# SPDX-License-Identifier: GPL-2.0-or-later
# NConvex: A Gap package to perform polyhedral computations
#
# Declarations
#
# to solve the system
# 3x+5y=8
# 4x-2y>=2
# 3x+y>=3
# --> SolveEqualitiesAndInequalitiesOverIntergers( [ [3,5] ], [ 8 ], [ [4,-2], [3,1] ], [2,3] );
#
# to get only solution where y is positive then add a 5'th argument for signs: 0 stands for free, 1 stands for positive
# --> SolveEqualitiesAndInequalitiesOverIntergers( [ [3,5] ], [ 8 ], [ [4,-2], [3,1] ], [2,3], [0,1] );
# to get only solution where x and y are positive then add a 5'th argument for signs: 0 stands for free, 1 stands for positive
# --> SolveEqualitiesAndInequalitiesOverIntergers( [ [3,5] ], [ 8 ], [ [4,-2], [3,1] ], [2,3], [1,1] );
#! @Chapter Polyhedrons
#! @Section ZSolve
#! @Description
#! This function produces a basis of the system <A>eqs</A> = <A>eqs_rhs</A>
#! and <A>ineqs</A> >= <A>ineqs_rhs</A>. It outputs a list containing three matrices.
#! The first one is a list of points in a polytope, the second is the hilbert basis
#! of a cone. The set of solutions is then the minkowski sum of the polytope
#! generated by the points in the first list and the cone generated by the hilbert
#! basis in the second matrix. The third one is the free part of the solution polyhedron.
#! The optional argument <A>signs</A> must be a list of zeros and ones which length is
#! the number of variables. If the ith entry is one, the ith variable must be >= 0.
#! If the entry is 0, the number is arbitraty. Default is all zero.
#! @Returns a list of three matrices
#! @Arguments eqs,eqs_rhs,ineqs,ineqs_rhs[,signs]
DeclareGlobalFunction( "SolveEqualitiesAndInequalitiesOverIntergers" );
#! @InsertChunk zsolve
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