Weighted Jacobi method

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

For the five-point discretization of $-\Delta +c$ on the unit square, the weighted iteration eigenvalues are $1-\omega [4{\sin }^2(\pi kh/2)+4{\sin }^2(\pi lh/2)+c{h}^2]/(4+c{h}^2)$. A fixed parameter converges for every mesh when $0<\omega \le 1$, but $\omega =1$ poorly damps the highest frequencies.

For the centered second-difference matrix with diagonal $-2$ and off-diagonals $1$, the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when $0<\omega \le 1$.

For modes with $k/(n+1)\in [1/2,1]$ and large $n$, the worst attenuation factor isIt is minimized by $\omega =2/3$, for which ${\mu }_{\omega }=1/3$.