Gaussian Quadrature

Numerical Analysis · Axiom Academy

Optimal node placement for maximum accuracy Newton-Cotes uses equally-spaced points. But what if we could choose WHERE to sample? Gauss's insight: Optimize both node positions and weights to maximize accuracy. The optimal nodes on [-1,1] are the roots of Legendre polynomials: The key advantage: n-point Gaussian quadrature is exact for polynomials up to degree 2n-1 ! Gauss formulas work on [-1,1]. To integrate on [a,b], transform: The transformation: x = (b-a)t/2 + (a+b)/2, where t ∈ [-1,1] 5. When to Use Gaussian Quadrature Maximum accuracy for given number of points Excellent for smooth functions Irrational node positions (need tables) Less intuitive than Newton-Cotes

This is the written version of the interactive lesson above. See the full Numerical Analysis course.