Fixed Point Method

The number $p$ is a fixed point of the function $f$ iff $f(p) = p$.

Existence and uniqueness

If $g \in C[a,b]$ and $g(x) \in [a,b]\;\forall x \in [a,b]$, then $g$ has at least one fixed point in $[a,b]$.

If in addition, $g'(x)$ exists on $(a,b)$ and $\exists k\ \in (0,1)$ such that:

Then $g$ has a unique fixed point in $(a,b)$.

Iteration algorithm

Start with $p_0$ (arbitrary point). Iterate using $p_{n+1} = g(p_n)$ until convergence. Convergence is guaranteed if the fixed point exists and is unique.

Implementation

def fixed_point(g, p0, tolerance=1e-6, max_iteration_count=100):
    p = p0
    for _ in range(max_iteration_count):
        p_new = g(p)
        if abs(p_new - p) < tolerance:
            return p_new
        p = p_new
    raise ValueError("Fixed point not found")

Fixed point theorem

If:

• $g \in C[a,b]$
• $\forall x \in [a,b],\; g(x) \in [a,b]$
• $g'$ exists on $(a,b)$
• $\exists k \in (0,1)$ and $\big\lvert g'(x) \big\rvert \le k\; \forall x \in (a,b)$

Then both equations mentioned below hold: