14.5 Iteration and numerical methods — Learn

Prerequisites: Substitute into formulae, use a calculator accurately, and round to a stated accuracy.

Objectives: Use an iteration rule repeatedly and report the required approximation. This page covers A20.

Repeat a rule

An iteration repeats a calculation using the previous answer. A rule such as $x_{n+1}=g(x_n)$ tells you to find the next value from the current value. The starting value is $x_0$.

Example. Use $x_{n+1}=\frac12(x_n+6/x_n)$ with $x_0=2$. Find $x_3$.

Use the rule to find the next value each time:

Therefore $x_3\approx2.449$. Keep enough digits during the working, then round the final value as requested.

An iteration can also solve an equation after rearranging it into a suitable form $x=g(x)$. Different rearrangements can behave differently, so follow the rule given in the question.

Watch out. The subscript tells you the step number. Do not reuse $x_0$ each time: calculate $x_1$, then use that result to calculate $x_2$, and continue.

Progress check

The rule starts from $x_0=2$. Predict whether $x_2$ is nearer to $2$ or to $2.5$, then calculate it and compare.

You should now be able to

• identify the starting value and iteration rule;
• calculate each new value from the previous one;
• state the requested approximation with suitable accuracy.