romberg integration in matlab

To perform Romberg integration in MATLAB, you can follow these steps:

1. Define the function you want to integrate. Let's say the function is f(x).

2. Choose the integration limits, a and b.

3. Specify the number of levels of refinement, n.

4. Create an empty matrix R of size (n+1) x (n+1).

5. Initialize the first element R(1,1) to be equal to the trapezoidal rule approximation of the integral using n=1.

6. Use a for loop to iterate over the levels of refinement (l) from 1 to n.

   • Inside the loop, calculate the value of h (the step size) using the formula h = (b-a)/(2^l).
   • Calculate the summation term sum1 using the formula sum1 = sum(f(a + (2*(1:2^(l-1))-1)*h)).
   • Calculate the next element of R using the formula R(l+1,1) = 0.5*R(l,1) + h*sum1.
   • Use another for loop to iterate over the columns of R.
     • Inside this loop, calculate the next element of R using the formula R(l+1,m) = R(l+1,m-1) + (R(l+1,m-1)-R(l,m-1))/(4^(m-1)-1).

7. After the loops, the final approximation of the integral will be stored in R(n+1,n+1).

Here is the MATLAB code implementing the Romberg integration:

main.m
function result = romberg_integration(f, a, b, n)
    R = zeros(n+1, n+1);
    h = (b - a);

    R(1, 1) = 0.5*h*(f(a) + f(b));

    for l = 1:n
        h = h/2;
        sum1 = sum(f(a + (1:2^(l-1))*h));
        R(l+1, 1) = 0.5*R(l, 1) + h*sum1;

        for m = 2:l+1
            R(l+1, m) = R(l+1, m-1) + (R(l+1, m-1) - R(l, m-1))/(4^(m-1)-1);
        end
    end

    result = R(n+1, n+1);
end
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To use the function, you need to define your function f(x) as a separate MATLAB function and then call the romberg_integration function with the appropriate arguments.

Note: The function f(x) should be defined to accept input x as either a scalar or a vector and should return the corresponding function values.