A 5-point Gauss-Legendre rule is compared to a 50-point trapezoidal rule for integrating a smooth, analytic function. Which claim best describes the likely outcome?
AThe 50-point trapezoidal rule is more accurate because it uses more evaluation points
BThe 5-point Gauss-Legendre rule may well outperform the 50-point trapezoidal rule because it converges exponentially for smooth integrands
CBoth achieve similar accuracy because both are quadrature rules using weighted sums
DThe trapezoidal rule is better because equally spaced nodes avoid aliasing errors
Gaussian quadrature converges exponentially for smooth analytic functions — doubling the number of nodes can square the accuracy. Newton-Cotes rules (trapezoidal, Simpson's) converge only algebraically at rate O(h^p). In practice, a 5-point Gauss-Legendre rule routinely outperforms much larger trapezoidal rules on smooth integrands. The key is that Gaussian quadrature optimizes both node locations and weights, squeezing maximum polynomial exactness from each evaluation.
Question 2 Multiple Choice
What is the fundamental freedom that allows Gaussian quadrature to achieve exactness for polynomials of degree up to 2n−1 using only n points?
AGaussian quadrature evaluates the integrand at complex-valued points, accessing more information
BGaussian quadrature uses adaptive refinement to concentrate points where the function varies most
CGaussian quadrature optimizes both node locations and weights simultaneously, doubling the free parameters compared to fixed-node rules
DGaussian quadrature applies a correction term derived from the function's derivatives at the boundary
Newton-Cotes rules fix nodes at equally spaced points first, then solve for weights — only n+1 free parameters (the weights) are used to match polynomial conditions. Gaussian quadrature treats both node locations and weights as free, giving 2n free parameters for n nodes. This is enough to enforce exactness for all polynomials up to degree 2n−1, roughly double what equally-spaced nodes can achieve.
Question 3 True / False
Gaussian quadrature with n points is exact for polynomials of degree up to n.
TTrue
FFalse
Answer: False
Gaussian quadrature with n points is exact for polynomials of degree up to 2n−1 — roughly twice what you might expect. This is the central payoff of optimizing node locations: with n nodes and n weights (2n free parameters), you can match the 2n conditions required to integrate all polynomials through degree 2n−1 exactly. A common misconception is to expect only degree n exactness, conflating Gaussian quadrature with Newton-Cotes rules.
Question 4 True / False
One practical disadvantage of Gaussian quadrature is that increasing the node count from n to n+1 requires computing an entirely new set of nodes and weights, discarding the previous n function evaluations.
TTrue
FFalse
Answer: True
Gaussian nodes (roots of orthogonal polynomials) are not nested — the n-point rule and the (n+1)-point rule use completely different node locations. This means you cannot reuse prior function evaluations when refining accuracy, unlike equally-spaced rules where you can insert midpoints. This is a real practical cost when adaptive accuracy control is needed. Gauss-Kronrod rules partially address this by nesting extended rule families.
Question 5 Short Answer
Why are the optimal nodes for Gaussian quadrature placed at the roots of orthogonal polynomials rather than at equally spaced points?
Think about your answer, then reveal below.
Model answer: The roots of orthogonal polynomials are precisely the node locations that maximize the degree of polynomial exactness for a given number of evaluations. Orthogonality ensures that the n-th degree polynomial (whose roots give the nodes) is orthogonal to all lower-degree polynomials, which is the algebraic condition that guarantees the resulting quadrature rule is exact for polynomials up to degree 2n−1. Equally spaced nodes fix locations before optimizing weights, sacrificing roughly half the potential exactness.
The connection between orthogonal polynomials and optimal quadrature is one of the beautiful results of approximation theory. The orthogonality condition is not merely convenient — it is precisely what makes the node placement optimal. Different weight functions and domains (Gauss-Hermite for ∫e^{-x²}f dx, Gauss-Laguerre for ∫₀^∞ e^{-x}f dx) require different orthogonal polynomial families, but the principle is always the same: orthogonal roots = optimal nodes.