For a giant system of equations, factoring is too expensive — so guess the answer and REFINE it. Gauss–Seidel updates each unknown in turn using the newest values of the others, sweeping again and again until nothing changes. It never inverts anything, uses almost no memory, and converges fast when the diagonal dominates — which is why it (and its cousins) solve the huge sparse systems in physics and graphics. Slide to sweep toward the solution.
Gauss–Seidel is an ITERATIVE linear solver: solve each equation i for its own unknown using the latest available values of the others, xᵢ←(bᵢ−Σ₌ₖₖᵢ aᵢ₌x₌)/aᵢᵢ, sweeping repeatedly. It uses updated values immediately (unlike Jacobi), converges faster, and needs no matrix inversion or factorization — only O(nonzeros) per sweep and almost no extra memory. It CONVERGES for strictly diagonally-dominant or symmetric-positive-definite A. It (with SOR/multigrid acceleration) solves the huge sparse systems from PDEs, circuits, and graphics. A fail-loud self-check throws unless a diagonally-dominant 2×2 converges to its known solution. ◆ real numerical methods, node-verified.
Convergence is only GUARANTEED for diagonally-dominant / SPD systems — otherwise it can stall or diverge — and it’s linear (often accelerated by SOR or multigrid). The sweep update and diagonal-dominance condition are exact.