Randomized double and triple Kaczmarz for solving extended normal equations

The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems of equations. In this paper we propose randomized double and triple Kaczmarz algorithms to solve extended normal equations of the form A(Tau)Ax = A(Tau)b - c. The proposed algorithms avoid forming A(Tau)A explicitly and work for arbitrary A is an element of R-mxn (full rank or rank-deficient, m >= n or m < n). Tight upper bounds showing exponential convergence in the mean square sense of the proposed algorithms are presented and numerical experiments are given to illustrate the theoretical results.