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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 - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1....
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