Maths IA · Algebra and functions · topic 6 of 40
Newton's method: how fast does it converge, and when does it fail? · IB Maths IA idea: guide, sample graph and step-by-step plan
great if you prove the rate yourselfYour 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.
- ✓ you like seeing why a method works
- ✓ you can run an iteration on a spreadsheet or in code
- ✓ you enjoy finding the case that breaks the rule
The big idea
← predict this firstNewton'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.
Make it yours: pick one twist
Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.
Swipe for more twists →
Run the method on functions with the same root but different multiplicity.
→ You see two different rates of convergence, and you must explain the difference using derivatives.Use a Taylor expansion about the root to show how the next error depends on the current one.
→ Your convergence claim is proved rather than observed. This is the route that shows HL-level rigour.Search systematically for starting values that cycle, diverge or land on a different root.
→ Your analysis turns into a classification of bad starting points, supported by a diagram of the iterations.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.
→ The picture is a map of which start goes where. The complex version links to complex numbers and is a non-routine extension.Run bisection, the secant method and Newton on the same equations and count the steps and the function evaluations.
→ A fair comparison needs a definition of 'cost'; the evaluation compares speed with reliability.How a calculator finds square roots, or an old method for cube roots.
→ Gives the introduction a real purpose while the maths stays the same.Your Maths IA research question: run the check
define 'fails' exactlyWrite your own question first. Then make sure it ticks every box:
- 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.
The picture to draw
One iteration is a tangent line meeting the axis. Draw yours for your own function, with the tangent steps and the formula for each step beside them.
How your Maths IA could be laid out
The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.
Your question, why root-finding interests you, and your prediction for how fast the method will converge.
The formula derived from the tangent line (not just quoted), and one clear worked iteration with a diagram.
A table of errors at each step for your chosen functions and roots, with the step count and how you computed it.
A derivation of the rate near a simple root with a justified Taylor argument; the repeated-root case.
Starting points that cycle, diverge or find the wrong root, with a clear picture and an explanation using derivatives.
A second method, or the basin map for several roots. A judgement on which is better and why.
Limits of your tests, the effect of computer precision, and what you changed after a surprising result.
Back to your aim: what you found about speed and about failure.
Sources for the method and any software used.
What your Maths IA graph could look like
log scale = pattern visibleMade-up data, to show you the form of a strong graph, not the answer. Yours will come from your own iterations. example only
- Axis labels say what is plotted (a logarithm of the error), which makes the pattern visible.
- The log scale makes different rates of convergence look different shapes, so the reader can tell them apart at a glance.
- Two functions on one graph, so the comparison is the point.
- The error cannot be shown below the accuracy of the computer; say so in the text.
- Another graph to consider: plot the next error against the current error on a log-log plot. The slope shows the order of convergence.
Your game plan
run 5 steps by hand first!Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictRead 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 rootsPick two or three functions with different roots (simple, repeated, close together). Say why each earns its place.
- Collect convergence dataRecord the error at each step, with a true root found independently. Decide how many digits you keep and say why.
- Find the rateLook for a pattern in the errors. Write down the claim you are testing before you fit anything.
- Explain with calculusUse a Taylor expansion or another justified argument to explain your observed rate. If a step needs an assumption, state it.
- Search for failuresChoose starting values systematically, not randomly. Classify what you find, and explain each type using the derivative.
- Compare or extendTry a second method or map which start finds which root. Decide what you will use as a fair comparison.
- ReflectWhich assumptions did the computer hide? What changed in your plan after your first surprising result?
- Write, cut, checkWrite 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.
Maths you'll need
Traps to avoid in this Maths IA
- 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.
More Maths IA ideas like this
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This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.