Numerical Methods

Numerical methods

Some equations can't be solved exactly by algebra. Numerical methods find approximate solutions to any required accuracy.

Locating roots

If f(x) is continuous and f(a) and f(b) have opposite signs (a sign change between a and b), then there is a root between a and b (where f(x) = 0).

• Caution: this fails if the function is discontinuous in the interval (e.g. an asymptote), or if there are an even number of roots (no sign change).

Iteration

Rearrange f(x) = 0 into the form x = g(x), then iterate:

x_(n+1) = g(xₙ)

starting from an estimate x₀. If the iteration converges, successive values home in on the root. A staircase or cobweb diagram on the graph of y = g(x) and y = x shows the convergence (or divergence). Not every rearrangement converges — some diverge away from the root.

The Newton–Raphson method

A faster method using the derivative:

x_(n+1) = xₙ − f(xₙ) ÷ f′(xₙ)

It usually converges very quickly, but can fail if f′(xₙ) = 0 (a stationary point) or if the starting value is poor (it may diverge or find a different root).

Numerical integration — the trapezium rule

Estimates the area under a curve by dividing it into trapezia:

∫[a to b] y dx ≈ (h/2)[y₀ + yₙ + 2(y₁ + y₂ + … + y_(n−1))]

where h = (b − a)/n (strip width) and y₀…yₙ are the y-values at the strip boundaries.

• More strips → better accuracy.
• Whether it overestimates or underestimates depends on whether the curve is concave/convex over the interval.

Worked example

Show that a root of f(x) = x³ − x − 3 lies between x = 1 and x = 2.

• f(1) = 1 − 1 − 3 = −3 (negative); f(2) = 8 − 2 − 3 = 3 (positive).
• There is a sign change and f is continuous, so a root lies between 1 and 2. ✓

Common mistakes

• Concluding "no root" from no sign change (there could be an even number of roots).
• Ignoring discontinuities when using the sign-change method.
• Rounding too early in iterations — keep full accuracy until the final answer.

Exam tips

• For root location, evaluate f at both ends and state the sign change and continuity.
• Show iteration values to enough decimal places and state convergence.
• For the trapezium rule, list the y-values clearly and use the correct h.

Key facts to remember

• A sign change of a continuous f between a and b ⇒ a root in (a, b) (beware discontinuities/even roots).
• Iteration x_(n+1) = g(xₙ) converges to a root if the rearrangement is suitable; Newton–Raphson x_(n+1) = xₙ − f(xₙ)/f′(xₙ) converges fast but can fail.
• Trapezium rule: ∫ ≈ (h/2)[y₀ + yₙ + 2(middle y-values)]; more strips = more accurate.