GaussSeideli

Gauss-Seidel is an iterative method for solving linear systems of the form Ax = b. It improves on the Jacobi method by using updated component values as soon as they become available within each iteration, which often leads to faster convergence.

In each iteration, the components of the solution vector x are updated sequentially. For a system with n equations, the i-th component is computed as:

x_i^(k+1) = (b_i − sum_{j=1}^{i−1} a_{ij} x_j^(k+1) − sum_{j=i+1}^n a_{ij} x_j^(k)) / a_{ii},

assuming a_{ii} ≠ 0. Equivalently, with A = L + U where L is the lower triangular part including the diagonal and U is the strictly upper part, the iteration can be written as x^(k+1) = −(L)^{-1} U x^(k) + (L)^{-1} b.

Convergence of Gauss-Seidel depends on the properties of A. It is guaranteed to converge for matrices that are strictly diagonally dominant by rows, and for symmetric positive definite A it converges for any initial guess. More generally, convergence occurs when the spectral radius ρ((D+L)^{-1} (−U)) is less than 1, where D is the diagonal of A. Sassenfeld’s criterion provides a practical sufficient condition for convergence.

Gauss-Seidel is related to Jacobi as a stationary iterative method and is a special case of successive over-relaxation (SOR) with relaxation factor ω = 1. It is widely used for solving sparse or structured linear systems in numerical linear algebra, engineering, and scientific computing.