1. Initialization: a. Choose an interval [a, b] that brackets the root of the equation. b. Calculate the function values at the endpoints: f(a) and f(b). c. Set the previous estimate of the root, x_prev, as one of the bracketing points.
2. Convergence Criteria: a. Define a tolerance value, epsilon, to determine the desired level of accuracy. b. Set the maximum number of iterations, max_iter, to avoid infinite looping.
3. Iteration: a. Check if the function value at any endpoint is close to zero within the tolerance: |f(a)| < epsilon or |f(b)| < epsilon. b. If true, return the corresponding endpoint as the root approximation and exit the algorithm. c. Check if the function values at the endpoints have opposite signs: f(a) * f(b) < 0.
   • If true, proceed to the next step.
   • If false, continue with bisection.
4. Bisection: a. Calculate the midpoint of the interval: c = (a + b) / 2. b. Check if the function value at the midpoint is close to zero within the tolerance: |f(c)| < epsilon.
   • If true, return c as the root approximation and exit the algorithm. c. Check if the function value at the midpoint has the same sign as the function value at point a: f(a) * f(c) > 0.
   • If true, update point a with the midpoint: a = c.
   • If false, update point b with the midpoint: b = c.
5. Interpolation: a. Perform inverse quadratic interpolation to estimate a new point, d, as a better approximation of the root. b. Check if the interpolated point d falls within the current interval [a, b].
   • If true, continue to the next step.
   • If false, perform bisection again.
6. Update and Repeat: a. Update the interval based on the new approximation: if |d – x_prev| < epsilon, set [a, b] = [d – epsilon, d + epsilon]. b. Update the previous estimate of the root: x_prev = d. c. Repeat steps 3 to 6 until the desired level of accuracy is achieved or the maximum number of iterations is reached.
7. Convergence Check: a. Monitor the convergence by checking if |b – a| < epsilon or |f(d)| < epsilon. b. If true, return the current approximation, d, as the root. c. If the maximum number of iterations is reached without convergence, consider it a failure to find the root within the desired accuracy.

These steps outline the iterative process of Brent’s Method for approximating the roots of equations. By combining bracketing, bisection, and interpolation techniques, Brent’s Method provides a robust and efficient approach to root-finding.

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