Course Summary

Numerical Analysis · Axiom Academy

SUMMARY Numerical Analysis Course Review A comprehensive review of all course topics and key insights Methods for solving f(x) = 0. Bisection is reliable but slow (linear convergence). Newton's method converges quadratically but requires derivatives and a good initial guess. Secant method approximates Newton without derivatives. Fixed-point iteration transforms the problem to x = g(x) and converges when |g'(x)| < 1. Constructing functions through data points. Lagrange interpolation gives an explicit formula but is expensive to modify. Newton's divided differences enable efficient computation and updating. Cubic splines avoid oscillation (Runge phenomenon) by using piecewise polynomials with smooth joins. Approximating definite integrals. Trapezoidal rule is simple with O(h²) error. Simpson's rule uses parabolas for O(h⁴) accuracy. Gaussian quadrature achieves maximum accuracy for polynomial degree with optimal node placement. Composite rules apply basic rules to subintervals for better accuracy. Unit 4: Numerical Differentiation Estimating derivatives from discrete data. Forward/backward differences have O(h) error. Central differences achieve O(h²) by symmetric sampling. Richardson extrapolation combines approximations at different step sizes to eliminate leading error terms. Beware: smaller h increases roundoff error.

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