 Weighted Jacobi method

With relaxation parameter $\omega$, weighted Jacobi updates byIts iteration matrix is $H_{\omega }=I-\omega D^{-1}A$.

Equivalently, the weighted Jacobi method takes a weighted average of the old vector and its full Jacobi method update: $x^{(k+1)}=(1-\omega )x^{(k)}+\omega x_{\mathrm{Jacobi}}^{(k+1)}$. If $Bv=\lambda v$, its iteration matrix multiplies that eigenvector by $1-\omega +\omega \lambda$. For the diagonal-4 Jacobi iteration for a three-variable tridiagonal system, the factors are $1-\omega$ and $1-\omega \pm \omega \sqrt{2}/4$; a Poisson equation smoothing parameter must not be inferred from this different matrix.