solving for one solution to a system of polynomials

Question

I'm trying to solve a system of equations:

Solve[ A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 &&
       C1 F1 - E1 G1 == 0 && C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 &&
        A2 - B2 + C2 == 0 &&  A3 - B3 + C3 == 0,      
       {A1,A2,A3,B1,B2,B3,C1,C2,C3,D1,D2,E1,E2,F1,F2,G1,G2,H1,H2}] 

Since we have more variables than equations, we have more than 1 solution satisfying the above equations but I don't want Mathematica to give me all possible solutions -- I would like just only one.

1. How do I get Mathematica to give me just one possible solution?

One way I thought about doing this is to plug in random numbers such as A2 = 1/2, B2 = 1, C2 = 1/2, etc. so that all other variables are determined. But this isn't a very effective strategy in case the numbers I plug in do not satisfy all of the above equations.

(One reason why I don't want all possible solutions to a system of equations is because suppose I am working with 30 equations and 80 variables. Then this is crashing Mathematica.)

Answer 1

You can use FindInstance :

FindInstance[A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 && 
  C1 F1 - E1 G1 == 0 && C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 && 
  A2 - B2 + C2 == 0 && A3 - B3 + C3 == 0, 
  {A1, A2, A3, B1, B2, B3, C1,C2, C3, D1, D2, E1, E2, F1, F2, G1, G2, H1, H2}]

(* {{A1 -> 0, A2 -> 0, A3 -> 0, B1 -> 0, B2 -> 0, B3 -> 0, C1 -> 0, 
     C2 -> 0, C3 -> 0, D1 -> 0, D2 -> 0, E1 -> 0, E2 -> 0, F1 -> 0, 
     F2 -> 0, G1 -> 0, G2 -> 0, H1 -> 0, H2 -> 0}} *)

Answer 2

FindInstance

FindInstance[eqns,vars] gives you only a trivial solution (A1 == A2 == ...== H2 == 0), which is not what one reallly wants. FindInstance[eqns, vars, n] helps in finding n solutions, e.g. for n == 2 yields two non-trivial solutions :

FindInstance[ A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 && 
              C1 F1 - E1 G1 == 0 && C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 &&
              A2 - B2 + C2 == 0 && A3 - B3 + C3 == 0,
             {A1, A2, A3, B1, B2,  B3, C1, C2, C3, D1, D2, E1, E2, F1, F2, G1, G2, H1, H2}, 2]

Of course there are infinitely many such instances of solutions, so they aren't too interesting as well.

What one really would like is a symbolic solution. Thus one should make use of Solve or Reduce.

Solve

Working with Solve you can find what and how many symbolic solutions there are adding this option MaxExtraConditions -> All :

sols = Solve[ A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 && 
              C1 F1 - E1 G1 == 0 && C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 && 
               A2 - B2 + C2 == 0 && A3 - B3 + C3 == 0, 
             {A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, E1, E2, F1, F2, G1, G2, H1, H2}, 
             MaxExtraConditions -> All] // Quiet; 

so you can check how many solutions there are :

Length @ sols

27

and select n-k solutions, for 1 <= k < n <= 27 : sols[[ k;;n ]], e.g. the first one

sols[[1]]

 {B1 -> ConditionalExpression[A1 + C1, E1 != 0 && E2 != 0 && A1 D1 != 0], 
  B2 -> ConditionalExpression[A2 + C2, E1 != 0 && E2 != 0 && A1 D1 != 0], 
  C3 -> ConditionalExpression[-A3 + B3, E1 != 0 && E2 != 0 && A1 D1 != 0], 
  G1 -> ConditionalExpression[(C1 F1)/E1, E1 != 0 && E2 != 0 && A1 D1 != 0], 
  G2 -> ConditionalExpression[(C2 F2)/E2, E1 != 0 && E2 != 0 && A1 D1 != 0], 
  H1 -> ConditionalExpression[-((A1 D1)/E1), E1 != 0 && E2 != 0 && A1 D1 != 0], 
  H2 -> ConditionalExpression[-((A2 D1 + A1 D2 - (A1 D1 E2)/E1)/E1),
                               E1 != 0 && E2 != 0 && A1 D1 != 0]}

this means e.g. that B1 == A1 + C1 under conditions E1 != 0 && E2 != 0 && A1 D1 != 0.

One can observe that if we omit MaxExtraConditions or we add MaxExtraConditions -> Automatic then solutions will not be represented in terms of ConditionalExpressions and therefore some troubles can apear potentially.

Sometimes it will be handy to specify only a few variables. Then we can use also MaxExtraConditions in Solve, and specifying e.g. {A1, A2, A3, B1, B2, B3} we get only one symbolic solution :

Solve[ A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 && C1 F1 - E1 G1 ==0 &&
       C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 && A2 - B2 + C2 == 0 && A3 - B3 + C3 == 0, 
      {A1, A2, A3, B1, B2, B3}, MaxExtraConditions -> Automatic] // Quiet

{{A1 -> ConditionalExpression[-((E1 H1)/D1), F2 == (E2 G2)/C2 && F1 == (E1 G1)/C1], 
  A2 -> ConditionalExpression[((D2 E1 H1)/D1 - E2 H1 - E1 H2)/D1, 
                               F2 == (E2 G2)/C2 && F1 == (E1 G1)/C1], 
  B1 -> ConditionalExpression[C1 - (E1 H1)/D1, F2 == (E2 G2)/C2 && F1 == (E1 G1)/C1], 
  B2 -> ConditionalExpression[C2 + ((D2 E1 H1)/D1 - E2 H1 - E1 H2)/D1, 
                               F2 == (E2 G2)/C2 && F1 == (E1 G1)/C1], 
  B3 -> ConditionalExpression[A3 + C3,  F2 == (E2 G2)/C2 && F1 == (E1 G1)/C1]}}

Reduce

Reduce finds all solutions

r = Reduce[ A1 D1 + E1 H1 == 0 && A2 D1 + A1 D2 + E2 H1 + E1 H2 == 0 && 
            C1 F1 - E1 G1 == 0 && C2 F2 - E2 G2 == 0 && A1 - B1 + C1 == 0 && 
            A2 - B2 + C2 == 0 && A3 - B3 + C3 == 0, 
            {A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, E1, E2, F1, F2, G1, G2, H1, H2}];

being implicitly ConditionalExpression's. To select only one solution we just evaluate r[[n]] for 1<= n <= 25.

Warning

Comparing with sols, found by Solve the number of solutions may be slightly different because certain ConditionalExpression's repeat some identical solutions under different conditions :

Length @ r

25