calculating the error of integral using the repeated simpsons rule in matlab

Here's a MATLAB function that implements the repeated Simpson's rule for approximating the integral of a function and also calculates the error estimate:

main.m
function [I, err] = repeated_simpson(f, a, b, n)
    % Inputs:
    %   f: the function to integrate, as a function handle
    %   a, b: the limits of integration
    %   n: the number of intervals to use
    % Outputs:
    %   I: the approximate value of the integral
    %   err: the estimated error of the approximation

    if mod(n, 2) == 1
        error('Error: n must be an even integer');
    end

    h = (b - a) / n;
    x = linspace(a, b, n+1);
    y = f(x);

    I = h/3 * (y(1) + y(end) + 4*sum(y(2:2:end-1)) + 2*sum(y(3:2:end-2)));
    
    % calculate the error estimate
    err = -h^4*(b-a)/180 * max(abs(f''''(x)));

end
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Here, f is the function to integrate, a and b are the limits of integration, and n is the number of intervals to use (which must be even). The function uses the formula for Simpson's rule to approximate the integral over each pair of adjacent intervals, and then sums up the results. To calculate the error estimate, we use the estimate from the error term for Simpson's rule, which is based on the fourth derivative of the function being integrated. We approximate this derivative by taking the maximum absolute value of the fourth derivative over the interval of integration. Note that this implementation assumes that the fourth derivative exists and is continuous over the interval of integration.