  The method of alternation projections (MAP) is an iterative procedure for
finding the projection of a point on the intersection of closed subspaces of an
Hilbert space. The convergence of this method is usually slow, and several
methods for its acceleration have already been proposed. In this work, we
consider a special MAP, namely Kaczmarz' method for solving systems of linear
equations. The convergence of this method is discussed. After giving its matrix
formulation and its projection properties, we consider several procedures for
accelerating its convergence. They are based on sequence transformations whose
kernels contain sequences of the same form as the sequence of vectors generated
by Kaczmarz' method. Acceleration can be achieved either directly, that is
without modifying the sequence obtained by the method, or by restarting it from
the vector obtained by acceleration. Numerical examples show the effectiveness
of both procedures.
