Richardson iteration ($x_{n+1}=x_n-\tau (A{x}_n-b)$)

For a real symmetric positive definite matrix $A$, the error in each eigenvector direction is multiplied by $1-\tau \lambda$. Convergence from every initial vector is therefore equivalent to $0<\tau <2/{\lambda }_{\max }$. At either endpoint there is a nondecaying error direction. The optimal constant parameter is $2/({\lambda }_{\min }+{\lambda }_{\max })$, giving contraction factor $({\lambda }_{\max }-{\lambda }_{\min })/({\lambda }_{\max }+{\lambda }_{\min })$.