Numerical Methods
Core concepts
Know how to locate a root by sign change and how to refine it using interval bisection, linear interpolation (false position), and the Newton–Raphson iteration. Understand the fixed-point iteration x = g(x) and its convergence condition |g'(x)| < 1 near the root, and the trapezium and Simpson's rules for approximate integration.
Traps
- A sign change guarantees a root only if the function is continuous on the interval; state continuity.
- Newton–Raphson can fail if f'(x) is zero or nearly zero at an iterate, or if the starting value is poorly chosen.
- Simpson's rule requires an even number of strips (odd number of ordinates); the trapezium rule works with any number.
- Rounding too early in iterative methods can cause the answer to drift; carry full precision between steps.
Worked example
Use one Newton–Raphson step to refine a root of f(x) = x² − 2 starting from x₀ = 1. f'(x) = 2x, so x₁ = x₀ − f(x₀)/f'(x₀) = 1 − (−1)/2 = 1.5. Rule of thumb: iterate until two successive values agree to the required number of decimal places, then quote the answer to that precision.