Gauss-Seidel needs half the iterations of Jacobi, but modern compilers eat the gains

eigensteve · x · 2026-09-08

Dr. Jean-Christophe Loiseau benchmarks Gauss-Seidel vs. Jacobi iterative solvers on the 2D Poisson equation.

Key finding: Gauss-Seidel does require roughly half the iterations, both theoretically and empirically — but that does not translate into a 2x speedup, because modern compilers are so good at leveraging CPU instructions (e.g. vectorization) that the Jacobi solver's per-iteration efficiency narrows the gap. A follow-up to his earlier post on writing an efficient Jacobi solver in modern Fortran.

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