Exploiting the Composite Step Strategy to the Biconjugate A-Orthogonal Residual Method for Non-Hermitian Linear Systems
Exploiting the Composite Step Strategy to the Biconjugate A-Orthogonal Residual Method for Non-Hermitian Linear Systems
复制标题
DOI：
10.1155/2013/408167
复制
发表时间：
2013-02
期刊：
影响因子：
--
通讯作者：
Y. Jing;Tingzhu Huang;B. Carpentieri;Y. Duan
中科院分区：
文献类型：
--
作者：
Y. Jing;Tingzhu Huang;B. Carpentieri;Y. Duan
双共轭A-正交残差（BiCOR）方法通过双共轭A-正交归一化过程在有限精度算术中进行，可能倾向于遭受两种来源的数值不稳定性，称为两种故障，类似于双共轭梯度（BCG）方法。本文自然地利用复合步骤BCG（CSBCG）方法发展到BiCOR方法中所采用的复合步骤策略来解决称为枢轴故障的故障之一。类似于CSBCG方法，所得到的有趣的变体，只有一个小的修改，通常的实现的BiCOR方法，能够避免近枢轴故障和计算所有定义良好的BiCOR迭代稳定的假设下，基本的双共轭A-正交归一化过程不中断。获得的另一个好处是，它似乎是一个可行的算法，提供了一些进一步的实际所需的平滑的残差的范数的收敛历史，这是合理的数值实验。此外，所展示的方法继承了BiCOR方法的经验观察到的稳定性和快速收敛速度的BCG方法，使其在一定程度上优于CSBCG方法的有前途的优点。
The Biconjugate A-Orthogonal Residual (BiCOR) method carried out in finite precision arithmetic by means of the biconjugate A-orthonormalization procedure may possibly tend to suffer from two sources of numerical instability, known as two kinds of breakdowns, similarly to those of the Biconjugate Gradient (BCG) method. This paper naturally exploits the composite step strategy employed in the development of the composite step BCG (CSBCG) method into the BiCOR method to cure one of the breakdowns called as pivot breakdown. Analogously to the CSBCG method, the resulting interesting variant, with only a minor modification to the usual implementation of the BiCOR method, is able to avoid near pivot breakdowns and compute all the well-defined BiCOR iterates stably on the assumption that the underlying biconjugate A-orthonormalization procedure does not break down. Another benefit acquired is that it seems to be a viable algorithm providing some further practically desired smoothing of the convergence history of the norm of the residuals, which is justified by numerical experiments. In addition, the exhibited method inherits the promising advantages of the empirically observed stability and fast convergence rate of the BiCOR method over the BCG method so that it outperforms the CSBCG method to some extent.