Numerical Analysis Questions Medium
The bisection method is a numerical technique used to solve nonlinear equations. It is an iterative method that repeatedly bisects an interval and selects a subinterval where the function changes sign, guaranteeing the existence of a root within that subinterval.
The steps involved in the bisection method are as follows:
1. Select an initial interval [a, b] such that f(a) and f(b) have opposite signs, indicating a root exists within the interval.
2. Calculate the midpoint c = (a + b) / 2.
3. Evaluate the function at the midpoint, f(c).
4. If f(c) is close enough to zero (within a specified tolerance), then c is considered the root and the process terminates.
5. If f(c) and f(a) have opposite signs, then the root lies within the subinterval [a, c]. Set b = c and go to step 2.
6. If f(c) and f(b) have opposite signs, then the root lies within the subinterval [c, b]. Set a = c and go to step 2.
7. Repeat steps 2-6 until the root is found within the desired tolerance.
The bisection method is relatively simple and guaranteed to converge to a root as long as the initial interval is chosen properly and the function is continuous. However, it may require a large number of iterations to achieve the desired accuracy, especially for functions with multiple roots or when the root is located near the boundaries of the interval.