  We propose a simple doubly stochastic block Gauss--Seidel algorithm for
solving linear systems of equations. By varying the row partition parameter and
the column partition parameter of the coefficient matrix, we recover the
Landweber algorithm, the randomized Kaczmarz algorithm, the randomized
Gauss--Seidel algorithm, and the doubly stochastic Gauss--Seidel algorithm. For
general (consistent or inconsistent) linear systems, we show the exponential
convergence of the {\it norms of the expected iterates} via exact formulas. For
consistent linear systems, we prove the exponential convergence of the {\it
expected norms of the error and the residual}. Numerical experiments are given
to illustrate the efficiency of the proposed algorithm.

Numerical Analysis;Numerical Analysis