Math Lesson 9.4.4 - Recursive Iteration and Iteration Machines

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Welcome to our Math lesson on Recursive Iteration and Iteration Machines, this is the fourth lesson of our suite of math lessons covering the topic of Iterative Methods for Solving Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Recursive Iteration. Iteration Machines

Sometimes, you will be given a formula to use in questions involving iteration. Such a formula - known as the "iteration machine" - is applied a number of times for variables unit until you obtain two same results in a row after rounding it to the desired number of decimal places. According to this procedure, if x_n + 1 = x_n when the result is rounded at the desired number of decimal places, then this result represents the root of the original equation. This method is known as the recursive iteration.

The general procedure used in iteration machines is as follows:

Let's clarify this procedure through an example.

Example 6

Using the iteration machine, find a solution for the equation

to one decimal place, by starting from x⁰ = 0

Solution 6

First, let's isolate x by finding a suitable transformation of the original equation. We will stop after obtaining two same results after rounding. Thus, we have:

Now, we have to use the formula

to calculate the value of x rounded to one decimal place. Thus, for x⁰ = 0, we have

For x¹ = - 1.4422, we have

For x² = - 2.16949, we have

For x³ = - 2.401359, we have

For x⁴ = - 2.46658, we have

Thus, since we obtained the same result (x = - 2.5) in the last two steps after rounding them to one decimal place, we have x = - 2.5 as a solution for our original equation.

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