Simpson's rule uses a quadratic (degree-2) interpolating polynomial, yet it integrates cubic polynomials exactly. Why?
AA quadratic polynomial can implicitly represent a cubic through the midpoint evaluation, which doubles its effective degree
BThe error term in Simpson's rule involves a fourth derivative, which vanishes for cubic polynomials—a consequence of the symmetric three-point arrangement
CSimpson's rule uses adaptive step sizing that automatically improves for polynomial integrands
DThis is a coincidence specific to the interval [−1, 1] and does not hold for general intervals
This 'superconvergence' of Simpson's rule is the key insight of this topic. The error term for Simpson's rule (derived via Taylor series analysis of the residual) involves the fourth derivative f⁽⁴⁾(ξ) for some ξ in the interval. For any cubic polynomial, the fourth derivative is identically zero—so the error term vanishes exactly, even though the interpolating polynomial is only quadratic. This symmetry-driven cancellation means Simpson's rule integrates one degree higher than expected. Option A is the most tempting misconception: there is no adaptive mechanism; it is purely a consequence of the error term structure.
Question 2 Multiple Choice
A student wants very high accuracy for a smooth integrand over [0, 1] and decides to use a single 10-point Newton-Cotes formula instead of applying Simpson's rule repeatedly over many small subintervals. What problem are they likely to encounter?
AThe 10-point formula requires solving a 10×10 linear system, making it computationally prohibitive
BHigher-order Newton-Cotes formulas with equally-spaced points suffer from Runge's phenomenon: large oscillations near the endpoints cause the weights to become large and alternating in sign, potentially making accuracy worse
CThe 10-point formula has lower convergence order than Simpson's rule, so it always produces less accurate results
DNewton-Cotes formulas are only defined for intervals of the form [0, 1] and cannot be applied to general domains
Runge's phenomenon is the central practical limitation of high-order Newton-Cotes formulas. With equally-spaced quadrature points, the Lagrange interpolating polynomial oscillates wildly near the endpoints as the degree grows, regardless of how smooth the integrand is. This causes the Newton-Cotes weights to alternate in sign and grow large, leading to catastrophic cancellation in the weighted sum. For smooth functions, composite Simpson's rule with many small subintervals outperforms a single high-order Newton-Cotes formula while remaining numerically stable.
Question 3 True / False
Halving the step size h in the trapezoidal rule reduces the integration error by a factor of 4, because the trapezoidal rule has O(h²) global accuracy.
TTrue
FFalse
Answer: True
Correct. The global error of the composite trapezoidal rule is O(h²): if you halve h (doubling the number of subintervals), the error decreases by a factor of 2² = 4. For composite Simpson's rule, the global error is O(h⁴), so halving h reduces the error by a factor of 16. Understanding these convergence rates is essential for choosing between methods and for error analysis—it tells you how much work (in terms of function evaluations) is needed to achieve a desired accuracy.
Question 4 True / False
Higher-order Newton-Cotes formulas—those using more equally-spaced evaluation points—typically produce more accurate results than lower-order formulas like Simpson's rule for the same interval.
TTrue
FFalse
Answer: False
False. This is the misconception that Runge's phenomenon refutes. For high-degree Newton-Cotes formulas, the Lagrange interpolating polynomial oscillates wildly near the endpoints of equally-spaced nodes, causing the quadrature weights to become large and alternating in sign. The resulting formula can produce wildly inaccurate results even for smooth functions. In practice, Newton-Cotes formulas beyond degree 4 (Simpson's 3/8 rule) are rarely used directly. The preferred approach is composite quadrature: applying low-degree formulas over many small subintervals.
Question 5 Short Answer
Why are composite quadrature methods preferred over single high-order Newton-Cotes formulas in practice, even when high accuracy is required?
Think about your answer, then reveal below.
Model answer: High-order Newton-Cotes formulas with equally-spaced points suffer from Runge's phenomenon: the Lagrange interpolating polynomial oscillates increasingly near the endpoints as degree grows, making the integration weights large and alternating in sign. This causes numerical instability and potentially worse accuracy than lower-order formulas. Composite methods instead apply a stable low-degree formula (trapezoidal or Simpson's) repeatedly over many small subintervals of width h, achieving global accuracy of O(h²) or O(h⁴) that improves predictably as h decreases—without the instability of high-degree interpolation over the full interval.
The deeper lesson is that 'higher degree' does not automatically mean 'more accurate' in numerical methods. Stability and error behavior must both be considered. Composite quadrature exploits the fact that a low-degree formula applied over small intervals can be both accurate and numerically stable, whereas a single high-degree formula over the whole interval may be neither.