calculate error by using repeated simpsons rule in matlab

Repeated Simpson's rule can be used to approximate definite integrals. The error of this approximation can also be calculated.

To calculate the error using repeated Simpson's rule in MATLAB, you can use the following code:

main.m
syms x;
f = <function>; % Define the function to integrate
a = <lower limit>; % Define the lower limit of integration
b = <upper limit>; % Define the upper limit of integration
n = <number of intervals>; % Define the number of intervals
h = (b-a)/(2*n);

% Calculate the integral using repeated Simpson's rule
sumOdd = 0;
for i = 1:n
    sumOdd = sumOdd + subs(f, x, a+(2*i-1)*h);
end
sumEven = 0;
for i = 1:n-1
    sumEven = sumEven + subs(f, x, a+2*i*h);
end
integral = h/3*(subs(f, x, a) + 4*sumOdd + 2*sumEven + subs(f, x, b));

% Calculate the error
diff4 = diff(diff(diff(diff(f))));
error = abs(((b-a)/180)*h^4*subs(diff4, x, (a+b)/2));

fprintf('Integral: %f\n', integral);
fprintf('Error: %f\n', error);
713 chars
25 lines

Replace <function>, <lower limit>, <upper limit>, and <number of intervals> with their respective values.

The code defines syms x to create a symbolic variable x that will be used for differentiation. It then calculates the integral using repeated Simpson's rule and stores the result in the variable integral. Finally, it calculates the error using the formula and stores the result in the variable error.

Note that the error calculation assumes that the fourth derivative of the function is continuous on the interval of integration.