Relaxation (iterative method)

In numerical mathematics splitting methods are iterative methods for solving linear systems of equations with a matrix and right-hand side in contrast to the direct process one approaches it from a initial guess gradually the desired solution, and aborts if the accuracy is high enough.

Description

The process results in a splitting of the system matrix with an invertible matrix.

This yields the fixed point equation

With results in the fixed point iteration

You can abort the iteration if the norm of the residual falls below a given error bound. The method converges if and only if the spectral radius of the matrix M is less than 1. With the help of the Banach fixed point theorem also follows the linear convergence speed of the entire process class. The lower the spectral, the faster the process converges. Can If B and A differ only slightly one point the Störungslemma that also the spectral radius of M is small. Thus a contrast results of faster convergence (B approximates A very good) at a low cost per iteration ( B is simply invertible ). Overall, these methods are too slow for many practical problems. However, they provide due to their simple applicability good preconditioner dar. In addition, many splitting methods as smoothers in a multigrid algorithm suitable.

Examples

• Jacobi method: is the diagonal of A.
• Richardson method: with a parameter.
• Gauss - Seidel method: the lower left triangular matrix diagonal.
• Others are the SOR method and SSOR.
• A way of iterative refinement for the Gaussian elimination method.

Modifications

A distinction is made between stationary process with constant iteration and non-stationary processes, where the matrices may depend on the index i.