Numerical solution of equations — common mistakes

When performing iterations, you must show your working clearly. Write down the value of each iterate ($x_1, x_2, ...$) to at least two more decimal places than required in the final answer. For example, if the answer must be to 3 d.p., write your iterates to at least 5 d.p. This provides evidence for your final rounded answer.

Examinations for this syllabus require you to demonstrate your understanding of the numerical methods themselves. You will get marks for showing the sign change test and for writing down the sequence of iterates. Relying solely on a calculator's solver function will not earn you these marks.

The question will almost always give you the specific rearrangement to use. If you have a choice, you would ideally pick one where the derivative of $g(x)$ has a magnitude less than 1 near the root, as this guarantees convergence. However, for P2, you are typically guided to the correct formula.

To 'show the root lies between 1.5 and 1.6', you test for a sign change using $f(1.5)$ and $f(1.6)$. To 'show the root is 1.5 correct to 1 d.p.', you must test for a sign change on the interval that rounds to 1.5, which is $[1.45, 1.55]$. You would test $f(1.45)$ and $f(1.55)$.

This is called divergence. It means the iterative formula you are using will not lead you to the root from your chosen starting point. This can happen if the rearrangement $x=g(x)$ is not suitable. In an exam, this is unlikely unless they are specifically asking you to show that an iteration fails.