Category:Gauss-Seidel Method

This category contains results about Gauss-Seidel Method.
Definitions specific to this category can be found in Definitions/Gauss-Seidel Method.

The Gauss-Seidel method is an iterative method of solving a system of simultaneous equations $\mathbf A \mathbf x = \mathbf b$.

For $3$ equations in $3$ unknowns:

\(\ds a_{1 1} x_1 + a_{1 2} x_2 + a_{1 3} x_3\)

\(=\)

\(\ds b_1\)

\(\ds a_{2 1} x_1 + a_{2 2} x_2 + a_{2 3} x_3\)

\(=\)

\(\ds b_2\)

\(\ds a_{3 1} x_1 + a_{3 2} x_2 + a_{3 3} x_3\)

\(=\)

\(\ds b_3\)

we may start with arbitrary solutions.

In many practical problems, $a_{1 1}$, $a_{2 2}$ and $a_{3 3}$ are large in comparison with $a_{i j}$ for $i \ne j$.

It is then convenient to take:

$x_1 = \dfrac {b_1} {a_{1 1} }$

\(\ds x_1\)

\(=\)

\(\ds \dfrac {b_1} {a_{1 1} }\)

\(\ds x_2\)

\(=\)

\(\ds \dfrac {b_2} {a_{2 2} }\)

\(\ds x_3\)

\(=\)

\(\ds \dfrac {b_3} {a_{3 3} }\)

as starting values.

Writing $\mathbf x_n$ for the column vector of values of $x_1$, $x_2$ and $x_3$ after the $n$th iteration, and:

\(\ds \mathbf L\)

\(=\)

\(\ds \begin {pmatrix} a_{1 1} & 0 & 0 \\ a_{2 1} & a_{2 2} & 0 \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}\)

\(\ds \mathbf U\)

\(=\)

\(\ds \begin {pmatrix} 0 & a_{1 2} & a_{1 3} \\ 0 & 0 & a_{2 3} \\ 0 & 0 & 0 \end {pmatrix}\)

the iterative relationship is:

$\mathbf x_{n + 1} = \mathbf L^{-1} \paren {\mathbf b - \mathbf U \mathbf x_n}$

where $n = 0, 1, 2, \ldots$

Note that:

$\mathbf L + \mathbf U = \mathbf A$

and:

$\mathbf x_0$ is the column vector of starting values.

The iterations are continued to convergence.