GuideLinear systemsintermediate

Jacobi method

The splitting A=L+D+U, the choice M=D that defines Jacobi, its component-wise iterative scheme and a worked example.

Taking M = D

$A$ is split into its strictly lower part $L$ , its diagonal $D$ and its strictly upper part $U$ . Jacobi picks the simplest preconditioner, $M=D$ , with $N=-(L+U)$ :

A=L+D+U,\qquad x^{(k+1)}=-D^{-1}(L+U)x^{(k)}+D^{-1}b

Component-wise, each unknown is solved from its equation using the previous iteration's values:

x_i^{(k+1)}=\frac{1}{a_{ii}}\left(b_i-\sum_{j\ne i} a_{ij}x_j^{(k)}\right),\qquad a_{ii}\ne 0

Derivation of the iteration matrix

DerivationDerivation: Jacobi iteration matrixView as its own page →

In each equation $i$ we solve for $x_i$ (possible because $a_{ii}\ne 0$ ):
$x_i=\frac{1}{a_{ii}}\left(b_i-\sum_{j\ne i}a_{ij}x_j\right)$
Writing this for all rows and separating the diagonal D from the parts L and U, the matrix form appears:
$x=D^{-1}\bigl(b-(L+U)x\bigr)=-D^{-1}(L+U)x+D^{-1}b$
As an iteration, it is the Jacobi scheme with iteration matrix H_J=−D⁻¹(L+U):
$x^{(k+1)}=-D^{-1}(L+U)x^{(k)}+D^{-1}b$