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Using NSolveValues on certain equations returns lists of solutions of the following kind:
enter image description here

For each of these rules I want to select only one value, namely the one absolutely closest to $2\pi/10$, but without selecting this manually. How would I find this?

Edit:
I have many of these equations which depend on a lot of different parameters, so I won't put all of that here. But the point is that I want to solve a large number of equations iteratively, where each equation depends on the solution of the previous one. However each equation results in a list ofinfinite solutions as above, but I only want to use the solution that is absolutely nearest to $2\pi/10$.

f[x_]:=ArcCos[0.5 (-0.00756467 + 
     0.749491 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
     0.503279 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) + 
     0.538571 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
     0.430085 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
     0.0857465 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) - 
     0.838206 (-0.423397 Cos[45. x] + 
        0.841604 Sin[45. x]))] ((0.566265 (0. + 
        0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
        0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
        0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
        0.538571 (0.227498 Cos[45. x] + 
           0.457042 Sin[45. x])))/(\[Sqrt](Abs[
         0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
          0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
          0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
        Abs[0.112051 + 
          0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
          0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
          0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
        Abs[0.050167 - 
          0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
          0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
          0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
          0.538571 (-0.423397 Cos[45. x] + 
             0.841604 Sin[45. x])]^2)) + (0.336803 (0.112051 + 
        0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
        0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
        0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
        0.0857465 (-0.423397 Cos[45. x] + 
           0.841604 Sin[45. x])))/(\[Sqrt](Abs[
         0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
          0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
          0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
        Abs[
         0.112051 + 
          0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
          0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
          0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
        Abs[0.050167 - 
          0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
          0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
          0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
          0.538571 (-0.423397 Cos[45. x] + 
             0.841604 Sin[45. x])]^2)) - (0.752268 (0.050167 - 
        0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
        0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
        0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
        0.538571 (-0.423397 Cos[45. x] + 
           0.841604 Sin[45. x])))/(\[Sqrt](Abs[
         0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
          0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
          0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
        Abs[0.112051 + 
          0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
          0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
          0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
          0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
        Abs[0.050167 - 
          0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
          0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
          0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
          0.538571 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2)))

NSolveValues[f[x] == 2 \[Pi]/10, x, Reals]
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  • $\begingroup$ Can you put the code that produced this result? $\endgroup$
    – polfosol
    Dec 9, 2021 at 9:57
  • $\begingroup$ @polfosol Thank you for your comment, please see my edit. $\endgroup$
    – Michael Henchard
    Dec 9, 2021 at 10:04
  • $\begingroup$ Also I was thinking about using something like NearestTo, but that doesn't work for the kind of object that I have from NSolveValues. $\endgroup$
    – Michael Henchard
    Dec 9, 2021 at 10:08
  • $\begingroup$ What is the meaning of argument 45. x ? $\endgroup$
    – Ulrich Neumann
    Dec 9, 2021 at 10:28
  • 2
    $\begingroup$ You could try to give an additional constrain like: NSolveValues[{f[x] == 2 \[Pi]/10, (x - 2 Pi/10)^2 < .005}, x] $\endgroup$
    – Daniel Huber
    Dec 9, 2021 at 10:49

2 Answers 2

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If I understand you right, you are looking for a solution f[x]==2Pi/10 with constraint (x-2Pi/10)^2 minimal?

Try NMinimize

 mini=NMinimize[{(x - 2 Pi/10)^2, f[x] == 2 Pi/10}, x]
 (*{0.0327761, {x -> 0.80936}}*)
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3
  • $\begingroup$ Thank you for your answer, I tried it however it as you say it give 0.80936, but there is another solution of f[x]==2Pi/10, which is 0.669734. Shouldn't it return that instead? $\endgroup$
    – Michael Henchard
    Dec 9, 2021 at 10:33
  • 1
    $\begingroup$ Interessting, don't know why! If you restrict the solution range mini = NMinimize[{(x - 2 Pi/10)^2, f[x] == 2 Pi/10, 0.6 < x < 1 }, x ] the correct result follows $\endgroup$
    – Ulrich Neumann
    Dec 9, 2021 at 10:53
  • $\begingroup$ Or FindRoot[f[x] == 2 Pi/10, {x, 2 Pi/10}] seems to be more reliable. $\endgroup$
    – Akku14
    Dec 9, 2021 at 15:48
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Since the desired root is near 2Pi/10 use FindRoot with an initial estimate of 2Pi/10

Clear["Global`*"]

f[x_] = ArcCos[
     0.5 (-0.00756467 + 
        0.749491 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
        0.503279 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) + 
        0.538571 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
        0.430085 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
        0.0857465 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) - 
        0.838206 (-0.423397 Cos[45. x] + 
           0.841604 Sin[45. x]))] ((0.566265 (0. + 
           0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
           0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
           0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
           0.538571 (0.227498 Cos[45. x] + 
              0.457042 Sin[45. x])))/(\[Sqrt](Abs[
             0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
              0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
              0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
           Abs[
             0.112051 + 
              0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
              0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
              0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
           Abs[
             0.050167 - 
              0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
              0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
              0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
              0.538571 (-0.423397 Cos[45. x] + 
                 0.841604 Sin[45. x])]^2)) + (0.336803 (0.112051 + 
           0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
           0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
           0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
           0.0857465 (-0.423397 Cos[45. x] + 
              0.841604 Sin[45. x])))/(\[Sqrt](Abs[
             0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
              0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
              0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
           Abs[
             0.112051 + 
              0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
              0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
              0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
           Abs[
             0.050167 - 
              0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
              0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
              0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
              0.538571 (-0.423397 Cos[45. x] + 
                 0.841604 Sin[45. x])]^2)) - (0.752268 (0.050167 - 
           0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
           0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
           0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
           0.538571 (-0.423397 Cos[45. x] + 
              0.841604 Sin[45. x])))/(\[Sqrt](Abs[
             0. + 0.503279 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) - 
              0.749491 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.0857465 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) - 
              0.538571 (0.227498 Cos[45. x] + 0.457042 Sin[45. x])]^2 + 
           Abs[
             0.112051 + 
              0.430085 (0.457042 Cos[45. x] - 0.227498 Sin[45. x]) - 
              0.503279 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) - 
              0.838206 (0.227498 Cos[45. x] + 0.457042 Sin[45. x]) + 
              0.0857465 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2 + 
           Abs[
             0.050167 - 
              0.430085 (0.287778 Cos[45. x] - 0.876915 Sin[45. x]) + 
              0.838206 (0.876915 Cos[45. x] + 0.287778 Sin[45. x]) + 
              0.749491 (0.841604 Cos[45. x] + 0.423397 Sin[45. x]) + 
              0.538571 (-0.423397 Cos[45. x] + 0.841604 Sin[45. x])]^2))) // 
   Rationalize[#, 0] & // Simplify

(* -((ArcCos[(1/
      2000000000000)(-7564670000 + 1615335655046 Cos[45 x] - 
        1179276556501 Sin[45 x])] (-84197000000 + 
       1139633204749573157 Cos[45 x] + 
       1561033413550243039 Sin[45 x]))/(500000 \[Sqrt](Abs[
         224102000000 - 907979420789 Cos[45 x] - 1243722090718 Sin[45 x]]^2 + 
        Abs[790864917931 Cos[45 x] + 1083301448052 Sin[45 x]]^2 + 
        4 Abs[
          50167000000 + 1014011691237 Cos[45 x] + 
           1388960982854 Sin[45 x]]^2))) *)

Solving,

sol = FindRoot[f[x] == 2 Pi/10, {x, 2 Pi/10}]

(* {x -> 0.669734} *)

Graphically,

Plot[f[x] - 2 Pi/10, {x, 0.4, 0.9},
 PlotPoints -> 200,
 MaxRecursion -> 5,
 GridLines -> {{2 Pi/10}, None},
 GridLinesStyle -> Directive[Gray, Thick, Dashed],
 Epilog -> {Red, AbsolutePointSize[4], Point[{x /. sol, 0}]}]

enter image description here

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