Write a MATLAB function, called fixed_point_iteration that inputs a function

Fixed point iteration function solutions submitted (max: Unlimited) Write a MATLAB function, called fixed_point_iteration that inputs a function, g, an initial guess Xo, an error tolerance, tol, and a maximum number of iterations, N, and outputs the fixed point of g, obtained using the fixed point iteration, starting with Xo. Your function should have an error defined by E = (x2 – Xn-1), and stop when the error is less than the tolerance, or if the number of iterations exceeds N – whichever happens first. Your function header should look something like:

The Correct Answer and Explanation is :

Here is the MATLAB function for performing fixed point iteration based on the problem description:

function [c, n, err] = fixed_point_iteration(g, xe, tol, N)
    % Fixed point iteration method to find the root of g(x) = x
    % INPUTS:
    %   g   - the function g(x)
    %   xe  - initial guess
    %   tol - error tolerance
    %   N   - maximum number of iterations
    % OUTPUTS:
    %   c   - the computed fixed point
    %   n   - the number of iterations made
    %   err - the final error

    % Initialize values
    x0 = xe; % initial guess
    n = 0;    % iteration counter
    err = Inf; % initialize error with a large value

    % Start the iteration loop
    while n < N
        % Compute the next approximation using the iteration g(x)
        x1 = g(x0);

        % Compute the error
        err = abs(x1 - x0);

        % Check if the error is below the tolerance
        if err < tol
            break; % exit the loop if tolerance is satisfied
        end

        % Update x0 for the next iteration
        x0 = x1;

        % Increment iteration counter
        n = n + 1;
    end

    % Output the result
    c = x1; % final value is the computed fixed point
end

Explanation:

The fixed_point_iteration function is used to find the fixed point of a given function ( g(x) ). It starts with an initial guess, iteratively updates the guess using the fixed-point formula, and continues until either the error becomes smaller than a specified tolerance or the maximum number of iterations is reached.

Inputs:

g: The function for which we’re solving ( g(x) = x ).
xe: The initial guess ( x_0 ).
tol: The error tolerance. When the error between successive guesses is less than this value, the iteration stops.
N: The maximum number of iterations.

Outputs:

c: The estimated fixed point (root of ( g(x) = x )).
n: The number of iterations performed.
err: The final error after the last iteration.

Process:

The function initializes the first guess and enters a loop to perform the fixed-point iteration.
For each iteration, the new approximation is computed using ( x_1 = g(x_0) ), and the error ( \text{err} = |x_1 – x_0| ) is calculated.
The loop continues until the error is smaller than the tolerance, or the maximum number of iterations is reached.

The plot above illustrates how the method works: it finds the point where the function ( g(x) ) intersects the line ( y = x ), which represents the fixed point.

Explanation:

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