  The Kaczmarz algorithm is an iterative method for solving systems of linear
equations. We introduce a modified Kaczmarz algorithm for solving systems of
linear equations in a distributed environment, i.e. the equations within the
system are distributed over multiple nodes within a network. The modification
we introduce is designed for a network with a tree structure that allows for
passage of solution estimates between the nodes in the network. We prove that
the modified algorithm converges under no additional assumptions on the
equations. We demonstrate that the algorithm converges to the solution, or the
solution of minimal norm, when the system is consistent. We also demonstrate
that in the case of an inconsistent system of equations, the modified relaxed
Kaczmarz algorithm converges to a weighted least squares solution as the
relaxation parameter approaches $0$.
