Newton's method with no real roots

Question

So as the title would suggest I'm currently reading about Newton's method for finding roots. I'm having trouble understanding the reasoning for a function without a root.

It reads as following:

"Consider the function $f(x) =1+x^2$. Clearly f has no real roots though it does have complexroots $x\pm i$.The Newton method formula for f is:

$x_{n+1} = x_{n} - \frac{1+x^2}{2x_{n}}=\frac{x^2-1}{2x_{n}}$"

What is happening here?

Many thanks to whomever might expand this a little for me!

Answer 1

You have to choose complex starting values, otherwise the method cannot converge to complex roots.

With the correct iteration formula $$x_{n+1}=x_n - \frac{f(x)}{f'(x_n)} = x_n - \frac{x_n^2+1}{2x_n} = \frac{2x_n^2-x_n^2 -1}{2x_ n}=\frac{x_n^2 - 1}{2x_n}$$ and a complex starting value you get e.g.

  1.0              + 1.0 i
  0.2500000000     + 0.7500000000 i
 -0.07500000000    + 0.9750000000 i
  0.001715686274   + 0.9973039215 i
 -0.46418462831e-5 + 1.000002160 i
 -0.1002647834e-19 + 1.000000000 i
  0.0              + 1.000000000 i

and for the other root

  3.0              - 1.0 i
  1.350000000      - 0.5500000000 i
  0.3573529412     - 0.4044117647 i
 -0.4348049736     - 0.8964750065 i
  0.001593678319   - 0.8997608310 i
 -0.0001874328610  - 1.005581902 i
 -0.1037539915e-5  - 1.000015475 i
 -0.1605575154e-10 - 1.000000000 i
  0.0              - 1.000000000 i

Answer 2

There's a typo in your displayed equation. The $x$ in the numerator should be $x_n$. Then one just writes

$$x_{n+1} = x_n -\frac{1+x_n^2}{2x_n} = x_n\frac{2x_n}{2x_n} - \frac{1+x_n^2}{2x_n} =\frac{2x_n^2-x_n^2-1}{2x_n}. $$

Answer 3

In some cases, the Newton's Method does not work. For example, if you encounter a stationary point in the process (division by zero).

Your example is another one in which Newton's Method does not work, because starting with a real number you only go through real numbers in the process. BUT you could start with a complex number, and (with a bit of luck) you will converge to the correct complex root.

Answer 4

Newton's Method works only if the sequence of iterates converges. This is not always the case.

For example if you choose $f(x)=\sqrt[3] x$ and try Newton's Method to find the root $x=0$

We get $$ x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^{1/3}}{(1/3) x_n^{-2/3}}=-2x_n $$ The sequence of iterates starting at $x=1$ is $$1,-2,4,-8,...$$ which does not converge to $x=0$