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Newton's method is an iterative method that helps find the root (zero) of a real-valued function. The method starts with an initial guess, which we will denote as x鈧�, and then successively refines this guess by iterating the following formula: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} where x_n is the current guess, x_{n+1} is the next guess, f(x) is the function for which we want to find the root, and f'(x) is its derivative.

To decide when to terminate Newton's method, consider the following criteria: 1. Maximum number of iterations: Set a maximum number of iterations, N, and terminate when you reach that count. This will prevent the method from running indefinitely if it cannot find a satisfactory root. 2. Function value tolerance: Set a tolerance level for the function value, such as 系鈧�, and terminate when you have |f(x_n)| < 系鈧�. This criterion checks if the current guess is sufficiently close to the root, based on its function value. 3. Convergence rate tolerance: Set another tolerance level for the difference in successive root estimates, such as 系鈧�, and terminate when you have |x_n - x_{n-1}| < 系鈧�. This criterion checks for convergence based on any significant changes in the estimates of the root.

Select the termination criteria that best suit your specific problem and requirements. You may consider combining any of the criteria mentioned in step 2, or even use all of them. For example, you could choose to terminate when: - the maximum number of iterations is reached - the current guess is close enough to the root, based on function value, AND - the convergence rate is acceptable, based on the difference between successive root estimates.

Finally, implement the chosen termination criteria in your Newton's method algorithm. At each iteration, check if any of the chosen criteria have been met. If so, terminate the method and return the current guess as the root of the function. As an example, if you're using all the termination criteria described in step 2, your algorithm would check if (N is the maximum number of iterations, 系鈧� is the function value tolerance, and 系鈧� is the convergence rate tolerance): - the current iteration count is equal to or greater than N - |f(x_n)| < 系鈧�, and - |x_n - x_{n-1}| < 系鈧� If any of these conditions are met, the method is terminated, and the current estimate x_n is returned as the root of the function. Ultimately, deciding when to terminate Newton's method is a balance between the desired accuracy of the solution and computational resources.