Successive over-relaxation

Successive over-relaxation (SOR) is an iterative method for solving linear systems that scales each Gauss–Seidel update by a relaxation factor. Choosing the factor well can speed convergence.

Gauss–Seidel method: An iterative method for solving linear systems that updates each variable using the newest available values. SOR takes each Gauss–Seidel update and scales its change by a relaxation factor.
Linear system: A set of linear equations that can be written as a matrix equation. SOR approximates solutions to systems of this form.
Sparse matrix: A matrix in which most entries are zero. SOR can update sparse systems without storing or processing most zero entries.
Successive under-relaxation: A Gauss–Seidel-based iteration that uses a relaxation factor between zero and one. It damps updates, whereas SOR uses a factor greater than one to accelerate progress.
Relaxation method: An iterative approach that repeatedly adjusts an approximate solution to reduce an equation’s residual. SOR is a relaxation method whose updates are performed one variable at a time.
Matrix splitting: A decomposition of a matrix into parts used to construct an iterative solver. SOR derives its updates by splitting the system matrix into diagonal, lower, and upper parts.
Poisson equation: A partial differential equation relating the Laplacian of a function to a specified source term. Finite-difference Poisson problems produce systems often solved with SOR.
Conjugate gradient method: An iterative method for symmetric positive-definite linear systems based on conjugate search directions. It can converge much faster than SOR on large positive-definite systems, without coordinate-by-coordinate updates.
Relaxation parameter: A scalar that controls the size of an iterative method’s update. Its value determines how far SOR moves beyond or toward each Gauss–Seidel update.
Jacobi method: An iterative linear solver that computes each new coordinate using only values from the previous iterate. Its simultaneous updates contrast with SOR’s sequential, immediately reused updates.

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