Fixed Point Method

The Fixed Point Method is a numerical technique used to find approximate solutions to equations of the form $x = g(x)$ . It iteratively refines an initial guess until convergence is achieved.

Given a function:

f(x) = 0

x = g(x)

How the Fixed Point Method Works

Rearrangement: Rearrange the equation to the form $x = g(x)$ .
Initial Guess: Choose an initial approximation $x_0$ .
Iteration: Use the function to compute:
$x_{n+1} = g(x_n)$
Repeat until convergence.
Convergence Check: Stop when $|x_{n+1} - x_n| < ext{Error margin}$ .

Example

Given Function:

x^2 - 4

Error Margin is 0.01

Step 1: Initial Guess:

x_0 = 1

Step 2: Rearranged Function:

x_0 = 1 \\\\ \text{to} \\\\ g(x_0) = \sqrt{4} = 2

Step 3: First Iteration:

x_1 = g(x_0) = \sqrt{4} = 2

|x_1 - x_0| = |2 - 1| = 1

\text{Since} \ |x_1 - x_0| > 0.01, \text{ continue iterating.}

Step 4: Second Iteration:

x_2 = g(x_1) = \sqrt{4} = 2

|x_2 - x_1| = |2 - 2| = 0

\text{Since} \ |x_2 - x_1| < 0.01, \text{ we have converged.}

Result:

x \approx 2

Conclusion

The Fixed Point Method is an effective approach for solving equations of the form $f(x) = 0$ . In this example, we showed that the solution converges to $x = 2$ . The method guarantees convergence under suitable conditions on the function $g(x)$ .

Fixed-Point (Iteration) Method Solver