A Comparison of Adaptive Chebyshev and Least Squares Polynomial Preconditioning for Hermitian Positive Definite Linear Systems
A Comparison of Adaptive Chebyshev and Least Squares Polynomial Preconditioning for Hermitian Positive Definite Linear Systems
复制标题
DOI：
10.1137/0913001
复制
发表时间：
1992
期刊：
影响因子：
--
通讯作者：
S. Ashby;T. Manteuffel;J. S. Otto
中科院分区：
文献类型：
--
作者：
S. Ashby;T. Manteuffel;J. S. Otto
本文探讨了Hermitian正定线性方程组Ax = B$的自适应多项式预处理方法。这种预处理器易于使用，并且非常适合于矢量和/或并行机器。在考察了多项式预处理在共轭梯度法中的作用后，讨论了最小二乘和切比雪夫预处理多项式。特征值分布，其中每一个是非常适合的，然后确定。动态计算最佳的切比雪夫多项式预条件的自适应过程也被描述。最后，在Cray X-MP/48和Alliant FX/8上的各种数值实验中证明了自适应多项式预处理的有效性。结果表明，相对较低的次数（2-16）多项式通常是最好的。
This paper explores the use of adaptive polynomial preconditioning for Hermitian positive definite linear systems, $Ax = b$. Such preconditioners are easy to employ and well suited to vector and/or parallel machines. After examining the role of polynomial preconditioning in conjugate gradient methods, the least squares and Chebyshev preconditioning polynomials are discussed. Eigenvalue distributions for which each is well suited are then determined. An adaptive procedure for dynamically computing the best Chebyshev polynomial preconditioner is also described. Finally, the effectiveness of adaptive polynomial preconditioning is demonstrated in a variety of numerical experiments on a Cray X-MP/48 and Alliant FX/8. The results suggest that relatively low degree (2–16) polynomials are usually best.