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 \(\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}\). The proposed algorithms avoid forming \(\mathbf{A}^\top \mathbf{A}\) explicitly and work for arbitrary \(\mathbf{A}\in \mathbb {R}^{m\times n}\) (full rank or rank-deficient, \(m\ge n\) 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.

随机化的Kaczmarz算法最近由于其简单性，速度和近似求解大型线性方程组的能力而备受关注。在本文中，我们提出了随机的双重和三重Kaczmarz算法来求解形式为\（\ mathbf {A} ^ \ top \ mathbf {Ax} = \ mathbf {A} ^ \ top \ mathbf {b}-\\ mathbf {c} \）。所提出的算法避免显式地形成\（\ mathbf {A} ^ \ top \ mathbf {A} \）并为\ mathbb {R} ^ {m \ times n} \中的任意 \（\ mathbf {A} \（满等级或等级不足，\（m \ ge n \）或\（m <n \））。紧的 给出了算法均方指数收敛的上限，并进行了数值实验验证了理论结果。