Iteration and numerical methods · Grade 12

In Grade 12 you learn: iteration to solve equations.

Many equations, such as x³ + x − 5 = 0, cannot be solved exactly with algebra. Numerical methods find the solution approximately, as accurately as you need: first trap the root between two values, then close in on it with an iteration formula or the Newton–Raphson method.

Key facts

• Change of sign: if f(a) and f(b) have opposite signs (and f is continuous), there is a root between a and b.
• Iteration: rearrange f(x) = 0 into x = g(x), then use xₙ₊₁ = g(xₙ) again and again.
• An iteration converges near a root α when |g′(α)| < 1.
• To show a root is 1.52 to 2 decimal places, show that f changes sign between 1.515 and 1.525.
• Keep full calculator accuracy (use the ANS key) and round only at the end.