  This paper studies the convergence rate of a message-passing distributed
algorithm for solving a large-scale linear system. This problem is generalised
from the celebrated Gaussian Belief Propagation (BP) problem for statistical
learning and distributed signal processing, and this message-passing algorithm
is generalised from the well-celebrated Gaussian BP algorithm. Under the
assumption of generalised diagonal dominance, we reveal, through painstaking
derivations, several bounds on the convergence rate of the message-passing
algorithm. In particular, we show clearly how the convergence rate of the
algorithm can be explicitly bounded using the diagonal dominance properties of
the system. When specialised to the Gaussian BP problem, our work also offers
new theoretical insight into the behaviour of the BP algorithm because we use a
purely linear algebraic approach for convergence analysis.
