A convergence accelerator of a linear system of equations based upon the power method
A convergence accelerator of a linear system of equations based upon the power method
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DOI：
10.1002/1097-0363(20010330)35:6
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发表时间：
2000-06
影响因子：
1.8
通讯作者：
A. Dagan
中科院分区：
文献类型：
--
作者：
A. Dagan
我们考虑迭代数值格式的收敛速度作为后处理器阶段加速的一种方法。本文采用的方法是：(1)消除包含在凸包原点中的残差特征模态；(2)采用主收敛算法对剩余残差项进行平滑处理。为此，采用多项式矩阵法推导两种不同方法的特征方程。第一种方法是基于向量缩放，第二种方法是基于标准方程方法。这两种方法的输入都是从主程序获得的两个连续迭代/周期级别之间的解差。由于矩阵的病态结构，这两种方法都采用了奇异值分解。在目前的工作中，理查森外推的显式形式的使用推翻了使用理查森迭代与Leja排序的需要。在二维拉普拉斯方程边值问题、三维多网格问题、球面上的势解问题和一维稳态Burger方程3个代表性问题上，将这些方法与GMRES算法的性能进行了比较
We consider the convergence rate of an iterative numerical scheme as a method for accelerating at the post-processor stage. The methodology adapted here is: (1) residual eigenmodes included in the origin of the convex hull are eliminated; (2) remaining residual terms are smoothed away by the main convergence algorithm. For this purpose, the polynomial matrix approach is employed for deriving the characteristic equation by two different methods. The first method is based on vector scaling and the second is based on the normal equations approach. The input for both methods is the solution difference between two consecutive iteration/cycle levels obtained from the main program. The singular value decomposition was employed for both methods due to the ill-conditioned structure of the matrices. The use of the explicit form of the Richardson extrapolation in the present work overrules the need to employ the Richardson iteration with a Leja ordering. The performance of these methods was compared with the GMRES algorithm for three representative problems: two-dimensional boundary value problem using the Laplace equation, three-dimensional multi-grid, potential solution over a sphere and the one-dimensional steady state Burger equation