  The Kaczmarz method is an iterative numerical method for solving large and
sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have
developed an acceleration technique for the Kaczmarz method that minimizes the
distance to the desired solution in the direction of a full Kaczmarz step.
  The present paper generalizes this technique to an acceleration scheme that
minimizes the Euclidean norm error over an affine subspace spanned by a number
of previous iterates and one additional cycle of the Kaczmarz method. The key
challenge is to find a formulation in which all parameters of the least-squares
problem defining the unique minimizer are known, and to solve this problem
efficiently.
  A numerical experiment demonstrates that the proposed affine search has the
potential to clearly outperform the Kaczmarz and the randomized Kaczmarz
methods with and without the Gearhart-Koshy/Tam line-search.
