A function has a sharp spike in a small region and is nearly flat everywhere else. To achieve high accuracy with the fewest function evaluations, which approach is best?
AApply a very high-degree polynomial rule to the entire interval
BUse composite Simpson's rule with uniform subinterval spacing
CUse adaptive composite quadrature that refines subintervals near the spike
DUse the basic trapezoidal rule with a large fixed number of uniform panels
Adaptive composite quadrature concentrates evaluations where the function varies rapidly and uses coarser panels where it is smooth. A uniform mesh wastes evaluations on the flat regions. A single high-degree polynomial over the whole interval fits the spike poorly (Runge's phenomenon is a related failure). The key insight is that uniform spacing is optimal only for uniformly well-behaved functions; adapting the mesh to the function's local behavior delivers the same accuracy with far fewer evaluations.
Question 2 Multiple Choice
You apply composite Simpson's rule to an integral, then double the number of subintervals. By approximately what factor does the error decrease?
A2
B4
C8
D16
Composite Simpson's rule has error proportional to h⁴(b−a), where h is the subinterval width. Doubling n halves h, so the error scales as (h/2)⁴ = h⁴/16 — a factor of 16 reduction. This is why composite rules achieve high accuracy quickly: each doubling of the number of panels produces a large error reduction. Composite trapezoidal rule achieves only a factor of 4 per doubling (error ∝ h²). This rapid convergence is the main reason composite rules dominate practical integration.
Question 3 True / False
Using a single high-order polynomial rule over an entire interval is generally more accurate than a composite rule with the same total number of function evaluations.
TTrue
FFalse
Answer: False
This is a common misconception. For smooth, well-behaved functions, high-order rules can be very accurate. But for functions with local features, rapid variation, or near-singularities, high-degree polynomials over large intervals fit poorly — the Runge phenomenon is an extreme example. Composite rules use low-order formulas on small panels, so local behavior is well-captured everywhere. In practice, composite rules often win on accuracy-per-evaluation precisely because real-world integrands are rarely uniform in their smoothness.
Question 4 True / False
Doubling the number of subintervals in composite Simpson's rule reduces the error by approximately a factor of 16.
TTrue
FFalse
Answer: True
Composite Simpson's rule has global error O(h⁴), where h = (b−a)/n is the subinterval width. Doubling n halves h, so error scales as h⁴ → (h/2)⁴ = h⁴/16. This fourth-order convergence is what makes Simpson's rule far preferable to the trapezoidal rule (which has O(h²) error, giving only a factor of 4 improvement per doubling). The rapid convergence rate is the core reason to prefer composite Simpson's rule for smooth integrands.
Question 5 Short Answer
Why do composite quadrature rules typically outperform single high-order polynomial rules for integrands with local features or rapid variation? What is the structural reason for this advantage?
Think about your answer, then reveal below.
Model answer: Composite rules keep the polynomial degree low and reduce the panel width h, concentrating accuracy where it is needed. Error in a composite rule depends on h raised to a power, so smaller panels dramatically reduce local errors. A single high-order polynomial over the full interval must simultaneously represent all the function's variation — a task it fails when the function has sharp local features, since high-degree polynomial fits tend to oscillate (Runge's phenomenon). Composite rules sidestep this by never asking any single panel to represent more than a locally smooth piece of the function.
The structural insight is that dividing the domain trades polynomial degree for panel width, and panel width enters the error formula at a high power. Adaptive methods take this further: they automatically apply more panels (smaller h) where the integrand is difficult and fewer where it is easy, so the error budget is spent efficiently. This is why modern numerical integration libraries use adaptive composite schemes rather than fixed high-order rules.