论文标题
重新审视奇异系统的转基因
GMRES on singular systems revisited
论文作者
论文摘要
在[Hayami K，Sugihara M. Numer线性代数应用。 2011; 18：449--469]，作者分析了最小二乘问题的广义最小残差（GMRE）方法的收敛行为$ \ min_ {{\ bf x} \ in {\ bf r}^n} {\ bf b} - a {\ bf x} \ | _2}^2 $，其中$ a \ in {\ bf r}^{n \ times n} $可能是单数，$ {\ bf b b} \ in {\ bf r}正交补充$ {\ cal r}（a）^\ perp $组件。但是，我们发现，如果$ {\ cal r}（a）= {\ cal r}（a^{\ scriptssize t}）$未完成，则gmres给出了最小二乘解决方案的事实。在本文中，我们将提供一个完整的证明。
In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449--469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem $ \min_{ {\bf x} \in {\bf R}^n} {\| {\bf b} - A {\bf x} \|_2}^2$, where $ A \in {\bf R}^{n \times n}$ may be singular and $ {\bf b} \in {\bf R}^n$, by decomposing the algorithm into the range $ {\cal R}(A) $ and its orthogonal complement $ {\cal R}(A)^\perp $ components. However, we found that the proof of the fact that GMRES gives a least squares solution if $ {\cal R}(A) = {\cal R}(A^{\scriptsize T} ) $ was not complete. In this paper, we will give a complete proof.
