Singapore School Mathematics Operating Manual · Chapter 19
Some mathematical answers are not obtained in one exact algebraic step. They are approached.
A calculator repeatedly improves an estimate. A graphing process narrows a root. A numerical method generates a sequence of approximations. A recursive formula updates one value from the previous value. A trial-and-improve method homes in on a target. In every case, the learner must decide whether the process is moving toward a stable answer or merely producing more decimals.
This chapter develops four linked ideas: iteration, convergence, error control and stopping criteria. The purpose is not to turn school Mathematics into numerical analysis. It is to build a disciplined answer to a common question: when is an approximation trustworthy enough to stop?
Iteration · Convergence · Fixed points · Error control · Stopping criteria · Failure modes · Practice · Worked answers
1. Iteration means using an output to generate a new input
Suppose x₀ = 1 and xₙ₊₁ = (xₙ + 3)/2.
Then:
x₁ = 2, x₂ = 2.5, x₃ = 2.75, x₄ = 2.875, and so on.
Each new value is generated from the previous one using the same rule.
An iterative process is therefore a controlled recurrence. The sequence of approximations is part of the mathematical object, not just scratch work.
2. Repetition alone is not evidence of convergence
Consider xₙ₊₁ = xₙ + 1 starting from x₀ = 0.
The values are 0, 1, 2, 3, 4, … .
The rule repeats perfectly, but the sequence does not settle toward a finite limit.
Iteration describes the mechanism. Convergence describes the long-run behaviour.
Never infer convergence simply because a recurrence exists.
3. Convergence means the approximations approach a stable limit
For xₙ₊₁ = (xₙ + 3)/2, the sequence approaches 3.
The distance from 3 halves at every step:
|xₙ₊₁ − 3| = |(xₙ + 3)/2 − 3| = |xₙ − 3|/2.
This gives a strong convergence statement. The error shrinks geometrically.
A convergent sequence need not ever equal its limit exactly. It only needs to get arbitrarily close.
4. Stable-looking decimals can still mislead
Suppose a calculator produces 1.414, 1.4142, 1.41421, 1.414214.
The values appear to stabilise near √2.
That is useful evidence, but the display alone does not explain why the process converges or how much error remains.
A stopping rule based only on “the digits stopped changing on my screen” depends on display length and may hide unresolved error.
Whenever possible, combine numerical stability with a mathematical error bound or independent check.
5. Convergence can be monotonic
A sequence can approach its limit from one side.
For xₙ = 1 − 1/n with n ≥ 1, the sequence increases toward 1.
Every term is below 1 and each new term is larger than the previous term.
A bounded increasing sequence has strong structural evidence of convergence in more advanced analysis.
At school level, the important observation is that direction and bounds can help assess behaviour.
6. Convergence can oscillate
Consider xₙ = (−1)ⁿ/n.
The signs alternate, but the magnitude shrinks toward zero.
The sequence converges to 0 while repeatedly crossing the limit.
Therefore “the values must move in one direction” is not a requirement for convergence.
Oscillation is compatible with convergence if the oscillation amplitude shrinks.
7. Oscillation can also fail to converge
For xₙ = (−1)ⁿ, the sequence alternates between 1 and −1 forever.
The oscillation does not shrink.
There is no single limit.
A solver should distinguish “alternating but contracting” from “alternating indefinitely at fixed size”.
8. Divergence can occur by growth without bound
The sequence xₙ = n² grows without bound.
There is no finite limiting value.
In some contexts we say it tends to infinity, but that is different from convergence to a real number.
If an iterative solver is supposed to approximate a finite root and its values grow rapidly in magnitude, that is a warning sign rather than progress.
9. A fixed point is a value unchanged by the iteration rule
If xₙ₊₁ = g(xₙ), a fixed point L satisfies L = g(L).
For g(x) = (x + 3)/2, solve L = (L + 3)/2.
Then 2L = L + 3, so L = 3.
If the iteration converges to a limit and the update function behaves appropriately, the limit often must be a fixed point.
This gives a useful candidate for the long-run value.
10. A fixed point need not be attracting
Consider xₙ₊₁ = 2xₙ.
The value L = 0 is a fixed point because 2(0) = 0.
But starting from x₀ = 0.1 gives 0.2, 0.4, 0.8, 1.6, … . The sequence moves away from zero.
A fixed point can exist without nearby iterations converging to it.
Existence of a target and stability of the iteration are separate questions.
11. Contraction explains many successful school-level iterations
For a simple fixed-point iteration xₙ₊₁ = g(xₙ), if nearby points are pulled closer together, convergence is plausible.
In calculus language, a sufficient local condition is often |g′(L)| < 1 near a fixed point L.
If |g′(L)| > 1, nearby errors tend to expand.
This derivative test is an extension for learners who have studied calculus. The operating-manual idea is broader: does the update shrink error or amplify it?
12. The same equation can produce good and bad rearrangements for iteration
Suppose we want to solve x² = 3.
One rearrangement is x = 3/x, giving iteration xₙ₊₁ = 3/xₙ.
Starting from x₀ = 1 gives 3, 1, 3, 1, … . It oscillates and fails to converge.
Another iteration, xₙ₊₁ = (xₙ + 3/xₙ)/2, converges rapidly to √3 for positive starting values.
Algebraic equivalence of fixed points does not guarantee numerical equivalence of iterative behaviour.
13. Newton-style averaging can improve a root estimate
To approximate √S, a classic iteration is
xₙ₊₁ = 1/2(xₙ + S/xₙ).
For S = 10 and x₀ = 3:
x₁ = (3 + 10/3)/2 ≈ 3.1666667.
x₂ ≈ (3.1666667 + 10/3.1666667)/2 ≈ 3.1622807.
x₃ ≈ 3.16227766.
The value stabilises rapidly near √10.
The method combines the current guess with the reciprocal correction required by x² = S.
14. Recursive sequences can converge without being root solvers
Consider a population-style model Pₙ₊₁ = 0.8Pₙ + 20.
A steady state satisfies P = 0.8P + 20, so 0.2P = 20 and P = 100.
If P₀ = 40, then P₁ = 52, P₂ = 61.6, P₃ = 69.28, and so on, approaching 100.
The same convergence concepts apply to recurrence models and iterative numerical solvers.
15. Absolute error measures distance from the true value
If the true value is T and approximation is A, absolute error is |A − T|.
If π is approximated by 3.14, the absolute error is about 0.00159265.
Absolute error is expressed in the same unit as the quantity being approximated.
It answers “how far away?” in raw magnitude.
16. Relative error scales error to the size of the quantity
Relative error is |A − T|/|T| when T ≠ 0.
A 1-unit error is enormous if the true value is 2 and tiny if the true value is 10,000.
Percentage error multiplies relative error by 100%.
Relative error is useful when comparing approximation quality across different scales.
17. Successive-difference error is not the same as true error
Suppose |xₙ₊₁ − xₙ| is very small.
That shows the iteration is changing slowly.
It does not automatically prove xₙ is close to the true solution.
A process can stagnate at a wrong value, cycle with small amplitude or move slowly while still far from the target.
Successive difference is a practical stopping indicator, not a universal proof of accuracy.
18. Residual error checks the original equation
Suppose we are approximating a root of f(x) = 0.
The residual is |f(xₙ)|.
If xₙ is close to a true root and the function is well behaved, the residual should often be small.
For x² − 10 = 0 and x = 3.1623, residual ≈ |3.1623² − 10| ≈ 0.00014.
Residual checks the original problem rather than only the internal movement of the iteration.
19. A small residual can still coexist with sizeable input error in poorly conditioned problems
Consider f(x) = 0.000001(x − 1000).
At x = 900, the residual is only 0.0001, which looks small.
But the x-error is 100.
The scale of the function matters.
This is an advanced caution: residual size should be interpreted relative to the problem’s sensitivity.
20. Bracketing gives a strong error certificate
If a root is known to lie between a and b, then any point inside the interval is at most b − a away from either endpoint, and the midpoint is at most (b − a)/2 from the unknown root.
For example, if √2 is bracketed between 1.4142 and 1.4143, the interval width is 0.0001.
The midpoint has absolute error at most 0.00005.
Bracketing turns uncertainty into an explicit interval rather than an impression of stability.
21. Bisection halves a bracket repeatedly
Suppose f is continuous and f(a) and f(b) have opposite signs, so a root lies between a and b.
Evaluate the midpoint. Keep the half-interval that still brackets a sign change.
Each step halves the interval width.
After n bisections, the width is the original width divided by 2ⁿ.
This produces a predictable stopping rule based on desired accuracy.
22. Iteration can converge quickly or slowly
If the error halves each step, ten extra iterations reduce error by about a factor of 1024.
If the error only multiplies by 0.99 each step, convergence is much slower.
The number of stable digits gained per iteration depends on convergence rate.
This explains why some numerical methods feel efficient while others require many updates.
23. More iterations are not always better
In exact arithmetic, a convergent process may keep improving indefinitely.
On calculators and computers, finite precision can eventually limit improvement.
Repeated computation can also waste time after the requested accuracy has already been reached.
A stopping criterion should match the question’s precision requirement.
24. Stop when the mathematical requirement is met
If a question asks for 3 decimal places, the aim is not necessarily to make successive approximations identical to 12 displayed digits.
It is enough to establish the value rounds correctly to 3 decimal places.
If an interval bracket lies entirely inside one 3-decimal rounding cell, the rounded answer is guaranteed.
This is a stronger stopping rule than simply observing repeated digits.
25. Stopping by successive difference
A common practical rule is stop when |xₙ₊₁ − xₙ| < ε.
This is easy to compute.
It is useful when the method is known to converge reliably.
But ε should be chosen in relation to the desired answer accuracy, and the method’s convergence properties should support the inference.
26. Stopping by residual
For solving f(x) = 0, another rule is stop when |f(xₙ)| < ε.
This checks how closely the original equation is satisfied.
It is often a better independent test than successive difference alone.
Using both can be stronger: the estimate is stable and the original equation residual is small.
27. Stopping by interval width
For a bracketed root, stop when b − a is below a chosen tolerance.
If the midpoint is reported, its maximum absolute error is at most half the interval width.
This gives a direct error guarantee.
Whenever a bracket is available, it is one of the cleanest stopping methods.
28. Stopping for significant figures requires relative thinking
To guarantee a certain number of significant figures, the absolute tolerance depends on the magnitude of the number.
An error of 0.001 may be tiny relative to 1000 but large relative to 0.01.
Relative error or a correct rounding interval is more appropriate than a fixed absolute tolerance across all scales.
29. A repeated decimal can be exact or approximate depending on context
1/3 = 0.333… exactly as a recurring decimal.
0.333 is an approximation.
Iteration may generate longer decimal approximations, but the exact mathematical object may have a symbolic representation.
Do not confuse “many digits” with “more exact” when an exact form is available and required.
30. Exact arithmetic should often be preserved until approximation is needed
If a problem gives √3 and later multiplies by 2√3, exact manipulation gives 6.
Replacing √3 early with 1.732 can introduce avoidable rounding error.
Numerical iteration belongs where exact solution is unavailable, impractical or not required—not as a default replacement for exact structure.
This complements the BTT guidance on preserving exact form where appropriate.
31. Iteration can diverge
If xₙ₊₁ = 3xₙ and x₀ = 0.1, the magnitude grows rapidly.
A solver should not continue indefinitely hoping that “more iterations” will eventually fix a divergent rule.
Inspect the update formula, starting value and fixed-point stability.
32. Iteration can cycle
xₙ₊₁ = 1 − xₙ with x₀ = 0 gives 0, 1, 0, 1, … .
The process enters a 2-cycle.
No single limit is reached.
Detecting repeated patterns can reveal cycling quickly.
33. Iteration can stagnate
A numerical process may stop changing because calculator precision has been exhausted.
That does not necessarily mean the true mathematical error is zero.
Stagnation at machine precision and mathematical convergence are different concepts.
For school calculations, the distinction usually matters only when many iterations are used, but the principle is important.
34. Starting values can matter
Some iterative methods converge from one starting value and fail from another.
Newton-type methods can jump to a different root, move into an undefined region or diverge if the starting point is poor.
A numerical answer should therefore record the method and starting value where the route depends on them.
35. Multiple roots can attract different starting values
For an equation with several roots, an iterative method may converge to whichever root lies in the attraction region of the starting value.
Finding one root numerically does not establish that the complete solution set has been found.
This connects iteration directly to Exhaustive Solutions and Completeness.
36. Numerical answers should return to the original problem
After an iteration converges, substitute the final approximation into the original equation or model.
Check domain restrictions, units, feasibility and requested precision.
A numerically stable estimate that violates the original context is not a valid answer.
Convergence belongs inside the larger operating manual, not outside it.
37. A practical convergence audit
Ask:
What sequence is being generated? What limit or root is expected? Is the sequence approaching one value or cycling/diverging? Is the error shrinking? What independent residual or bracket check is available? What accuracy does the question require? What precise rule tells me to stop?
A good numerical solution should be able to answer every one of these questions in proportion to the level of the task.
38. Independent practice
1. Starting from x₀ = 0, use xₙ₊₁ = (xₙ + 4)/2 to find x₁, x₂, x₃ and the fixed point.
2. Does xₙ₊₁ = xₙ + 2 converge for x₀ = 1?
3. Does xₙ = (−1)ⁿ/n converge? State the limit.
4. Does xₙ = (−1)ⁿ converge?
5. Find the fixed point of xₙ₊₁ = 0.6xₙ + 8.
6. Use one step of xₙ₊₁ = 1/2(xₙ + 10/xₙ) starting from x₀ = 3.
7. A root is known to lie in [1.4142, 1.4143]. What is the maximum error of the midpoint estimate?
8. Explain why |xₙ₊₁ − xₙ| being small does not always prove the true error is small.
9. For solving f(x) = 0, what is a residual?
10. Why can xₙ₊₁ = 3/xₙ fail when used to solve x² = 3?
11. Give one advantage of a bracket-width stopping rule.
12. Explain why finding one numerical root does not prove all roots have been found.
39. Worked answers
1. x₁ = 2, x₂ = 3, x₃ = 3.5. Fixed point L satisfies L = (L + 4)/2, so L = 4.
2. No. The sequence is 1, 3, 5, 7, … and grows without bound.
3. Yes. The magnitude 1/n tends to 0, so the alternating sequence converges to 0.
4. No. It alternates between 1 and −1 without shrinking.
5. L = 0.6L + 8, so 0.4L = 8 and L = 20.
6. x₁ = 1/2(3 + 10/3) = 19/6 ≈ 3.1666667.
7. Interval width is 0.0001, so midpoint maximum error is 0.00005.
8. The sequence may be stagnating or moving slowly while still far from the true solution. Successive difference measures movement between iterates, not direct distance to truth.
9. The residual is the magnitude |f(xₙ)|, measuring how closely the current estimate satisfies the original equation.
10. Starting at 1 gives 3, then 1, then 3, creating a 2-cycle rather than convergence.
11. It provides an explicit bound on the remaining root uncertainty.
12. An equation may have several roots and the chosen iteration may converge to only one depending on the starting value or method.
40. Continue through the School Mathematics manual
Use Mathematical Type Checking to ensure the iterates remain inside the legal domain, Exhaustive Solutions and Completeness when several roots may exist, and Construction, Verification and Impossibility for the evidence required to trust a numerical candidate.
Return to the BTT Mathematics Hub for Batch 05.