Gauss–Seidel method

An iterative method for solving a linear system by updating each variable in turn, using newly computed values immediately. Its convergence depends on the system matrix and the initial approximation.

Jacobi method: An iterative method for linear systems that computes every new variable from the previous iteration’s values. It uses the same componentwise equations but delays all updates until the next iteration.
Sparse matrix: A matrix in which most entries are zero. Sparse systems let implementations perform Gauss–Seidel sweeps without processing every matrix entry.
Linear system: A set of linear equations that must hold simultaneously for a shared set of unknowns. The method seeks an approximate solution to this system by repeated sweeps.
Conjugate gradient method: An iterative method for symmetric positive-definite linear systems that minimizes error over expanding Krylov subspaces. It can converge much faster than stationary Gauss–Seidel sweeps on suitable large systems.
Successive over-relaxation: An iterative method that accelerates Gauss–Seidel updates by applying a relaxation factor. It modifies each Gauss–Seidel correction to improve convergence when a suitable factor is known.
Red-black ordering: A grid-based ordering that partitions unknowns into two groups, often by checkerboard color. This ordering exposes parallel updates for certain Gauss–Seidel schemes on grid problems.
Residual (numerical analysis): The difference between a system’s right-hand side and the product of its matrix with an approximate solution. Residual size measures how well a current Gauss–Seidel iterate satisfies the equations.
GMRES: A Krylov subspace method for solving nonsymmetric linear systems by minimizing the residual over a growing subspace. It handles many nonsymmetric systems for which Gauss–Seidel may converge poorly.
Matrix splitting: A decomposition of a matrix into parts used to construct iterative methods for linear systems. Gauss–Seidel derives its iteration by splitting the system matrix into diagonal, lower, and upper parts.
Multigrid method: A family of solvers that reduces error across multiple spatial scales using grids of different resolutions. Gauss–Seidel often smooths oscillatory error between multigrid restriction and interpolation steps.

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Direct methodCompared with: Its convergence depends on repeated updates, unlike the prescribed elimination steps of a direct solver.