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equation solving - Solve answer optimization


I have an equation system (here is the simplified version for illustration)


    eq = Thread[
   Equal[{a[

        0]^2 (a[47]^2 (2 - 2 u[2] - 2 u[5] + 6 u[7] - 2 u[9] + 
            6 u[10] - 2 u[11] + 6 u[13] - 2 u[15] - 2 u[17]) + 
         a[49]^2 (2 - 2 u[2] - 2 u[5] + 6 u[7] - 2 u[9] + 6 u[10] - 
            2 u[11] + 6 u[13] - 2 u[15] - 2 u[17]) + 
         a[52]^2 (2 - 2 u[2] - 2 u[5] + 6 u[7] - 2 u[9] + 6 u[10] - 
            2 u[11] + 6 u[13] - 2 u[15] - 2 u[17])) + 
      a[47]^2 (a[
           49]^2 (2 + 2 u[1] - 2 u[2] + 2 u[3] - 6 u[4] + 6 u[5] - 
            6 u[6] - 2 u[7] + 2 u[8] - 2 u[9] + 6 u[10] - 2 u[11] + 
            2 u[12] - 2 u[13] - 6 u[14] + 6 u[15] + 2 u[16] - 

            2 u[17] + 2 u[18] - 6 u[19]) + 
         a[52]^2 (2 + 2 u[1] - 2 u[2] + 2 u[3] - 6 u[4] + 6 u[5] - 
            6 u[6] - 2 u[7] + 2 u[8] - 2 u[9] + 6 u[10] - 2 u[11] + 
            2 u[12] - 2 u[13] - 6 u[14] + 6 u[15] + 2 u[16] - 
            2 u[17] + 2 u[18] - 6 u[19])) + 
      a[0] a[47] a[49] a[
        52] (8 - 12 u[1] + 12 u[3] + 12 u[4] - 12 u[6] + 12 u[8] - 
         24 u[9] + 24 u[10] + 24 u[11] - 12 u[12] + 12 u[14] - 
         12 u[16] + 12 u[18] - 12 u[19]) + 
      a[49]^2 a[

        52]^2 (2 + 2 u[1] - 2 u[2] + 2 u[3] - 6 u[4] + 6 u[5] - 
         6 u[6] - 2 u[7] + 2 u[8] - 2 u[9] + 6 u[10] - 2 u[11] + 
         2 u[12] - 2 u[13] - 6 u[14] + 6 u[15] + 2 u[16] - 2 u[17] + 
         2 u[18] - 6 u[19]) + 
      a[47]^4 (-1 - u[1] + u[2] - u[3] - u[4] + u[5] - u[6] + u[7] - 
         u[8] + u[9] + u[10] + u[11] - u[12] + u[13] - u[14] + u[15] -
          u[16] + u[17] - u[18] - u[19]) + 
      a[49]^4 (-1 - u[1] + u[2] - u[3] - u[4] + u[5] - u[6] + u[7] - 
         u[8] + u[9] + u[10] + u[11] - u[12] + u[13] - u[14] + u[15] -
          u[16] + u[17] - u[18] - u[19]) + 

      a[52]^4 (-1 - u[1] + u[2] - u[3] - u[4] + u[5] - u[6] + u[7] - 
         u[8] + u[9] + u[10] + u[11] - u[12] + u[13] - u[14] + u[15] -
          u[16] + u[17] - u[18] - u[19]) + 
      a[0]^4 (-1 + u[1] + u[2] + u[3] + u[4] + u[5] + u[6] + u[7] + 
         u[8] + u[9] + u[10] + u[11] + u[12] + u[13] + u[14] + u[15] +
          u[16] + u[17] + u[18] + u[19]), 
     a[0]^2 a[49] a[
        52] (-8 u[1] + 4 u[2] + 8 u[3] + 12 u[4] - 4 u[6] - 4 u[7] - 
         4 u[9] + 8 u[11] - 4 u[12] - 8 u[13] + 4 u[15] + 4 u[16] + 
         4 u[18]) + 

      a[0]^3 a[
        47] (2 u[1] - 2 u[3] - 2 u[4] + 2 u[6] + 4 u[7] + 2 u[8] + 
         4 u[10] + 2 u[12] + 4 u[13] + 2 u[14] - 2 u[16] + 2 u[18] - 
         2 u[19]) + 
      a[49]^3 a[
        52] (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 4 u[15] + 
         4 u[19]) + 
      a[49] a[52]^3 (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 
         4 u[15] + 4 u[19]) + 
      a[47]^2 a[49] a[

        52] (4 u[1] - 4 u[2] + 4 u[3] - 4 u[7] + 4 u[8] - 4 u[9] + 
         8 u[11] - 8 u[12] + 8 u[13] + 12 u[14] - 12 u[15] - 
         8 u[16] + 8 u[17] - 8 u[18] + 12 u[19]) + 
      a[0] (a[47]^3 (-2 u[1] + 2 u[3] + 2 u[4] - 2 u[6] + 4 u[7] - 
            2 u[8] + 4 u[10] - 2 u[12] + 4 u[13] - 2 u[14] + 
            2 u[16] - 2 u[18] + 2 u[19]) + 
         a[47] (a[
              49]^2 (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 
               2 u[6] - 2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 
               4 u[13] - 6 u[14] + 6 u[16] - 6 u[18] + 6 u[19]) + 

            a[52]^2 (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 
               2 u[6] - 2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 4 u[13] 
               - 6 u[14] + 6 u[16] - 6 u[18] + 6 u[19]))), 
     a[0]^2 a[47] a[
        52] (-8 u[1] + 4 u[2] + 8 u[3] + 12 u[4] - 4 u[6] - 4 u[7] - 
         4 u[9] + 8 u[11] - 4 u[12] - 8 u[13] + 4 u[15] + 4 u[16] + 
         4 u[18]) + 
      a[0]^3 a[
        49] (2 u[1] - 2 u[3] - 2 u[4] + 2 u[6] + 4 u[7] + 2 u[8] + 
         4 u[10] + 2 u[12] + 4 u[13] + 2 u[14] - 2 u[16] + 2 u[18] - 

         2 u[19]) + 
      a[47]^3 a[
        52] (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 4 u[15] + 
         4 u[19]) + 
      a[47] a[52]^3 (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 
         4 u[15] + 4 u[19]) + 
      a[47] a[49]^2 a[
        52] (4 u[1] - 4 u[2] + 4 u[3] - 4 u[7] + 4 u[8] - 4 u[9] + 
         8 u[11] - 8 u[12] + 8 u[13] + 12 u[14] - 12 u[15] - 
         8 u[16] + 8 u[17] - 8 u[18] + 12 u[19]) + 

      a[0] (a[49]^3 (-2 u[1] + 2 u[3] + 2 u[4] - 2 u[6] + 4 u[7] - 
            2 u[8] + 4 u[10] - 2 u[12] + 4 u[13] - 2 u[14] + 
            2 u[16] - 2 u[18] + 2 u[19]) + 
         a[49] (a[
              47]^2 (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 
               2 u[6] - 2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 
               4 u[13] - 6 u[14] + 6 u[16] - 6 u[18] + 6 u[19]) + 
            a[52]^2 (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 
               2 u[6] - 2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 
               4 u[13] - 6 u[14] + 6 u[16] - 6 u[18] + 6 u[19]))), 

     a[0]^2 a[47] a[
        49] (-8 u[1] + 4 u[2] + 8 u[3] + 12 u[4] - 4 u[6] - 4 u[7] - 
         4 u[9] + 8 u[11] - 4 u[12] - 8 u[13] + 4 u[15] + 4 u[16] + 
         4 u[18]) + 
      a[0]^3 a[
        52] (2 u[1] - 2 u[3] - 2 u[4] + 2 u[6] + 4 u[7] + 2 u[8] + 
         4 u[10] + 2 u[12] + 4 u[13] + 2 u[14] - 2 u[16] + 2 u[18] - 
         2 u[19]) + 
      a[47]^3 a[
        49] (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 4 u[15] + 

         4 u[19]) + 
      a[0] (a[52]^3 (-2 u[1] + 2 u[3] + 2 u[4] - 2 u[6] + 4 u[7] - 
            2 u[8] + 4 u[10] - 2 u[12] + 4 u[13] - 2 u[14] + 
            2 u[16] - 2 u[18] + 2 u[19]) + 
         a[47]^2 a[
           52] (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 2 u[6] - 
            2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 4 u[13] - 6 u[14] + 
            6 u[16] - 6 u[18] + 6 u[19]) + 
         a[49]^2 a[
           52] (6 u[1] - 4 u[2] + 2 u[3] - 6 u[4] + 4 u[5] - 2 u[6] - 

            2 u[8] + 4 u[9] - 8 u[11] + 2 u[12] + 4 u[13] - 6 u[14] + 
            6 u[16] - 6 u[18] + 6 u[19])) + 
      a[47] (a[
           49]^3 (4 u[4] - 4 u[5] + 4 u[6] - 4 u[10] + 4 u[14] - 
            4 u[15] + 4 u[19]) + 
         a[49] a[52]^2 (4 u[1] - 4 u[2] + 4 u[3] - 4 u[7] + 4 u[8] - 
            4 u[9] + 8 u[11] - 8 u[12] + 8 u[13] + 12 u[14] - 
            12 u[15] - 8 u[16] + 8 u[17] - 8 u[18] + 12 u[19]))}, 0]];

I want to solve them for any parameters a[_]. For this simple equation the answer is



SolveAlways[eq, {a[0], a[47], a[49], a[52]}]

Out[2]= {{u[2] -> -u[4] - u[6] - u[8] - 2 u[12] + 2 u[13] - 2 u[15] - u[16] + u[17] - 3 u[18] + 2 u[19],


u[5] -> u[4] + u[6] - u[10] + u[14] - u[15] + u[19],


u[7] -> -u[10] - u[13],


u[9] -> 2/3 + u[8] + u[10] + u[12] - u[13] + u[15] - u[17] + 2 u[18] - 2 u[19],


u[11] -> 1/3 + u[12] - u[13] - u[14] + u[15] + u[16] - u[17] + u[18] - u[19],


u[1] -> -u[6] - u[8] - u[12] - u[14] - u[18],


u[3] -> -u[4] - u[16] - u[19]}}


So, I have 7 solution, from which only two of them have free constants 1/3 and 2/3. All other u[_], I can take zeros, so my simple answer is just u[9]->2/3 && u[11]->1/3



Unfortunately I need to solve much large system, which contains more that 7000 u[], and SolveAlways[ ] simply cannot do that. Fortunately, I can use linearity in u[] and avoid SolveAlways[] completely using simple variable replacement


eqToUspace = (eq /. {y_Plus?(FreeQ[#, a[_]] &) :> uSpace[y]});
(toReplace = Union[Cases[eqToUspace, _uSpace, Infinity]]);
newvarsList = Table[ur[i], {i, Length[toReplace]}];
uReplRules = Thread[Rule[toReplace, newvarsList]];

Which yields the system


simpleEq = eqToUspace /. uReplRules

Out[2]= {a[0]^2 (a[47]^2 ur[1] + a[49]^2 ur[1] + a[52]^2 ur[1]) + a[0] a[47] a[49] a[52] ur[3] + a[49]^2 a[52]^2 ur[4] + a[47]^2 (a[49]^2 ur[4] + a[52]^2 ur[4]) + a[47]^4 ur[6] + a[49]^4 ur[6] + a[52]^4 ur[6] + a[0]^4 ur[7] == 0, a[0]^2 a[49] a[52] ur[2] + a[0]^3 a[47] ur[5] + a[49]^3 a[52] ur[9] + a[49] a[52]^3 ur[9] + a[0] (a[47]^3 ur[8] + a[47] (a[49]^2 ur[10] + a[52]^2 ur[10])) + a[47]^2 a[49] a[52] ur[11] == 0, a[0]^2 a[47] a[52] ur[2] + a[0]^3 a[49] ur[5] + a[47]^3 a[52] ur[9] + a[47] a[52]^3 ur[9] + a[0] (a[49]^3 ur[8] + a[49] (a[47]^2 ur[10] + a[52]^2 ur[10])) + a[47] a[49]^2 a[52] ur[11] == 0, a[0]^2 a[47] a[49] ur[2] + a[0]^3 a[52] ur[5] + a[47]^3 a[49] ur[9] + a[0] (a[52]^3 ur[8] + a[47]^2 a[52] ur[10] + a[49]^2 a[52] ur[10]) + a[47] (a[49]^3 ur[9] + a[49] a[52]^2 ur[11]) == 0}



Solution of which (can be tested by SolveAlways[] as well) is just zeroes: {{ur[1] -> 0, ur[2] -> 0, ur[3] -> 0, ur[4] -> 0, ur[5] -> 0, ur[6] -> 0, ur[7] -> 0, ur[8] -> 0, ur[9] -> 0, ur[10] -> 0, ur[11] -> 0}}


After back substitution of old variables I can then solve the new system


unOptimized = 
 Solve[Thread[Equal[toReplace /. {uSpace -> Identity}, 0]], 
  Cases[toReplace, _u, Infinity]]

During evaluation of In[]:= Solve::svars: Equations may not give solutions for all "solve" variables. >>


Out[167]= {{u[13] -> -u[7] - u[10], u[17] -> 1 - u[2] - u[5] - u[9] - u[11] - u[15], u[12] -> 1/3 - u[1] + u[3] + u[4] - u[6] - u[9] + u[10] + u[11], u[18] -> 1/3 + u[1] - u[2] - u[3] - 2 u[4] - u[7] - u[10] - u[11] - u[15] - u[16], u[8] -> -(2/3) - u[1] + u[2] - u[3] + u[4] - u[5] + u[6] + u[7] + u[9] - u[10], u[14] -> u[3] + u[5] - u[6] + u[10] + u[15] + u[16], u[19] -> -u[3] - u[4] - u[16]}}


It contains same number of solutions. The only problem is that it is unoptimal in the sense that instead of just 2 free constants I have 4. Of course this is related to which u[_] Solve decides to solve first.


My question is how to find optimal solution of the reformulated problem with smallest number of constant terms (i.e. which have no u[_]). I even cannot identify the type of problem here.




Answer



Here a way, may be not to reduce calculation time, but to reduce memory usage with large systems.


First define all possible parameter combinations of the a[i] and then find the corresponding equations for the u[i]. In order to be valid for all a[i], these equations have all to equal zero.


paracomb = (Outer[
 Times, {a[0], a[47], a[49], a[52]}, {a[0], a[47], a[49], 
  a[52]}, {a[0], a[47], a[49], a[52]}, {a[0], a[47], a[49], 
  a[52]}] // Flatten // Union)

(*   {a[0]^4, a[0]^3 a[47], a[0]^2 a[47]^2, a[0] a[47]^3, a[47]^4, 
a[0]^3 a[49], a[0]^2 a[47] a[49], a[0] a[47]^2 a[49], a[47]^3 a[49], 

a[0]^2 a[49]^2, a[0] a[47] a[49]^2, a[47]^2 a[49]^2, a[0] a[49]^3, 
a[47] a[49]^3, a[49]^4, a[0]^3 a[52], a[0]^2 a[47] a[52], 
a[0] a[47]^2 a[52], a[47]^3 a[52], a[0]^2 a[49] a[52], 
a[0] a[47] a[49] a[52], a[47]^2 a[49] a[52], a[0] a[49]^2 a[52], 
a[47] a[49]^2 a[52], a[49]^3 a[52], a[0]^2 a[52]^2, 
a[0] a[47] a[52]^2, a[47]^2 a[52]^2, a[0] a[49] a[52]^2, 
a[47] a[49] a[52]^2, a[49]^2 a[52]^2, a[0] a[52]^3, a[47] a[52]^3, 
a[49] a[52]^3, a[52]^4}   *)

coe = Coefficient[eq[[All, 1]], #, 1] & /@ paracomb;


coeu = Union[Flatten[DeleteCases[#, 0] & /@ coe] // Simplify, 
          SameTest -> (#1 === -#2 || #1 === #2 &)]

(*   {-2 (-1 + u[2] + u[5] - 3 u[7] + u[9] - 3 u[10] + u[11] - 3 u[13] + 
u[15] + u[17]), -4 (2 u[1] - u[2] - 2 u[3] - 3 u[4] + u[6] + 
u[7] + u[9] - 2 u[11] + u[12] + 2 u[13] - u[15] - u[16] - u[18]), 
2 (1 + u[1] - u[2] + u[3] - 3 u[4] + 3 u[5] - 3 u[6] - u[7] + u[8] - 
u[9] + 3 u[10] - u[11] + u[12] - u[13] - 3 u[14] + 3 u[15] + 
u[16] - u[17] + u[18] - 3 u[19]), -1 - u[1] + u[2] - u[3] - u[4] +

u[5] - u[6] + u[7] - u[8] + u[9] + u[10] + u[11] - u[12] + u[13] - 
u[14] + u[15] - u[16] + u[17] - u[18] - 
u[19], -2 (u[1] - u[3] - u[4] + u[6] - 2 u[7] + u[8] - 2 u[10] + 
u[12] - 2 u[13] + u[14] - u[16] + u[18] - u[19]), 
2 (u[1] - u[3] - u[4] + u[6] + 2 u[7] + u[8] + 2 u[10] + u[12] + 
2 u[13] + u[14] - u[16] + u[18] - u[19]), 
4 (u[4] - u[5] + u[6] - u[10] + u[14] - u[15] + u[19]), -1 + u[1] + 
u[2] + u[3] + u[4] + u[5] + u[6] + u[7] + u[8] + u[9] + u[10] + 
u[11] + u[12] + u[13] + u[14] + u[15] + u[16] + u[17] + u[18] + 
u[19], -4 (-2 + 3 u[1] - 3 u[3] - 3 u[4] + 3 u[6] - 3 u[8] + 

6 u[9] - 6 u[10] - 6 u[11] + 3 u[12] - 3 u[14] + 3 u[16] - 
3 u[18] + 3 u[19]), 
2 (3 u[1] - 2 u[2] + u[3] - 3 u[4] + 2 u[5] - u[6] - u[8] + 2 u[9] - 
4 u[11] + u[12] + 2 u[13] - 3 u[14] + 3 u[16] - 3 u[18] + 
3 u[19]), 
4 (u[1] - u[2] + u[3] - u[7] + u[8] - u[9] + 2 u[11] - 2 u[12] + 
2 u[13] + 3 u[14] - 3 u[15] - 2 u[16] + 2 u[17] - 2 u[18] + 
3 u[19])}   *)

SolveAlways with a arbitrary variable r gives you the desired result



SolveAlways[Thread[coeu == 0], r] // Timing

(*    {0.015, {{u[2] -> -u[4] - u[6] - u[8] - 2 u[12] + 2 u[13] - 2 u[15] - 
 u[16] + u[17] - 3 u[18] + 2 u[19], 
 u[5] -> u[4] + u[6] - u[10] + u[14] - u[15] + u[19], 
 u[7] -> -u[10] - u[13], 
 u[9] -> 2/3 + u[8] + u[10] + u[12] - u[13] + u[15] - u[17] + 
 2 u[18] - 2 u[19], 
 u[11] -> 
 1/3 + u[12] - u[13] - u[14] + u[15] + u[16] - u[17] + u[18] - 

 u[19], u[1] -> -u[6] - u[8] - u[12] - u[14] - u[18], 
 u[3] -> -u[4] - u[16] - u[19]}}}    *)

Despite expectation Solve doesn't give the minimal solution


Solve[Thread[coeu == 0], Array[u, 19]] // Timing

(*    {0., {{u[10] -> -(2/3) - u[1] + u[2] - u[3] + u[4] - u[5] + u[6] + 
 u[7] - u[8] + u[9], 
 u[12] -> -(1/3) - 2 u[1] + u[2] + 2 u[4] - u[5] + u[7] - u[8] + 
 u[11], u[13] -> 

 2/3 + u[1] - u[2] + u[3] - u[4] + u[5] - u[6] - 2 u[7] + u[8] - 
 u[9], u[16] -> 
 2/3 + u[1] - u[2] - u[4] - u[7] + u[8] - u[9] + u[14] - u[15], 
 u[17] -> 1 - u[2] - u[5] - u[9] - u[11] - u[15], 
 u[18] -> 
 1/3 + u[1] - u[2] - 2 u[4] + u[5] - u[6] - u[7] - u[11] - u[14], 
 u[19] -> -(2/3) - u[1] + u[2] - u[3] + u[7] - u[8] + u[9] - u[14] +
  u[15]}}}    *)

But Solve with reduced variable set does like SolveAlways.



Solve[Thread[coeu == 0], Array[u, 12]] // Timing

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I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...
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