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Feb 22

The bisection method is a numerical technique used to find the root of a function within a given interval. It works by repeatedly dividing the interval in half and then selecting the subinterval in which the root lies. This process is repeated until the interval becomes sufficiently small.

Here is an example to illustrate the bisection method:

Consider the function f(x) = x^2 - 4 and we want to find the root of this function within the interval [1, 3].

1. Calculate the midpoint of the interval [1, 3]:

- Midpoint = (1 + 3) / 2 = 2

2. Evaluate the function at the midpoint:

- f(2) = 2^2 - 4 = 0

3. Determine in which subinterval the root lies:

- Since f(2) = 0, the root lies in the subinterval [1, 2].

4. Repeat the process with the new interval [1, 2]:

- Calculate the midpoint: (1 + 2) / 2 = 1.5

- Evaluate the function at the midpoint: f(1.5) = 1.5^2 - 4 = -1.75

5. Determine in which subinterval the root lies:

- Since f(1.5) < 0, the root lies in the subinterval [1.5, 2].

6. Repeat the process with the new interval [1.5, 2]:

- Calculate the midpoint: (1.5 + 2) / 2 = 1.75

- Evaluate the function at the midpoint: f(1.75) = 1.75^2 - 4 = -0.4375

7. Determine in which subinterval the root lies:

- Since f(1.75) < 0, the root lies in the subinterval [1.75, 2].

8. Continue this process until the interval becomes sufficiently small or the desired level of accuracy is achieved.

The bisection method is a simple and robust technique for finding the root of a function within a given interval. It is guaranteed to converge to a solution as long as the function is continuous and changes sign within the interval.

Here is an example to illustrate the bisection method:

Consider the function f(x) = x^2 - 4 and we want to find the root of this function within the interval [1, 3].

1. Calculate the midpoint of the interval [1, 3]:

- Midpoint = (1 + 3) / 2 = 2

2. Evaluate the function at the midpoint:

- f(2) = 2^2 - 4 = 0

3. Determine in which subinterval the root lies:

- Since f(2) = 0, the root lies in the subinterval [1, 2].

4. Repeat the process with the new interval [1, 2]:

- Calculate the midpoint: (1 + 2) / 2 = 1.5

- Evaluate the function at the midpoint: f(1.5) = 1.5^2 - 4 = -1.75

5. Determine in which subinterval the root lies:

- Since f(1.5) < 0, the root lies in the subinterval [1.5, 2].

6. Repeat the process with the new interval [1.5, 2]:

- Calculate the midpoint: (1.5 + 2) / 2 = 1.75

- Evaluate the function at the midpoint: f(1.75) = 1.75^2 - 4 = -0.4375

7. Determine in which subinterval the root lies:

- Since f(1.75) < 0, the root lies in the subinterval [1.75, 2].

8. Continue this process until the interval becomes sufficiently small or the desired level of accuracy is achieved.

The bisection method is a simple and robust technique for finding the root of a function within a given interval. It is guaranteed to converge to a solution as long as the function is continuous and changes sign within the interval.