= Weighted Jacobi method
= Weighted Jacobi iteration
{synonym}
= Relaxed Jacobi method
{synonym}
= Weighted Jacobi
{synonym}
With relaxation parameter $\omega$, weighted Jacobi updates by
$$
x^{(\nu+1)}
=x^{(\nu)}+\omega D^{-1}(b-Ax^{(\nu)}).
$$
Its <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 $1-\omega\pm\omega\sqrt2/4$; a <Poisson equation> smoothing parameter must not be inferred from this different <matrix>.
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