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2.6 Numerical solution of equations

Syllabus
9709–2028–2029
Topic
2.6
Level
AS

Learning objectives

2.6.1Root location• locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change e.g. finding a pair of consecutive integers between which a root lies.2.6.2Iterative approximations• understand the idea of, and use the notation for, a sequence of approximations which converges to a root of an equation2.6.3Iteration formulae• understand how a given simple iterative formula of the form xn + 1 = F(xn) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy. Knowledge of the condition for convergence is not included, but an understanding that an iteration may fail to converge is expected.

Locate a root by a graph crossing or a sign-change bracket

Rewrite the equation as f(x)=0f(x)=0 and locate where the graph y=f(x)y=f(x) crosses the xx-axis, or plot the two sides separately and locate their intersection. The graph supplies an approximate root or a search interval.

If $f$ is continuous on $[a,b]$ and $f(a)f(b)<0$, then at least one root lies in $(a,b)$. Evaluate consecutive integers or progressively closer endpoints when requested.

For a continuous ff, f(1)<0f(1)<0 and f(2)>0f(2)>0 locates at least one root between 11 and 22. State the function values or their signs, not just the interval.

A graph gives visual approximate evidence; a sign-change bracket gives endpoint evidence. A narrower bracket gives a tighter location but remains an interval, not the exact root.

A sign change guarantees at least one root under continuity, not uniqueness. A repeated/touching root may have no sign change, so graphical evidence can still matter.

An iterative approximation is useful only when its sequence converges to the intended root

Iteration replaces x by x_{n+1}=g(x_n). A fixed point α satisfies g(α)=α, corresponding to a root of the rearranged equation.

Choose a starting value in the stated interval, compute enough figures during iteration, and stop using a tolerance on successive values or the residual. Different rearrangements can converge differently.

For x=cos x, starting x₀=0 gives a sequence approaching about 0.739; starting values should remain in a region where g behaves stably.

A few stable-looking digits do not prove convergence, and iteration can diverge or enter a cycle even when the equation has a root.

Relate, run and verify a fixed-point iteration

For xn+1=F(xn)x_{n+1}=F(x_n), a convergent limit α\alpha must satisfy α=F(α)\alpha=F(\alpha). Rearrange that fixed-point equation to confirm it is the original equation whose root is required, including any domain restrictions.

Use the stated starting value, keep guard digits, tabulate nn and xnx_n, apply the same formula repeatedly, watch for settling/divergence/cycling, and continue until successive values justify the prescribed rounded answer.

To solve $x^3+x-1=0$, the given rearrangement $x_{n+1}=(1-x_n)^{1/3}$ has fixed-point equation $x^3=1-x$, hence $x^3+x-1=0$. Run it only from the given/appropriate start.

For a requested number of decimal places, obtain successive values that round consistently at that precision and substitute the reported approximation into the original equation as a residual sense-check when practical.

An algebraically related iteration may fail to converge or may approach a different root. The derivative condition for convergence is explicitly not required in this syllabus; judge only from the given task and observed sequence behaviour.