Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain the problem is to compute a solution also on its interior. Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation.
They have also been developed for solving nonlinear systems of equations. They are also used for the solution of linear equations for linear least-squares problems and also for systems of linear inequalities, such as those arising in linear programming. Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. In numerical mathematics, relaxation methods are iterative methods for solving systems of equations, including nonlinear systems. For other uses, see Relaxation (disambiguation). This article is about iterative methods for solving systems of equations.
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