On the Consistency of Linear Stationary Iterative Methods

Abstract

For a given matrix G and a given vector k the linear stationary iterative method $u^{(n + 1)} = Gu^{(n)} + k$ is consistent with the linear system $Au = b$ (where A is a given matrix and b is a given vector) if $\mathcal {S}(A,b) \subseteq \mathcal {S}(I - G,k)$. Here, in general, $\mathcal {S}(A,b)$ denotes the set of solutions of $Au = b$. The iterative method is reciprocally consistent if $\mathcal {S}(1 - G,k) \subseteq \mathcal {S}(A,b)$ and is completely consistent if $\mathcal {S}(1 - G,k) = \mathcal {S}(A,b)$. If $\mathcal {S}(A,b)$ is nonempty, then the method is consistent if and only if $G = I - MA$ and $k = Mb$ for some matrix M. The method is completely consistent if $G = I - MA$ and $k = Mb$ for some nonsingular matrix M. This generalizes the elementary result that if A is nonsingular, then the method is completely consistent if and only if $I - G$ is nonsingular and $k = (I - G)A^{ - 1} b$. It is also shown that if $S(I - G,k)$ is nonempty, then the method is reciprocally consistent if and only if $A = Q(I - G)$ and $b = Qk$ for some matrix Q.