Exponential Convergence of a Randomized Kaczmarz Algorithm with Relaxation

The Kaczmarz method is a well-known iterative algorithm for solving linear system of equations Ax = b. Recently, a randomized version of the algorithm has been introduced. It was proved that for the system Ax = b or Ax approximate to b + r, where r is an arbitrary error vector, the randomized Kaczmarz algorithm converges with expected exponential rate. In the present paper, we study the randomized Kaczmarz algorithm with relaxation and prove that it converges with expected exponential rate for the system of Ax = b and Ax approximate to b + r. The numerical experiments of the randomized Kaczmarz algorithm with relaxation are provided to demonstrate the convergence results.