System of equations and inequalities

Hi, I am trying to solve a system of equations and inequalities where I have 6 variables and 8 equations, like this:

syms h1 h2 h3 j12 j13 j23 
                        
eq1 = + j12 + j23 + j13 - h1 - h2 - h3 == 3;
eq2 = - j12 - j23 + j13 - h1 + h2 - h3 == 3;
eq3 = - j12 + j23 - j13 + h1 - h2 - h3 == 3;
eq4 = + j12 + j23 + j13 + h1 + h2 + h3 == 3;
eq5 = - j12 + j23 - j13 - h1 + h2 + h3 <= 0;
eq6 = - j12 - j23 + j13 + h1 - h2 + h3 <= 0;
eq7 = + j12 - j23 - j13 - h1 - h2 + h3 <= 0;
eq8 = + j12 - j23 - j13 + h1 + h2 - h3 <= 0;

I tried many methods, but none of them seems to work, do you have any suggestions?

PS: A solution of the system is the following, but I expect to find many more.

j12 = - 1; j23 = 2; j13 = 2;

h1 = 1; h2 = 1; h3 = -2.

Thank you in advance.

Answer 1

[Dongyue]

Hi Luciano,

Please try the following codes, where k and l are variables. For your example, k=2 and l=2:

clear; clc;
syms h1 h2 h3 j12 j13 j23 integer
                        
eq1 = + j12 + j23 + j13 - h1 - h2 - h3 == 3;
eq2 = - j12 - j23 + j13 - h1 + h2 - h3 == 3;
eq3 = - j12 + j23 - j13 + h1 - h2 - h3 == 3;
eq4 = + j12 + j23 + j13 + h1 + h2 + h3 == 3;
eq5 = - j12 + j23 - j13 - h1 + h2 + h3 <= 0;
eq6 = - j12 - j23 + j13 + h1 - h2 + h3 <= 0;
eq7 = + j12 - j23 - j13 - h1 - h2 + h3 <= 0;
eq8 = + j12 - j23 - j13 + h1 + h2 - h3 <= 0;

eqs = [eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8];
vars = [h1, h2, h3, j12, j13, j23];

S = solve(eqs, vars,'Real',true,'ReturnConditions',true)

S = struct with fields:
            h1: 3 - l
            h2: 3 - k
            h3: k + l - 6
           j12: 3 - l - k
           j13: k
           j23: l
    parameters: [k    l]
    conditions: in(k, 'integer') & in(l, 'integer') & h2 + j12 + 3 <= h3 + j13 + j23 + l & h3 + j13 + 3 <= h2 + j12 + j23 + l & h3 + j12 + l <= h2 + j13 + j23 + 3 & h2 + h3 + j23 + l <= j12 + j13 + 3

Best,

Dongyue