modified regula falsi in matlab

The Regula Falsi method is a numerical method used to find the roots of a given function. The method works by drawing a straight line through the two end points of the interval containing the root and finding the intersection point of this line with the x-axis. This intersection point becomes the new estimate for the root by replacing the endpoint of the interval that has the same sign. However, if the line intersects the x-axis outside the interval then we take the intersection point of the line with the interval endpoint which is closer to the intersection point.

The Regula Falsi method can be modified in Matlab to increase its efficiency. One such modification is to use a convergence criterion to stop the algorithm when the desired level of accuracy is reached. This can be achieved by defining a tolerance value and stopping the algorithm when the error between the approximated root and the true root is within this tolerance value.

Here's an implementation of the Regula Falsi method in Matlab with the convergence criterion:

main.m
function root = regula_falsi(f, a, b, tol)
% f is the function for which we want to find the root
% a, b are the end points of the interval containing the root
% tol is the desired tolerance value

% Evaluate the function at the end points
fa = f(a);
fb = f(b);

% Check if the function has the same sign at both end points
if fa * fb > 0
    error('Function has the same sign at both end points');
end

while true
    % Find the slope of the line joining (a, fa) and (b, fb)
    slope = (fb - fa) / (b - a);
    
    % Find the x-intercept of the line
    c = b - fb / slope;
    fc = f(c);
    
    % Stop if the approximated root is within the tolerance
    if abs(fc) < tol
        root = c;
        break
    end
    
    % Replace the interval endpoint that has the same sign with fc
    if fa * fc > 0
        a = c;
        fa = fc;
    else
        b = c;
        fb = fc;
    end
end
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Note that this implementation assumes that the function has a root in the given interval and may not converge for some functions. It is also recommended to limit the number of iterations to prevent the algorithm from running indefinitely.