Search results
Results from the WOW.Com Content Network
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel .
A multigrid method with an intentionally reduced tolerance can be used as an efficient preconditioner for an external iterative solver, e.g., [7] The solution may still be obtained in () time as well as in the case where the multigrid method is used as a solver.
To solve the equations, we choose a relaxation factor = and an initial guess vector = (,,,). According to the successive over-relaxation algorithm, the following table is obtained, representing an exemplary iteration with approximations, which ideally, but not necessarily, finds the exact solution, (3, −2, 2, 1) , in 38 steps.
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...
Gauss–Seidel method. Successive over-relaxation (SOR) — a technique to accelerate the Gauss–Seidel method Symmetric successive over-relaxation (SSOR) — variant of SOR for symmetric matrices; Backfitting algorithm — iterative procedure used to fit a generalized additive model, often equivalent to Gauss–Seidel; Modified Richardson ...
It is similar to the Jacobi and Gauss–Seidel method. We seek the solution to a set of linear equations, expressed in matrix terms as =. The Richardson iteration is ...
Gauss–Jordan elimination: solves systems of linear equations; Gauss–Seidel method: solves systems of linear equations iteratively; Levinson recursion: solves equation involving a Toeplitz matrix; Stone's method: also known as the strongly implicit procedure or SIP, is an algorithm for solving a sparse linear system of equations
When approximating the constraints locally to first order, this is the same as the Gauss–Seidel method. For small matrices it is known that LU decomposition is faster. Large systems can be divided into clusters (for example, each ragdoll = cluster). Inside clusters the LU method is used, between clusters the Gauss–Seidel method is used. The ...