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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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plotting - Adding a thick curve to a regionplot

Suppose we have the following simple RegionPlot: f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}] Now I'm trying to change the curve defined by $y=g[x]$ into a thick black curve, while leaving all other boundaries in the plot unchanged. I've tried adding the region $y=g[x]$ and playing with the plotstyle, which didn't work, and I've tried BoundaryStyle, which changed all the boundaries in the plot. Now I'm kinda out of ideas... Any help would be appreciated! Answer With f[x_] := 1 - x^2 g[x_] := 1 - 0.5 x^2 You can use Epilog to add the thick line: RegionPlot[{y < f[x], f[x] < y < g[x], y > g[x]}, {x, 0, 2}, {y, 0, 2}, PlotPoints -> 50, Epilog -> (Plot[g[x], {x, 0, 2}, PlotStyle -> {Black, Thick}][[1]]), PlotStyle -> {Directive[Yellow, Opacity[0.4]], Directive[Pink, Opacity[0.4]],
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