A numerical analyst wants to find a root of f(x) = x³ − 2 on the interval [1, 2] to within error ε = 10⁻⁶. Approximately how many bisection iterations are required?
AAbout 6 iterations — one per decimal place of accuracy
BAbout 20 iterations — since (b − a)/2ⁿ ≤ ε gives 1/2ⁿ ≤ 10⁻⁶, so n ≥ log₂(10⁶) ≈ 20
CAbout 3 iterations — bisection converges quickly once near the root
DIt cannot be determined without knowing the derivative of f near the root
After n iterations, the interval width is (b − a)/2ⁿ = 1/2ⁿ. Setting 1/2ⁿ ≤ 10⁻⁶ gives 2ⁿ ≥ 10⁶, so n ≥ log₂(10⁶) ≈ 19.9, meaning about 20 iterations. This is bisection's linear convergence in action: each iteration buys only about 0.3 decimal digits of accuracy (since log₁₀(2) ≈ 0.301). The number of iterations can be computed in advance without any information about f's shape near the root — this predictability is one of bisection's strengths.
Question 2 Multiple Choice
Newton's method finds a root in 5 iterations; bisection requires 50 iterations on the same problem. A student concludes bisection is the inferior method in all practical situations. What is the critical flaw in this reasoning?
AThere is no flaw — Newton's method is strictly superior in all cases and bisection is obsolete
BBisection is faster for polynomial equations specifically, so the comparison is unfair
CNewton's method requires computing the derivative f'(x) and can diverge or cycle if the initial guess is poorly chosen; bisection requires no derivative and guarantees convergence from any valid bracket — making it indispensable when derivatives are unavailable or the function is ill-behaved
DBisection converges faster than Newton's for functions with multiple roots
Newton's method achieves quadratic convergence (doubling correct digits per step) but requires f'(x) and can fail spectacularly — cycling, diverging to infinity, or converging to the wrong root — if the starting point is unlucky. Bisection has no such failure modes: given a valid bracket, it always converges. In practice, hybrid methods like Brent's method combine bisection's safety with faster methods' speed. Understanding bisection is the conceptual anchor for all such strategies.
Question 3 True / False
After n bisection iterations starting from an interval [a, b], the interval containing the root is guaranteed to have width exactly (b − a)/2ⁿ, regardless of the function's behavior.
TTrue
FFalse
Answer: True
Each bisection step discards exactly half the remaining interval, no matter what the function looks like — whether it is steeply sloped, nearly flat, or has complicated curvature near the root. The only information used is the sign of f at the midpoint. This geometric halving is what makes bisection's convergence rate perfectly predictable: the width after n steps is always exactly (b − a)/2ⁿ. It is also what makes convergence 'linear' — the number of correct bits grows at exactly 1 per iteration.
Question 4 True / False
Bisection can be started from any two points a and b as long as f(a) ≠ f(b), without needing to check for a sign change.
TTrue
FFalse
Answer: False
Bisection requires f(a) and f(b) to have *opposite signs* — not merely different values. The Intermediate Value Theorem only guarantees a root in [a, b] when f(a) < 0 and f(b) > 0 (or vice versa). If f(a) and f(b) have the same sign, there may be zero roots or an even number of roots in [a, b], and bisection provides no guarantee. Finding a valid bracket [a, b] with a sign change is the human judgment step — bisection cannot automate it.
Question 5 Short Answer
Why is finding the initial bracket [a, b] the step that requires human judgment, and what mathematical condition must that bracket satisfy for bisection to be guaranteed to work?
Think about your answer, then reveal below.
Model answer: The initial bracket must satisfy f(a) · f(b) < 0 — that is, f must have opposite signs at the two endpoints. This is the condition required by the Intermediate Value Theorem to guarantee a root exists in [a, b]. Finding such a bracket requires knowledge of the function's behavior: one might plot f, evaluate it at several candidate points, or use domain knowledge about where roots should lie. Bisection cannot automate this step because it has no mechanism to search for sign changes — it can only refine a bracket once one is provided.
This is not a limitation of bisection but a fundamental division of labor: the IVT guarantees existence given the bracket; bisection converts that existence guarantee into a root estimate. The user's job is to supply the existence condition. Once a valid bracket is given, the algorithm takes over entirely and delivers a convergence guarantee that requires no further human input.