论文标题
用于求解扩展正常方程的随机双重和三重kaczmarz
Randomized double and triple Kaczmarz for solving extended normal equations
论文作者
论文摘要
随机Kaczmarz算法最近受到了广泛的关注，因为它的简单性，速度和近似求解大规模线性方程式的能力。在本文中，我们提出了随机的双重和三重kaczmarz算法来求解$ \ bf a^\ top ax a^\ top ax = a^\ top b-c $的扩展正常方程。所提出的算法避免形成$ \ bf a^\ top a $ applicitly，并为{\ it nutary} $ \ mbf a \ in \ mbbr^{m \ times n} $（完整等级或排名差，$ m \ geq n $或$ m <n $）。 {\ it紧密}上限显示了在提议的算法的均方根含义中显示指数收敛的，并给出了数值实验以说明理论结果。
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 $\bf A^\top Ax=A^\top b-c$. The proposed algorithms avoid forming $\bf A^\top A$ explicitly and work for {\it arbitrary} $\mbf A\in\mbbr^{m\times n}$ (full rank or rank deficient, $m\geq n$ or $m<n$). {\it 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.
