Gauss-Seidel method

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Introduction

We seek the solution to set of linear equations:

$A \phi = b$

In matrix terms, the the Gauss-Seidel iteration can be expressed as

$\phi^{(k)} = \left( {D - L} \right)^{ - 1} \left( {U\phi^{(k - 1)} + b} \right),$

where $D$ , $L$ and $U$ represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix $A$ and $k$ is the iteration count. This matrix expression is mainly of academic interest, and is not used to program the method. Rather, an element-based approach is used:

$\phi^{(k)}_i = \frac{1}{a_{ii}} \left(b_i - \sum_{j<i}a_{ij}\phi^{(k+1)}_j-\sum_{j>i}a_{ij}\phi^{(k)}_j\right),\, i=1,2,\ldots,n.$

Note that the computation of $\phi^{(k)}_i$ uses only those elements of $\phi^{(k+1)}$ that have already been computed. This means that no additional storage is required, and the computation can be done in place ( $\phi^{(k+1)}$ replaces $\phi^{(k)}$ ). While this might seem like a rather minor concern, for large systems it is unlikely that every iteration can be stored. Thus, unlike the Jacobi method, we do not have to do any vector copying should we wish to use only one storage vector. The iteration is generally continued until the changes made by an iteration are below some tolerance.