Category: Numerical Analysis

Professor: We’re going to talk about Taylor polynomials this quarter.
Student: Shit.
Professor: What was that? I thought I heard yippee!

If you ever take the inverse of a matrix and I find out, I will kill you.

If you ever take the inverse of a matrix and I find out, I will kill you.

If you all make me really moody, I will give you a 5×5 determinant to do by hand for the final – that’s 120 terms.

We cannot analyze ideas. We only analyze algorithms here.

“If you’ve seen The Matrix, you’ll understand binary.”

Numerical analysis professor

hyrodium:

Durand-Kerner method is 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]]]]