Romberg

Romberg

% Function for Romberg integration method

% Computes the integral of a given function using Romberg's method

% Renamed function: romberg_integration

function [R, quad, e, hr] = romberg_integration(func, a, b, n, toler)

mr = 1;

hr = b - a;

e = 1;

j = 0;

R = zeros(4, 4);

R(1, 1) = h * (feval(func, a) + feval(func, b)) / 2;

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while ((e > toler) && (j < n)) || (j < 4)

j = j + 1;

hr = hr / 2;

sum = 0;

for p = 1:m

w = a + hr * (2 * p - 1);

sum= sum + feval(func, x);

end

R(j + 1, 1) = R(j, 1) / 2 + hr * sum;

mr = 2 * mr;

for k = 1:j

R(j + 1, k + 1) = R(j + 1, k) + (R(j + 1, k) - R(j, k)) / (4^k - 1);

end

e = abs(R(j, j) - R(j + 1, k + 1));

end

quad = R(j + 1, j + 1);

end

% Comments added for better code readability and understanding:

% Function: romberg_integration

% Inputs:

% - func: the function to integrate

% - a: lower integration limit

% - b: upper integration limit

% - n: maximum number of iterations

% - toler: tolerance for convergence

% Outputs:

% - R: matrix containing the Romberg integration table

% - quad: final estimate of the integral

% - e: error estimate of the integral

% - hr: step size

% The function implements Romberg's method for numerical integration.

% Romberg's method improves upon the trapezoidal rule by successively

% refining the estimate of the integral using Richardson extrapolation.

% It is an iterative process that doubles the number of function evaluations

% in each iteration until the desired tolerance is reached or the maximum

% number of iterations is reached.