Numerical Linear Algebra

A weather model, a fluid simulation, a neural network with millions of weights — each reduces, ultimately, to solving gigantic linear systems. Gaussian elimination by hand is unthinkable; even by computer, naive elimination is too slow and unstable. Numerical linear algebra is the engineering that makes it possible: LU decomposition factors a matrix once for many solves, and iterative methods creep toward the answer for systems too vast to factor at all.