A Numerical Study for MrR Algorithm for Linear Equations with Symmetric Matrices

IF 0.4 Q4 ENGINEERING, MULTIDISCIPLINARY Journal of Advanced Simulation in Science and Engineering Pub Date : 2019-01-01 DOI:10.15748/JASSE.6.128
Kuniyoshi Abe, S. Fujino, S. Ikuno
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Abstract

We treat with Krylov subspace methods for efficiently solving linear equations. AZMJ variant of Orthomin(2) [1, 2] (abbreviated as AZMJ) has been proposed for solving the linear equations. In this paper, we redesign an alternative AZMJ variant, i.e., propose an alternative minimum residual method for symmetric matrices using the coupled twoterm recurrences formulated by Rutishauser. The recurrence coefficients are determined by imposing the A-orthogonality on the residuals as well as the Conjugate Residual (CR) method. The proposed variant is referred to as MrR. It is mathematically equivalent to CR and AZMJ, but the implementations are different; the recurrence formulas contain alternative expressions for the auxiliary vector and the recurrence coefficients. Moreover, we derive a preconditioned MrR algorithm. By numerical experiments on the linear equations with real symmetric matrices, we demonstrate that the residual norms of MrR converge faster than those of CG and AZMJ, and the preconditioned MrR algorithm is effective.
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对称矩阵线性方程MrR算法的数值研究
我们用Krylov子空间方法处理线性方程的有效解。提出了Orthomin(2)[1,2]的AZMJ变体(简称AZMJ)来求解线性方程。在本文中,我们重新设计了一种备选的AZMJ变体,即利用Rutishauser的耦合两项递推式,提出了对称矩阵的备选最小残差法。通过对残差施加a正交性和共轭残差(CR)方法确定递归系数。提议的变体被称为MrR。它在数学上等同于CR和AZMJ,但实现是不同的;递归公式包含辅助向量和递归系数的替代表达式。此外,我们还推导了一个预置的MrR算法。通过对实对称矩阵线性方程组的数值实验,证明了MrR的残差范数收敛速度快于CG和AZMJ的残差范数收敛速度,证明了预条件MrR算法是有效的。
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