Pure · Numerical methods
Chapter 1 · 3
The idea
The Newton–Raphson method
A fast iteration that uses the tangent — x_(n+1) = x_n − f(x_n)/f'(x_n) — its geometric meaning, why it converges so quickly, and the cases where it fails (a horizontal tangent or a poor starting value).
A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.
In this lesson — start anywhere
Pure · Numerical methods
The Newton–Raphson method
A fast iteration that uses the tangent — x_(n+1) = x_n − f(x_n)/f'(x_n) — its geometric meaning, why it converges so quickly, and the cases where it fails (a horizontal tangent or a poor starting value).
Why it works
Follow the tangent down
The iteration works, but slowly, and only for the right rearrangement. The Newton–Raphson method is a cleverer choice of step that homes in on a root astonishingly fast, using the tangent to the curve.Start at a guess . Draw the tangent to at the point . Unless the curve is doing something awkward, that tangent crosses the -axis much closer to the root than was — so take the crossing as your next estimate , and repeat.
Keep reading — free
The rest of the explanation, plus 3 worked examples you step through move by move.
Start freeTakes a minute — no card.