5 · IB Maths
Newton–Raphson method (enrichment)
Slide down the tangent to the axis, evaluate again, repeat. Watch the iterations race towards a root, or, from an unlucky start, cycle and refuse to converge. Drag on the graph to move the starting value.
Predict first
Start from the default settings (f(x) = x³ − 2x − 5, x₀ = 2.50, 3 iterations). Add one more iteration. How does the error in x_n change?
Newton–Raphson converges quadratically near a simple root: the new error is about a constant times the old error squared. Here f(x_n) drops from about 4×10⁻⁵ to about 9×10⁻¹¹.
Check it in the model: Move the Iterations slider from 3 to 4 (or press Next step) and compare f(x_n) in the last two rows of the table.
y = f(x) tangents latest iterate. Drag on the graph to set x₀.
Try this
- Use "Fast convergence" and step through the iterations. Compare f(xn) from one row of the table to the next. How quickly does it shrink?
- Choose "The famous 2-cycle" and add steps. Which two values does the method bounce between, and why can the tangent never lead out?
- Choose f(x) = x³ − 2x − 5 and drag x₀ close to 0.8, where f′ is small. What happens to x₁, and is the method still safe to use?
Exam tip: each step is the x-intercept of the tangent, so write xn+1 = xn − f(xn) ÷ f′(xn) and keep the full calculator value between steps. Newton–Raphson is not in the syllabus, but it gives a feel for the iterative methods you do meet.
More on 5: Chain and product rules · Second derivative and points of inflexion · Trapezoidal rule and Riemann sums · Displacement and distance travelled · Euler's method and differential equations · Implicit differentiation · Related rates of change · Area under a curve · Calculus of motion · Limits · Maclaurin series · Optimisation · Slope fields · Tangents and the derivative · Volume of revolution