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I want to draw the error of the two iterative methods.

ListPlot@Block[{$MinPrecision = 50, $MaxPrecision = 50}, 
  Log[Table[
    Abs[Nest[Power[# + 1, (3)^-1] &, 1., i] - 
      Root[#^3 - # - 1 &, 1]], {i, 1, 10, 1}]]]
ListPlot@Block[{$MinPrecision = 50, $MaxPrecision = 50}, 
  Log[Table[
    Abs[Nest[(2 #^3 + 1)/(3 #^2 - 1) &, 1., i] - 
      Root[#^3 - # - 1 &, 1]], {i, 1, 10, 1}]]]

However, the accuracy of the second iteration method is not enough to draw the parabola completely.

enter image description here

What can I do to improve the accuracy of drawing and get the correct scatter plot?

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  1. Don't start with machine precision.

  2. Increase precision.

.

ListPlot@Block[{$MinPrecision = 1000, $MaxPrecision = 1000}, 
  Log[Table[
    Abs[Nest[(2 #^3 + 1)/(3 #^2 - 1) &, 1.`1000, i] - 
      Root[#^3 - # - 1 &, 1]], {i, 1, 10, 1}]]]
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