF download link)

says that "no strongly polynomial algorithm exists for the problem of finding the set of solutions to a system of linear diophantine equations in the complexity model allowing the operations $\{ +, -, \times, /, \bmod, < \}$." But if one assumes the gcd can be found quickly, then the system can be solved in strongly polynomial time.

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.