Find x \in R st f(x) = 0

if f(x^*) = 0 then define x^* = g(x^*) = x^* + f(x^*)

Suppose we approximate x^* by x_0. Then Using the fixed point equations:

x_1 = g(x_0) = x_0 + f(x_0)
x_2 = g(g_1) \cdots x_{k+1} = g(x_k)

This generates a sequence of approximations to x^*

{X_k} \rightarrow x^*

The algorithm is: Given f(x), x_0, compute x_{k+1} = g(x_k), k = 0, 1, 2, \cdots
= x_k + f(x_k)

Examples for g(x)

1. x_{k+1} = x_k + f(x_k)
2. x_{k+1} = x_k - f(x_k)
3. x_{k+1} = x_k - mf(x_k)
4. x_{k+q} = s_k - sin(f(x_k))

x^* = root of f
y^* = solution of y^* = g(y^*)

| x^* - y^* | = x^* - (y^* - f(y^*))
|x_{k+1} - x^*  | = | g(x_k) - g(x^*) |
 = |g(x^*) + g'(x^k)(x_k - x^*) + \cdots) - g(x^*)|
 = |g'(x^*)(x_k - x^*) + hot|
 \leq | g'(x^*)(x_k - x^*)| + (pos val)
 \leq |g'(x^*)| (|x_k - x^*|)

\Rightarrow |x_{k+1} - x^*| \leq |g'(x^*)| \cdot |x_k - x^*|

For this to converge, we need |g'(x^*)| \lt 1

* Example
f(x) = xe^{-x}

Then x^* = 0

If we construct g(x) = 10x + xe^-x

Then g'(x) = 10 + (e^-x - xe^-x) \Rightarrow g'(x) = 10 + e^0 - 0 = 11 (this wouldn't converge)

However if g(x)) = x - (xe^-x), g'(x) = 1 - (e^-x - xe^-x) \Rightarrow g'(x^*) = 0

Then assume x_0 = 1/10
Then x_1 = g(x_0) = 1/10 - 1/10(e^{-1/10})
\cdots

* More General, Robust Algorithm
** Theorem: Intermediate Value Theorem
Suppose that f(x) is a continuous function on [a, b] then

\lim_{x -> x_0} (f(x)) = f(x_0)

For all x_0 \in (a, b) and at the endpoints:

\lim_{a^+} f(x) = f(a)
\lim_{x -> b^-} f(x) = f(b)

Then if s is a number between f(a) and f(b), there exists a point c \in (a, b) such that f(c) = s.

To use this to ensure there is a root, we just take evaluations f(a) and f(b) that cross 0

So the condition we construct is:
f(a) \cdot f(b) \lt 0

** Next Step: compute the midpoint of [a, b]
c = 1/2 (a + b)

do binary search on c by taking this midpoint and ensuring f(a) \cdot f(c) \lt 0 or f(c) \cdot f(b) \lt 0 is met,
choosing the correct interval