Error in Numerical Integration

Numerical Analysis · Axiom Academy

LESSON Error in Numerical Integration Understanding and bounding approximation errors 1. Where Does Error Come From? The error arises because we approximate f(x) with a simpler function (polynomial) and integrate that instead. The difference between the true integral and our approximation is the truncation error . For a single interval [a,b], the trapezoid rule error involves the second derivative: For the composite rule with n subintervals: Simpson's rule error involves the fourth derivative—two orders better! A quadrature rule has degree of precision d if it integrates all polynomials of degree ≤ d exactly. 5. A Priori vs A Posteriori Estimates A priori: Bound error before computing, using bounds on derivatives. A posteriori: Estimate error after computing, by comparing results with different h.

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