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Durand-Kerner methodis a root-finding algorithm for solving polynomial equations.The red points are the solution of P(z)=0, and the blue points is a initial values z_k^(0).

Here’s mathematica code, and you can try this code on Wolfram Programming Lab for free! (https://www.wolfram.com/programming-lab/):

m = 20;

pts2[pts_,

n_] := {(Complex @@@ pts)[[1 ;; n]], (Complex @@@ pts)[[-n ;; -1]]};

CompList[lz_] := Transpose@{Re[#], Im[#]} &@lz;

DKA[root_, lz0_] := (

Clear[f, rec, lz];

lz[0] = lz0;

f[z_] = Times @@ (z – #) &@root;

rec[lz_, n_] :=

lz[[n]] – f[lz[[n]]]/Times @@ Delete[lz[[n]] – lz, n];

lz[i_] := lz[i] = Table[rec[lz[i – 1], n], {n, Length[lz0]}];

p = CompList /@ lz /@ Range[0, m];

Graphics[{Point[Flatten[p, 1]], Line[Transpose[p]],

PointSize[0.02], Blue, Point[CompList@lz0], Red,

Point[CompList@root]}, PlotRange -> 3]);

Abeath[pts_, n_] :=

Table[Mean /@ Transpose[pts] +

2 {Cos[2 \[Pi] i/n], Sin[2 \[Pi] i/n]}, {i, n}];

Manipulate[pts = PadRight[pts, 2 n, RandomReal[{-1, 1}, {2 n, 2}]];

DKA @@ pts2[pts, n], {n, 1, 10, 1}, {{pts, {{0, 0}}}, Locator,

Appearance -> None},

Button[“Aberth”,

pts = Join[pts[[1 ;; n]], Abeath[pts[[1 ;; n]], n]]]]