Algorithm for Bisection Method

The steps to apply the bisection method to find the root of the equation are

1. Choose and as two guesses for the root such that , or in other words, changes sign between and .

2. Estimate the root, of the equation as the mid-point between and as

3. Now check the following

a. If , then the root lies between and ; then and .

b. If , then the root lies between and ; then and .

c. If ; then the root is _. Stop the algorithm if this is true.

4. Find the new estimate of the root

Find the absolute approximate relative error as

where

= estimated root from present iteration

= estimated root from previous iteration

5. Compare the absolute relative approximate error with the pre-specified relative error tolerance . If , then go to Step 3, else stop the algorithm. Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it.

Example 1:

You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the distance to which the ball will get submerged when floating in water.

Figure 5: Floating ball problem

The equation that gives the depth to which the ball is submerged under water is given by

Use the bisection method of finding roots of equations to find the depth to which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration, and the number of significant digits at least correct at the end of each iteration.

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