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 how you can implement the Fixed-Point Iteration method in MATLAB:

MATLAB Code

matlabCopyEdit% Define the fixed-point function
g = @(x) cos(x);  % Example: g(x) = cos(x)

% Fixed-point iteration method
maxIter = 100;  % Maximum number of iterations
tolerance = 1e-6;  % Convergence tolerance
x0 = 0.5;  % Initial guess

% Initialize variables
x = x0;

% Fixed-point iteration loop
for k = 1:maxIter
    x_new = g(x);  % Compute the next approximation
    if abs(x_new - x) < tolerance  % Check for convergence
        fprintf('Converged to %.6f after %d iterations.\n', x_new, k);
        break;
    end
    x = x_new;  % Update the current approximation
end

% If the loop completes without convergence
if k == maxIter
    fprintf('Did not converge after %d iterations.\n', maxIter);
end

Explanation

1. Define the Fixed-Point Function:
   The function g(x) represents the function that we use for fixed-point iteration. In this case, we choose the example g(x) = cos(x).
2. Set Parameters:
   The initial guess x0 is set to 0.5. The iteration will continue until either the maximum number of iterations (maxIter) is reached or the absolute difference between consecutive approximations is smaller than the tolerance (tolerance).
3. Iterative Process:
   We compute the next approximation x_new using the function g(x). If the absolute difference between x_new and x (the current approximation) is smaller than the tolerance, the process is considered to have converged, and the loop terminates.
4. Convergence:
   If the algorithm converges, a message with the final value and the number of iterations is displayed. If it doesn’t converge within the maximum iterations, an error message is printed.

Example

Let’s use the function g(x) = cos(x) and an initial guess of x0 = 0.5. The expected outcome is that the function will converge to the solution x ≈ 0.739085, which is the fixed point of the cosine function.

Analysis

The fixed-point iteration method is useful for solving equations of the form x = g(x). For it to converge, the function g(x) must satisfy certain conditions, such as the derivative of g(x) being less than 1 in magnitude at the fixed point.

In this case, since |g'(x)| < 1 near the fixed point, the method will converge. The convergence is guaranteed if these conditions hold, and the accuracy can be adjusted using the tolerance value.