Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method
Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method
复制标题
共轭梯度法存在的充分必要条件
DOI：
10.1137/0721026
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发表时间：
1984
影响因子：
2.9
通讯作者：
T. Manteuffel
中科院分区：
文献类型：
--
作者：
V. Faber;T. Manteuffel
我们刻画了矩阵A的$CG(s)$类，其中线性系统$A{\bf x} = {\bf b}$可以用s项共轭梯度法求解。我们证明，除了少数异常外，类$CG(s)$由已知共轭梯度方法的矩阵a组成。这些矩阵是厄米矩阵，$A^ * = A$，以及形式为$Ae^{i\theta} (dI + B)$的矩阵，其中$B^ * = - B$。
We characterize the class $CG(s)$ of matrices A for which the linear system $A{\bf x} = {\bf b}$ can be solved by an s-term conjugate gradient method. We show that, except for a few anomalies, the class $CG(s)$ consists of matrices A for which conjugate gradient methods are already known. These matrices are the Hermitian matrices, $A^ * = A$, and the matrices of the form $Ae^{i\theta} (dI + B)$, with $B^ * = - B$.