Numerical Analysis/Bisection Method Worked Example

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X^3=20 ,f(X), X4=4, X1=1 take four iterations .

Example[edit]

Find an approximation of correct to within 10-4 by using the bisection method on . starting on [1, 2].

Analysis of the Problem[edit]

The number of iterations we will use, n, must satisfy the following formula:
.
Thus, we will use 14 iterations of the bisection method.

Iteration #1[edit]

First, we find the midpoint of the interval [1, 2]:



Then we check if is positive or negative:



The value of is negative, which means that is less than .

We therefore use as the left endpoint of our new interval and keep 2 as the right endpoint.

yes

Iteration #2[edit]

Repeat the process from Iteration #1 to do Iteration #2:


Iteration #3 - Iteration #14[edit]

Complete Iteration #3 - Iteration #14:


the final answer is follow or describe below

Conclusion[edit]

Thus, we have found that

is an approximation of correct to within 10-4.