Numerical Analysis/Bisection Method Worked Example

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

Example

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

Analysis of the Problem

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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

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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

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


Iteration #3 - Iteration #14

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Complete Iteration #3 - Iteration #14:


the final answer is follow or describe below

Conclusion

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Thus, we have found that

is an approximation of   correct to within 10-4.