rombergs method in matlab

Here is an implementation of Romberg's method for numerical integration in MATLAB:

```main.m```function I = romberg(f, a, b, n)

% Romberg's method for numerical integration
% f: function to integrate
% a, b: limits of integration
% n: number of iterations (at least 2)

h = (b-a)./2.^(0:n-1);

% Compute the first column of the Romberg tableau
R = zeros(n);
R(1,1) = (b-a)*(f(a)+f(b))/2;

% Compute the rest of the tableau
for j = 2:n
% Compute the terms for this column
subtotal = 0;
for i = 1:2^(j-2)
subtotal = subtotal + f(a + (2*i-1)*h(j));
end
R(j,1) = 1/2 * (R(j-1,1) + h(j-1)*subtotal);

% Compute the rest of the column using Richardson extrapolation
for k = 2:j
R(j,k) = R(j,k-1) + (R(j,k-1) - R(j-1,k-1))/(4^(k-1) - 1);
end
end

% Return the most accurate estimate
I = R(n,n);

end
```751 chars33 lines```

To use this function, simply pass in the function you want to integrate, the limits of integration, and the desired number of iterations. For example, to integrate `sin(x)` from `0` to `pi` using 4 iterations:

```main.m```f = @(x) sin(x);
a = 0;
b = pi;
n = 4;
I = romberg(f, a, b, n);
```64 chars6 lines```

This should give you an estimate of `2.0000`, which is the exact answer in this case.

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