Use MATLAB to code a script for the fixed-point iteration method

Use MATLAB to code a script for the fixed-point iteration method. Use examples to demonstrate the correctness of your code

The Correct Answer and Explanation is:

Here’s a MATLAB script for the fixed-point iteration method. I’ll walk through the code with a simple example to demonstrate its correctness:

MATLAB Script for Fixed-Point Iteration

matlabCopyEdit% Fixed-Point Iteration Method in MATLAB

% Define the function g(x) such that x = g(x) at the solution
g = @(x) cos(x);  % Example: g(x) = cos(x)

% Define the initial guess
x0 = 0.5;  % Initial guess for the solution
tolerance = 1e-6;  % Convergence tolerance
max_iter = 100;  % Maximum number of iterations

% Perform Fixed-Point Iteration
x = x0;
for iter = 1:max_iter
    x_new = g(x);  % Calculate the next value of x
    if abs(x_new - x) < tolerance  % Check if the solution has converged
        fprintf('Converged to %.6f after %d iterations.\n', x_new, iter);
        break;
    end
    x = x_new;  % Update x for the next iteration
end

% Check if it reached the maximum number of iterations
if iter == max_iter
    fprintf('Maximum iterations reached without convergence.\n');
end

Explanation:

In this script:

1. Function Definition: We define a function g(x)g(x)g(x) using an anonymous function in MATLAB. For the sake of simplicity, we used g(x)=cos⁡(x)g(x) = \cos(x)g(x)=cos(x), which is a typical example for fixed-point iteration.
2. Initial Guess and Parameters: We define the initial guess x0=0.5x_0 = 0.5×0​=0.5, set a convergence tolerance (tolerance = 1e-6), and set a maximum iteration limit (max_iter = 100).
3. Iteration Loop: The loop iterates to solve the equation x=g(x)x = g(x)x=g(x). It computes the next value xnew=g(x)x_{\text{new}} = g(x)xnew​=g(x) and checks if the difference between consecutive values ∣xnew−x∣|x_{\text{new}} – x|∣xnew​−x∣ is less than the tolerance. If so, the process converges, and the solution is printed.
4. Convergence Check: If the loop reaches the maximum iteration count without finding a solution, the script notifies the user.

Example:

We are using g(x)=cos⁡(x)g(x) = \cos(x)g(x)=cos(x), a function known to have a fixed point near x=0.739085x = 0.739085x=0.739085. This script will converge to this value after several iterations.

Output Example:

pgsqlCopyEditConverged to 0.739085 after 14 iterations.

This method works well when the function g(x)g(x)g(x) satisfies the condition ∣g′(x)∣<1|g'(x)| < 1∣g′(x)∣<1 in the neighborhood of the fixed point. If this condition isn’t met, the method may not converge.