迭代法

Question

牛顿函数现在起作用了，我想展示网格中的哪些初始点产生收敛到-1的牛顿迭代，收敛到（1 (3)^1/2)/2我，鉴于：

f（x）=x^31

newton[x0_] := (
  x = x0;
  a1 = {};
  b1 = {};
  c1 = {};
  counter = 0;
  error  = Abs[f[x]];
  While[counter < 20 && error > 0.0001,
   If[f'[x] != 0, x = x - N[f[x]/f'[x]]];
   counter = counter + 1;
   error = Abs[f[x]]];
  x)

我创建了一个网格来显示bi的哪些初始点收敛到根。

grid = Table[a + b I, {a, -2, 2, 0.01}, {b, -2, 2, 0.01}];

Answer 1

委婉地说，你的代码不是最优的，但是为了让你领先，你为什么不从这样的东西开始：

f[x_] := x^3 + 1;

newton[x0_] := (x = x0;
  a1 = {};
  b1 = {};
  c1 = {};
  counter = 0;
  error = Abs[f[x]];
  While[counter < 20 && error > 0.0001, 
   If[f'[x] != 0, x = x - N[f[x]/f'[x]]];
   counter = counter + 1;
   error = Abs[f[x]]];
  {x, counter})

Table[Re@newton[a + b I], {a, -2, 2, 0.01}, {b, -2, 2, 0.01}] // Image