Gauss–Legendre Quadrature

Gauss–Legendre quadrature is a high-precision numerical integration method that optimizes both the positions of interior evaluation nodes (the roots of Legendre polynomials Pn(t) on [-1, 1]) and their weights so an n-node rule integrates every polynomial of degree up to 2n - 1 with zero truncation error.

Interactive Experiment

Verify degree exactness and interval orientation in the Gauss–Legendre Quadrature interactive workspace:

1. Start with Integrand = Quadratic, Nodes = 2, and Interval = Symmetric ([-1, 1]). Confirm Quadrature and Exact integral both equal 2/3 with Absolute error = 0.
2. Switch Integrand to Quartic while keeping Nodes = 2. Observe that Quadrature drops to 2/9 against Exact integral = 0.4 (error 8/45).
3. Change Nodes to 3 and Interval to Shifted ([0, 2]). Because degree 4 is below 2(3) - 1 = 5, Quadrature matches 32/5 = 6.4 with zero error.
4. Select Integrand = Sextic with Nodes = 3 on Symmetric to see where the 3-node rule first incurs truncation error (Q3 = 6/25 = 0.24 vs. 2/7, error 8/175), then test Collapsed ([1, 1], zero weights) and Reversed ([1, -1], negative weights summing to -2).