In this paper, we analyze the convergence behavior of the randomized extended Kaczmarz (REK) method for all types of linear systems (consistent or inconsistent, overdetermined or underdetermined, full-rank or rank-deficient). The analysis shows that the larger the singular value of $A$ is, the faster the error decays in the corresponding right singular vector space, and as $k\rightarrow\infty$, $x_{k}-x_{\star}$ tends to the right singular vector corresponding to the smallest singular value of $A$, where $x_{k}$ is the $k$th approximation of the REK method and $x_{\star}$ is the minimum $\ell_2 $-norm least squares solution. These results explain the phenomenon found in the extensive numerical experiments appearing in the literature that the REK method seems to converge faster in the beginning. A simple numerical example is provided to confirm the above findings.
翻译:在本文中,我们分析了所有类型的线性系统(一致或不一致、高定或低定、全位或低位)随机扩展卡茨马尔兹(REK)法(REK)的趋同行为。 分析表明,单价美元越大,错误在相应的右向单矢量空间中越快衰减,而以美元(rightarrow\infty$),美元($x ⁇ k}-x ⁇ ztar}$,其右单值相当于最小单价$($A$),美元($x ⁇ k})是REK方法的美元近似值,美元($x ⁇ star}$($x ⁇ _2美元)是最小最低平方。这些结果解释了在文献中出现的大量数字实验中发现的现象,即REK方法在开始时似乎会比较快。提供了一个简单的数字示例,以证实上述结果。