标题： 右预处理GMRES方法用于任意奇异系统 显示英文标题

标题： Right preconditioned GMRES for arbitrary singular systems

摘要： 布朗和沃克（1997）表明，GMRES在任意 $ b, x_0 \in {\bf R}^n $ 情况下，若 $A$ 是范围对称的，即 $ A \in {\bf R}^{n \times n} $ 不发生中断，则确定 $ A x = b $ 的最小二乘解。 $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. 这确定了在任意（奇异）矩阵$A \in {\bf R}^{n \times n}$和$ b \in {\bf R}^n $下的$ A x = b $的最小二乘解$ x = CA^{\rm T} z $，不会出现分解失败。为了使该方法数值稳定，我们建议使用带有适当阈值参数的伪逆，以抑制在求解严重病态的Hessenberg系统时微小奇异值的影响，这些系统出现在GMRES求解不一致的范围对称系统时的Arnoldi过程中。 数值实验表明，当将$C$取为单位矩阵，以及将对角线元素为$A A^{\rm T}$的对角矩阵的逆矩阵时，即使$A$不是范围对称的，包括$ {\rm index}(A) >1$的情况，也能给出一个最小二乘解。 显示英文摘要

摘要： Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = CA^{\rm T} z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $A \in {\bf R}^{n \times n}$ and $ b \in {\bf R}^n $. To make the method numerically stable, we propose using the pseudoinverse with an appropriate threshold parameter to suppress the influence of tiny singular values when solving the severely ill-conditioned Hessenberg systems which arise in the Arnoldi process of GMRES when solving inconsistent range-symmetric systems. Numerical experiments show that the method taking $C$ to be the identity matrix and the inverse matrix of the diagonal matrix whose diagonal elements are the diagonal of $A A^{\rm T}$ gives a least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.