complex-newtons-method.nb

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This method is taught in basic Calculus, and it is mostly of interest because \
it is (a) simple to define and understand, (b) easy to visualize, and (c) \
good enough to still be used for scientific computing, despite being over 300 \
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Many students will be surprised to know that this method does not just work \
for real-valued functions: it works for complex-valued functions, as well. \
Its behavior in the complex plane can be a bit more complicated to describe, \
and depending on the initial guess, the method may not necessarily converget \
to the nearest root.\
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    FrameLabel -> {None, None}, FrameTicks -> {{None, None}, {None, None}}, 
    GridLinesStyle -> Directive[
      GrayLevel[0.5, 0.4]], 
    Method -> {
     "DefaultBoundaryStyle" -> Automatic, "DefaultPlotStyle" -> Automatic}],
   FormBox[
    FormBox[
     TemplateBox[{"\"Divergent\"", 
       RowBox[{"-", "1"}], 
       RowBox[{
         RowBox[{"-", 
           RowBox[{"0.5`"}]}], "-", 
         RowBox[{"0.5`", " ", "\[ImaginaryI]"}]}], 
       RowBox[{
         RowBox[{"-", 
           RowBox[{"0.5`"}]}], "+", 
         RowBox[{"0.5`", " ", "\[ImaginaryI]"}]}]}, "SwatchLegend", 
      DisplayFunction -> (FormBox[
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         StyleBox[
          PaneBox[
           TagBox[
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               TagBox[
                GridBox[{{
                   GraphicsBox[{
                    Directive[
                    EdgeForm[
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                    GrayLevel[0]]], 
                    PointSize[0.5], 
                    AbsoluteThickness[1.6], 
                    RGBColor[0., 0., 0.]], 
                    RectangleBox[{0, 0}, {10, 10}, "RoundingRadius" -> 0]}, 
                    AspectRatio -> Full, ImageSize -> {10, 10}, 
                    PlotRangePadding -> None, ImagePadding -> Automatic, 
                    BaselinePosition -> (Scaled[0.1] -> Baseline)], #}, {
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                    GrayLevel[0]]], 
                    PointSize[0.5], 
                    AbsoluteThickness[1.6], 
                    RGBColor[1., 0., 0.]], 
                    RectangleBox[{0, 0}, {10, 10}, "RoundingRadius" -> 0]}, 
                    AspectRatio -> Full, ImageSize -> {10, 10}, 
                    PlotRangePadding -> None, ImagePadding -> Automatic, 
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                    GrayLevel[0]]], 
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                    AbsoluteThickness[1.6], 
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                    PlotRangePadding -> None, ImagePadding -> Automatic, 
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                    AspectRatio -> Full, ImageSize -> {10, 10}, 
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                 GridBoxAlignment -> {
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                 GridBoxDividers -> {
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                    FrontEnd`Private`$ColorSelectorInitialAlpha = 1; 
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                    FrontEnd`Private`$ColorSelectorUseMakeBoxes = True; 
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