论文标题
Kaczmarz和随机Kaczmarz方法的错误估计值
Error estimates of Kaczmarz and randomized Kaczmarz methods
论文作者
论文摘要
kaczmarz方法是一种迭代投影方案，用于求解连接系统$ ax = b $。后来，它扩展到了不一致且不良的线性问题。但是经典的kaczmarz方法对相邻方程的相关性很敏感。为了降低相关性对收敛速率的影响，分别提出了随机的kaczmarz方法和随机块kaczmarz方法。在当前文献中，这些方法的错误估计结果是根据错误$ \ | x_k-x _*\ | _2 $建立的，其中$ x _*$是线性系统的解决方案$ ax = b $。在本文中，我们根据Kunio Tanabe的收敛定理扩展了Kaczmarz和随机Kaczmarz方法的当前误差估计，并获得有关错误$ \ | x_k-P_ {n（a）} x_0-x_0-x_0-x^\ dgagger \ | _2 $ _2 $的误差的一般结果。
The Kaczmarz method is an iterative projection scheme for solving con-sistent system $Ax = b$. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the adjacent equations. In order to reduce the impact of correlation on the convergence rate, the randomized Kaczmarz method and randomized block Kaczmarz method are proposed, respectively. In the current literature, the error estimate results of these methods are established based on the error $\|x_k-x_*\|_2$, where $x_*$ is the solution of linear system $Ax=b$. In this paper, we extend the present error estimates of the Kaczmarz and randomized Kaczmarz methods on the basis of the convergence theorem of Kunio Tanabe, and obtain some general results about the error $\|x_k-P_{N(A)}x_0-x^\dagger\|_2$.
