in the root finding algorithms we studied, we stopped iterating when the absolute or relative error fell below a prespecified tolerance. briefly explain the difference between these stopping criteria. in matlab

In the root finding algorithms, the stopping criteria determine when to stop iterating and consider the current approximation as the root. The two commonly used stopping criteria are the absolute error and the relative error.

1. Absolute Error: The absolute error is the absolute difference between the current approximation and the previous approximation. The algorithm stops iterating when the absolute error falls below a specified tolerance. Mathematically, if x_i is the current approximation and x_{i-1} is the previous approximation, then the absolute error E_a can be calculated as:

main.m
E_a = |x_i - x_{i-1}|
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2. Relative Error: The relative error is the absolute difference between the current approximation and the previous approximation, divided by the absolute value of the current approximation. The algorithm stops iterating when the relative error falls below a specified tolerance. Mathematically, if x_i is the current approximation and x_{i-1} is the previous approximation, then the relative error E_r can be calculated as:

main.m
E_r = |(x_i - x_{i-1}) / x_i|
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The main difference between the two stopping criteria is that the absolute error considers only the magnitude of the error, while the relative error takes into account the magnitude of the current approximation. This makes the relative error more suitable when the scale of the problem or the variable being approximated varies significantly.

In MATLAB, the stopping criteria can be implemented using conditional statements such as if and while loops. For instance:

main.m
tolerance = 1e-6; % Set desired tolerance
maxIterations = 100; % Set maximum number of iterations
x0 = ...; % Initial approximation

x_prev = x0;
x_curr = ...; % Compute current approximation

iteration = 1;
while abs(x_curr - x_prev) > tolerance && iteration <= maxIterations
    % Update previous and current approximations
    x_prev = x_curr;
    x_curr = ...; % Compute new approximation
    
    iteration = iteration + 1;
end

% The final value of x_curr is the root approximation
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Note: The specific implementation of the stopping criteria in MATLAB may vary depending on the specific root finding algorithm being used (e.g., Newton's method, bisection method, etc.).