Numerical Integration Summary

Numerical Analysis · Axiom Academy

Key concepts from trapezoid rule to Gaussian quadrature Trapezoid Rule: Linear interpolation between endpoints. Error O(h²). Simple but less accurate. Simpson's Rule: Parabolic interpolation through 3 points. Error O(h⁴). Exact for cubics! Composite Rules: Divide interval into n subintervals and sum. More points = better accuracy. Trade-off: Higher-order Newton-Cotes (n>7) can have negative weights causing instability. Key Idea: Choose both nodes AND weights optimally, not just weights. Exactness: n points exact for polynomials up to degree 2n-1 (twice Newton-Cotes!). Standard Interval: Defined on [-1, 1]. Use change of variables for other intervals. Legendre Nodes: Optimal positions are roots of Legendre polynomials. Choosing a Method: Decision Process Can you choose evaluation points? Yes → Consider Gaussian quadrature for optimal accuracy. Are points already fixed? Yes → Use Newton-Cotes (Trapezoid or Simpson's). Is the function smooth? Yes → Simpson's typically beats Trapezoid. No → Trapezoid may be more robust. Need adaptive precision? Yes → Use adaptive quadrature with error estimation. High dimensions? Yes → Consider Monte Carlo or sparse grids (curse of dimensionality!). Romberg Integration: Richardson extrapolation on Trapezoid sequence. Efficient for smooth functions. Adaptive Quadrature: Subdivide where error is large. Optimal for varying complexity. Improper Integrals: Use substitution to transform infinite limits or singularities.

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