Jacobi Method

Numerical Analysis · Axiom Academy

The simplest iterative method for linear systems Solving One Equation at a Time The Jacobi method has a beautifully simple idea: solve each equation for its "own" variable, using the previous iteration's values for all other variables. In matrix form, we split A = D + L + U (diagonal, lower, upper): Let's solve this system using Jacobi, starting from x⁽⁰⁾ = (0, 0, 0): Jacobi converges if the iteration matrix G = D⁻¹(L+U) has spectral radius ρ(G) < 1. A sufficient (not necessary) condition: You've learned the Jacobi method—simple, parallel, but sometimes slow!

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