Comparing Numerical Methods

This lesson brings together all the numerical methods covered in the Edexcel A-Level Mathematics specification (9MA0) — fixed-point iteration, Newton-Raphson, and bisection/sign change — and compares their strengths, weaknesses, and appropriate uses. You also need to understand error bounds.

---

Overview of the Three Methods

Feature	Sign Change / Bisection	Fixed-Point Iteration	Newton-Raphson
Formula	Midpoint m = (a+b)/2, check sign	x_{n+1} = g(x_n)	x_{n+1} = x_n - f(x_n)/f'(x_n)
Requires	f(x) only	A rearrangement x = g(x)	f(x) and f'(x)
Convergence speed	Slow (linear)	Moderate (linear)	Fast (quadratic)
Reliability	Very reliable	Variable	Usually good
Always converges?	Yes (if continuous and sign change exists)	Only if	g'(α)
Error bounds	Easy to determine	Harder to determine	Harder to determine

---

Bisection / Sign Change Method

Advantages