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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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Is there a way to do conditional matrix loop using 'continue'

I have the following: n = 3; m = 5; ww = RandomReal[{0, 0.1}, {n, n}]; uu = RandomReal[{0, 1}, {m, n}]; pp = RandomReal[{0, 1}, {n, n}]; ss = RandomInteger[{0, 5}, {m, n}]; Grid[{{"ww", "uu", "pp", "ss"}, {ww // TableForm, uu // TableForm, pp // TableForm, ss // TableForm}}, Spacings -> {5, 2}, Dividers -> All] where I would like to look at every element of matrix ss and produce a matrix tt , with zeroes at the locations in ss which have zeroes, and in all other positions do the following: tt = (-1/Subscript[ww, m]) Log[(1 - uu)/(Subscript[pp, m - 1])], where Subscript[ww, m] is the value at index of ww matrix and where Subscript[pp, m - 1] is the value at index-1 of pp matrix. So for example if the first value ever read from matrix ss happens to be 2, then value taken from matrix ww would be from the row 2, but from pp would be from row 1. Also how to tell difference between a 0 as a valid value from within the matrix elemen...
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