Find if a fixed-point iteration converges for a certain root

Question

I'm asked to find if the fixed-point iteration

$$x_{k+1} = g(x_k)$$

converges for the fixed points of the function $$g(x) = x^2 + \frac{3}{16}$$ which I found to be $\frac{1}{4}$ and $\frac{3}{4}$.

In this short video by Wen Shen,

https://www.youtube.com/watch?v=3_PKDDnxgCo

it's explained how to find these fixed-points and to see if a fixed-point iteration converges. My doubt is related to find if a fixed point iteration converges for a certain fixed point.

At more or less half of the video, she comes up with the following relation for the error

$$e_{k+1} = |g'(\alpha)| e_k$$

where $\alpha \in (x_k, r)$, by the mean value theorem, and because $g$ is continuous and differentiable.

If $|g'(\alpha)| < 1$, then the fixed-point iteration converges.

I think I agree with this last statement, but when she tries to see if the fixed point iteration converges for a certain root of a certain function, she simply finds the derivative of that function and plugs in it the root.

I don't understand why this is equivalent to $$e_{k+1} = |g'(\alpha)| e_k$$

Can someone fire up some light on my brain? (lol, can I say this?)

Answer 1

The point behind the derivative is that locally a function looks linear and the derivative is the multiplier. Symbolically: $$f(x) \approx f(x_0) + f'(x_0)(x-x_0)$$ for $x$ near $x_0$. The statement "looks like" can be made more precise with the Mean Value Theorem.

Now, try iterating a function of the form $g(x) = a + m(x-a)$. You'll find that it converges to its fixed point of $x=a$ when $|m|<1$.

We can prove this since $$|g(x)-a| = |a+m(x-a) - a| = |m|\cdot |x-a|$$ so that $$|g^n(x)-a| = |m|^n |x-a|.$$

This is why we say the fixed point $x_0$ is attractive if $|f'(x_0)|<1$ and repulsive if $|f'(x_0)|>1$.

Answer 2

We can do this in a different way using the Contraction Mapping Theorem.

$$d(g(x), g(y)) = d(x^2, y^2) = d(x, y) \times |x+y|$$ so if we can ensure that there is $r$ such that $|x + y| \leq r < 1$, and if we can ensure that $g$ applied to $[-r, r]$ produces a set which lies within $[-r, r]$, then in fact $g$ is a contraction mapping on $[-r, r]$.

Therefore, as long as there's a fixed point of $g$ inside $[-r, r]$ with $0 < r < \frac{1}{2}$, the iteration will converge when started anywhere in $[-r, r]$ to that (necessarily unique) fixed point. (Indeed, $g$ restricts to a function $[-r, r] \to [-r, r]$ if $\frac{1}{4}$ lies in the interval.)

So starting the iteration off from any $x \in (-\frac{1}{2}, \frac{1}{2})$ will produce something which converges to $\frac{1}{4}$. (We say nothing about how the iteration behaves from other starting points.)