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mathematical optimization - Find the values of 3 variables that best fit 6 equations



I have 6 equations


0.344027==0.5 (a+b)-0.5 (a-b) Cos[2 (-1.3439-0.0174533 c)]
0.679511==0.5 (a+b)-0.5 (a-b) Cos[2 (0.20944 -0.0174533 c)]

0.436543==0.5 (a+b)-0.5 (a-b) Cos[2 (-0.733038-0.0174533 c)]
0.324024==0.5 (a+b)-0.5 (a-b) Cos[2 (1.18682 -0.0174533 c)]
0.304968==0.5 (a+b)-0.5 (a-b) Cos[2 (-1.5708-0.0174533 c)]
0.676049==0.5 (a+b)-0.5 (a-b) Cos[2 (-0.174533-0.0174533 c)]

I could pick 3 equations from the list and got exact answers. However, I want my variables (a,b,c) to have values such that the right hand sides of all 6 equations approach the left hand side as much as possible. I'm not really sure what I can use in this case to resolve this.


Edit: I used Kuba's method to solve my problem. However, using FindFit (per belisarius' helpful suggestion) also gave me the same result. This works because all 6 equations have a pair of x and y that could be fitted through multivariable FindFit-- though if I were to be given random equations with a,b,c, NMinimize might be the only method to find them.



FindFit example in MMA's documentation Mathematica FindFit multivariable




Using FindFit to my example


Original equations:


0.344027==0.5 (a+b)-0.5 (a-b) Cos[2 (-1.3439-0.0174533 c)]
0.679511==0.5 (a+b)-0.5 (a-b) Cos[2 (0.20944 -0.0174533 c)]
0.436543==0.5 (a+b)-0.5 (a-b) Cos[2 (-0.733038-0.0174533 c)]
0.324024==0.5 (a+b)-0.5 (a-b) Cos[2 (1.18682 -0.0174533 c)]
0.304968==0.5 (a+b)-0.5 (a-b) Cos[2 (-1.5708-0.0174533 c)]
0.676049==0.5 (a+b)-0.5 (a-b) Cos[2 (-0.174533-0.0174533 c)]

FindFit



yData = {0.344027, 0.679511, 0.436543, 0.324024, 0.304968, 0.676049};
xData = {-1.3439, 0.20944, -0.733038, 1.18682, -1.5708, -0.174533};
data = {xData, yData, Table[0, {6}]} // Transpose;
model = y - (0.5 (a + b) - 0.5 (a - b) Cos[2 (x - 0.0174533 c)]);
fit = FindFit[data, model, {a, b, c}, {x, y}]
model /. fit /. {y -> yData, x -> xData}
Show[Plot3D[model /. fit, {x, -2, 2}, {y, 0, 1},
  AxesLabel -> Automatic,
  MeshShading -> {{None, None}, {None, None}}], 
 ListPointPlot3D[data, PlotStyle -> Directive[PointSize[Medium], Red]],

 ListPlot3D[data, VertexColors -> Hue]]

Answer



ClearAll[a, b, c]
NMinimize[
 #.# &[Subtract @@@ {
    0.344027 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (-1.3439 - 0.0174533 c)],
    0.679511 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (0.20944 - 0.0174533 c)],
    0.436543 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (-0.733038 - 0.0174533 c)],
    0.324024 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (1.18682 - 0.0174533 c)],
    0.304968 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (-1.5708 - 0.0174533 c)],

    0.676049 == 0.5 (a + b) - 0.5 (a - b) Cos[2 (-0.174533 - 0.0174533 c)]
    }], {a, b, c}]


{0.0059057, {a -> 0.291325, b -> 0.677473, c -> 3.2186}}

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