Method – Gauss-Jacobi. – Math-reference.net

Method

The Gauss-Jacobi method is a numerical method for solving a system of linear equations. Given a set of $n$ equations with $n$ variables,

$\begin{array}{ccccccccc} a_1 x_{1} &+& a_2 x_{2} & + & ... & + & a_n x_{n} & = & A\\\\ b_1 x_{1} &+& b_2 x_{2} & + & ... & + & b_n x_{n} & = & B \\ \vdots& & \vdots & & & & \vdots \\ c_1 x_{1} &+& c_2 x_{2} & + & ... & + & c_n x_{n} & = & C \end{array}$

Where $A$ , $B$ and $C$ are real numbers. We rearrange the above system of equations into the form,

$\begin{array}{cccccccccc} x_{1} & = & \cfrac{A}{a_1} & - & \cfrac{a_2}{a_1} x_{2} & - & ... & - & \cfrac{a_n}{a_1} x_{n} \\ x_{2} & = & \cfrac{B}{b_2} & - & \cfrac{b_1}{b_2} x_{1} & - & ... & - & \cfrac{b_n}{b_2} x_{n} \\ \vdots& & & \vdots & & & \vdots \\ x_{n} & = & \cfrac{C}{c_n} & - & \cfrac{c_2}{c_n} x_{1} & - & ... & - & \cfrac{c_{n-1}}{c_n} x_{n} \end{array}$

The above system of rearranged equations can be rewritten in the form

$\mathbf{x_{i+1}} = \mathbf{Gx_i} + \mathbf{h}$

Where $\mathbf{x_{i+1}}$ , $\mathbf{h}$ , $\mathbf{x_i}$ are vectors, and $\mathbf{G}$ is a matrix. Using some initial guess vector $\mathbf{V} = (x'_1, x'_2, ..., x'_n)$ , the values of $\mathbf{x_{i+1}}$ converge to the solutions of the system of equations.

Convergence

The Gauss-Jacobi method for a set of linear equations of the form

$\mathbf{Ax} = \mathbf{b}$

is guaranteed to converge if $\mathbf{A}$ is diagonally dominant. This does not imply however that if $\mathbf{A}$ is not diagonally dominant that the method will fail, as diagonal dominance is a sufficient but not necessary condition.