Iterative Methods for Numerical Solution of Equations (9709 Pure Mathematics 3) — Mathematics

Introduction

Iterative methods provide a systematic way to find numerical approximations to roots of equations that cannot be solved algebraically. In the 9709 Paper 3 exam, iterative questions are extremely common: you are expected to set up a recurrence relation of the form $x_{n+1} = F(x_n)$ , apply it repeatedly, and state the root to a required accuracy. This subtopic builds directly on Locating Roots of an Equation, where a sign change in $f(x)$ over an interval is used to confirm a root exists — here, you then home in on that root.

Core Concept

What is an iterative sequence?

An iterative sequence is a list of values $x_1, x_2, x_3, \ldots$ generated by repeatedly applying a rule of the form

x_{n+1} = F(x_n)

starting from an initial approximation $x_1$ (sometimes called $x_0$ ). If the sequence converges — that is, the values settle towards a fixed limit $\alpha$ — then $\alpha$ satisfies

\alpha = F(\alpha)

which is equivalent to the original equation being solved.

From equation to iterative formula

To use iteration, you must first rearrange the equation $f(x) = 0$ into the form $x = F(x)$ . The iterative formula is then $x_{n+1} = F(x_n)$ .

For example, the equation $x^3 + 3x - 5 = 0$ can be rearranged as:

x = \frac{5 - x^3}{3} \quad \Longrightarrow \quad x_{n+1} = \frac{5 - x_n^3}{3}

Convergence and divergence

An iteration converges when successive terms get closer and closer to a limit. An iteration diverges when successive terms move further from the root. You are not required to state a formal condition for convergence, but you must appreciate that a given iteration may fail to converge — if the values are growing without bound or oscillating wildly, the iteration has failed and a different rearrangement should be tried.

Determining the root to a prescribed accuracy

To state a root correct to, say, 3 decimal places (3 d.p.), continue iterating until two consecutive values agree to 4 d.p. or more, then round. Always verify using a sign-change check: show $f(\alpha - 0.0005) < 0$ and $f(\alpha + 0.0005) > 0$ (or vice versa) to confirm the root lies in the required interval.

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