Jacobi iteration

The iterative method for solving a linear system that updates every unknown at once, each time using only the previous round's values for the others.

Jacobi iteration solves $A\mathbf{x}=\mathbf{b}$ by splitting $A=D+R$ ( $D$ diagonal, $R$ the rest) and iterating $\mathbf{x}^{(k+1)}=D^{-1}\big(\mathbf{b}-R\mathbf{x}^{(k)}\big)$ -- every entry of $\mathbf{x}^{(k+1)}$ uses only entries of the OLD iterate $\mathbf{x}^{(k)}$ , never an entry already updated this round.

Examples

Gauss-Seidel is Jacobi's close cousin: it uses newly updated entries immediately instead of waiting for the next round.