non-negative integer solutions to system of linear equations in mathematica

Question

Related to my previous question, just wonder how to solve a system of linear equations with non-negative integral solutions, for example:

c11*x+c12*y+c13*z=d1
c21*x+c22*y+c23*z=d2

Thanks a lot!

Edit

I meant efficiently. For example, I could have used FrobeniusSolve to get two solution lists and try to find the intersection. But sometimes, the individual solution list is probably hugely large. Or try to verify each individual solution returned by one FrobeniusSolve to see whether they satisfy all the remaining equations, but that suffers from the same drawback.

Simon: I don't see how it works. Can you please give an example on "x1+2x2+5x3+7x4=40; x1+x2+2x3+x4=20"? — Qiang Li, Nov 18 '11 at 21:54
Please note I want to find all solutions, without any omission — Qiang Li, Nov 18 '11 at 21:58

Simon · Accepted Answer · 2011-11-18T22:41:11.490

Reduce is able to solve these types of problems.

To answer the specific case in your comment above:

In[1]:= solns =  Reduce[x1 + 2 x2 + 5 x3 + 7 x4 == 40 &&
                        x1 + x2 + 2 x3 + x4 == 20 &&
                        x1 > 0 && x2 > 0 && x3 > 0 && x4 > 0, 
                       {x1, x2, x3, x4}, Integers]

Out[1]= (x1 == 6 && x2 == 11 && x3 == 1 && x4 == 1) ||
        (x1 == 7 && x2 == 8 && x3 == 2 && x4 == 1) ||
        (x1 == 8 && x2 == 5 && x3 == 3 && x4 == 1) ||
        (x1 == 9 && x2 == 2 && x3 == 4 && x4 == 1) ||
        (x1 == 11 && x2 == 5 && x3 == 1 && x4 == 2) ||
        (x1 == 12 && x2 == 2 && x3 == 2 && x4 == 2)

Edit:

You can check that this is the same solution you get by solving the two equations separately and taking the intersection of their solutions:

In[2]:= a = Reduce[x1 + 2 x2 + 5 x3 + 7 x4 == 40 && 
                   x1 > 0 && x2 > 0 && x3 > 0 && x4 > 0, 
                  {x1, x2, x3, x4}, Integers];

        b = Reduce[x1 + x2 + 2 x3 + x4 == 20 && 
                   x1 > 0 && x2 > 0 && x3 > 0 && x4 > 0, 
                  {x1, x2, x3, x4}, Integers];

In[4]:= solns == Intersection[a, b]

Out[4]= True

And you can extract the solutions by, e.g., turning the solutions into a list of replacement rules and applying to the variables:

In[5]:= {x1, x2, x3, x4} /. {ToRules[solns]}

Out[5]= {{6, 11, 1, 1}, {7, 8, 2, 1}, {8, 5, 3, 1}, 
         {9, 2, 4, 1}, {11, 5, 1, 2}, {12, 2, 2, 2}}

+1 and you can use ToRules[] to transform the answer to a replacement list — Dr. belisarius, Nov 18 '11 at 22:03
@belisarius: can you elaborate on how to use ToRules? Ideally I want to extract the solutions as a list like {{6,11,1,1}, {7, 8, 2, 1}, ...} — Qiang Li, Nov 18 '11 at 22:11
also, Simon, what is your way of getting the intersection of two lists in mma? — Qiang Li, Nov 18 '11 at 22:14
@QiangLi - it is a built-in function http://reference.wolfram.com/mathematica/ref/Intersection.html — Verbeia, Nov 18 '11 at 22:18
@QiangLi: See edit for use of `ToRules` and also `Intersection` acting on `Or` instead of `List`. — Simon, Nov 18 '11 at 22:42

non-negative integer solutions to system of linear equations in mathematica

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