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 equations. In this paper we propose double triple algorithms extended normal equations form $$\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}$$ . proposed avoid forming \mathbf{A}$$ explicitly work for arbitrary $$\mathbf{A}\in \mathbb {R}^{m\times n}$$ (full rank or rank-deficient, $$m\ge n$$ $$m<n$$ ). Tight upper bounds showing exponential convergence in mean square sense are presented numerical experiments given illustrate theoretical results.