IB Maths IA idea · Numbers, algorithms and computing

Newton's method: how fast does it converge, and when does it fail?

Your calculator finds roots in a blink. Newton's method is the idea inside. You're going to test how fast it works, and then go looking for the starting point where it breaks.

Numbers, algorithms and computing
SL good
HL ideal
Maths AA Maths AI Simulation
Syllabus: AA 5.4, AA 5.6, AA 2.10, AA 5.12, AA 5.13, AA 1.12

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

Newton's method replaces a curve by its tangent, again and again. It is in neither course, so you derive it from the tangent line yourself. Does it behave the same near a simple root and a repeated root? Predict first, then measure the convergence rate and explain it with calculus. Then look for the starting points that fail: cycles, divergence, a jump to a far root. A strong IA connects the numbers to a justified argument. At HL, running iterations and describing them is not HL-level use of mathematics: you need a derivation (Taylor's theorem), a proof, or a link to complex numbers.

The maths you will use

Tangent line as a model. Derive $x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$ from the tangent line to $y=f(x)$ meeting the axis.
Order of convergence. Compute errors $e_n$ and estimate $p$ from $\log\frac{e_{n+1}}{e_n}$ against $\log\frac{e_n}{e_{n-1}}$ , or fit $e_{n+1} \approx C e_n^{p}$ .
Taylor expansion. Expand $f$ about the root to prove $e_{n+1} \approx \frac{f''(r)}{2f'(r)}e_n^2$ for a simple root.
Repeated roots. Show $f=(x-r)^m g(x)$ gives $e_{n+1} \approx \left(1-\frac{1}{m}\right)e_n$ , so convergence is only linear.
Failure analysis. Find starting points that cycle, diverge or jump to a far root by following iterates, and describe them with algebra.

Key equations

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

e_{n+1} \approx \frac{f''(r)}{2f'(r)}\,e_n^{2}

e_{n+1} \approx \left(1 - \frac{1}{m}\right) e_n \quad (\text{root of multiplicity } m)

The maths in action

Left: log-scale errors against iteration, one line plunging to 1e-16 in six steps and one declining slowly and steadily. Right: a three-coloured plane of starting points with intricate boundaries between the three cube roots of unity. — (a) Newton errors from $x_0=3$ : for the simple root of $x^2-2$ the error is squared each step (constant 0.35, theory 0.35); for the repeated root of $(x-1)^2$ it only halves (ratio 0.50). (b) Starting points for $z^3-1$ , coloured by the root reached.

Example research question

To adapt, never to copy

How does the speed of Newton's method change between a simple and a repeated root, measured by the order of convergence from computed errors and by a derivation from Taylor's theorem, and for which starting points does it fail on $f(x)=x^3-2x+2$ ?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the function(s) and the roots you will study, and says why you chose them.
Says what you will measure: the number of iterations, the size of the error, or the order of convergence (and how you will define it).
Says what 'fails' means: no convergence, a cycle, or convergence to the wrong root.
Includes a way to justify the behaviour with calculus, not only to observe it.
Is narrow enough for 12 to 20 pages: two or three functions done in depth, not ten done quickly.

Which course it fits

AA SL good
AA HL ideal
AI SL stretch
AI HL good

Syllabus links in Maths AA:

AA 5.4 Tangents and normals
AA 5.6 Standard derivatives; chain, product and quotient rules
AA 2.10 Solving equations graphically and analytically
AA 5.12 Continuity, differentiability, first principles (HL only)
AA 5.13 Limits with l'Hopital's rule or Maclaurin series (HL only)
AA 1.12 Complex numbers in Cartesian form (HL only)

Steps

Read and predict. Read how Newton's method is derived. Before running anything, predict how the error should behave near a root, and whether it should matter if the root is repeated.
Trial run (1 lesson). Run five steps on a simple equation by hand and in a spreadsheet or program. Check that they agree and that you can control the number of digits shown.
Choose functions and roots. Pick two or three functions with different roots (simple, repeated, close together). Say why each earns its place.
Collect convergence data. Record the error at each step, with a true root found independently. Decide how many digits you keep and say why.
Find the rate. Look for a pattern in the errors. Write down the claim you are testing before you fit anything.
Explain with calculus. Use a Taylor expansion or another justified argument to explain your observed rate. If a step needs an assumption, state it.
Search for failures. Choose starting values systematically, not randomly. Classify what you find, and explain each type using the derivative.
Compare or extend. Try a second method or map which start finds which root. Decide what you will use as a fair comparison.
Reflect. Which assumptions did the computer hide? What changed in your plan after your first surprising result?
Write, cut, check. Write in the layout order. Show one worked iteration and put the rest in tables. Run the checklist, then get your question checked before your teacher's draft.

Make it yours

Simple root against repeated root. Run the method on functions with the same root but different multiplicity.
Derive the rate. Use a Taylor expansion about the root to show how the next error depends on the current one.
Hunt for failure. Search systematically for starting values that cycle, diverge or land on a different root.
Map the basins. For a function with several roots, colour each start by the root it finds. On the real line first; at HL, in the complex plane for $z^{3}$ − 1.
Compare with other methods. Run bisection, the secant method and Newton on the same equations and count the steps and the function evaluations.
Change the context. How a calculator finds square roots, or an old method for cube roots.

HL extension

Prove quadratic convergence for a simple root with Taylor's theorem, find the exact constant, show how a repeated root changes the rate, and connect the failure regions of $z^3-1$ to complex numbers and symmetry.

The data you need

Run the iteration in a spreadsheet or program for chosen functions and starting points and record the errors at each step.

Traps

Quoting the Newton formula without deriving it from the tangent line.
Calling the convergence 'fast' without a measure: an error table or a rate, not an impression.
Trusting the software when the derivative is zero or nearly zero at the start. Say what happens.
At HL, staying with iteration tables and graphs. Without a derivation or a proof, criterion E stays in the middle band.
Rounding the iterates early, so the errors that you study are rounding errors.
Choosing only friendly functions, such as a square root. A method that never fails is not much of an exploration.
A list of failure cases with no reasons.

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