Newton-Raphson iteration

For a differentiable function, xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ). Evaluate the function and derivative at the same estimate. A zero derivative makes the update undefined; a small derivative can send the estimate far from the intended root.

One successful step does not prove convergence. Keep enough precision between steps, state the starting estimate, and check the final residual. Different starting values can lead to different roots or fail to settle. This is dedicated numerical-methods practice on mapped routes, not a universal Further Maths requirement.

Worked example

For f(x) = x² − 2, perform one Newton step from x₀ = 1.5.

1. f(1.5) = 0.25 and f′(1.5) = 3.
2. x₁ = 1.5 − 0.25/3.
3. Keep the exact fraction or sufficient decimal precision.

Answer: x₁ = 17/12 ≈ 1.41667

Revise first: Differentiating powers.