Stochastic algorithm corresponding to a general linear iterative process

Let u’=Mu+Nf be a general linear iterative process for solving the system Lu=f, with L=${\mathrm{L}}^{\mathrm{T}}$ and with 1=M+NL. Provided that Γ≡1/2(${\mathit{L}}^{\mathrm{-}1}$-${\mathit{ML}}^{\mathrm{-}1}$ ${\mathit{M}}^{\mathrm{T}}$ ${)}^{\mathrm{-}1}$ is a positive-definite matrix, it is shown that one can explicitly construct a corresponding stochastic algorithm which satisfies the homogeneous-state condition with respect to the probability distribution exp(-βS), where S=(1/2${\mathrm{u}}^{\mathrm{T}\mathrm{Lu}\mathrm{-}{\mathrm{f}}^{\mathrm{T}\mathrm{u}}}$. When ${\mathrm{M}}^{\mathrm{T}\mathrm{L}=\mathrm{LM}}$, the algorithm also satisfies the detailed-balance condition.