Fixed Point Method

The number is a fixed point of the function if .

Existence and uniqueness of a fixed point

If and , then has at least one fixed point in .

If in addition, exists on and and:

Then has a unique fixed point in .

Note

Root finding () and fixed point problems () are equivalent classes if both functions have a linear relationship.

Iteration algorithm

Start with (arbitrary point). Iterate using 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

Suppose:

• exists on
• and

Then

and