用Chebyshev多项式加速的预处理子空间迭代法

Preconditioning Subspace Iteration Method Accelerated by Using Chebyshev Polynomial

研究了计算大型稀疏对称矩阵的若干个最大或最小特征值的问题的子空间迭代法.首先引入了加速子空间迭代法的Chebyshev迭代法和预处理技术.为了更好地加速子空间迭代法的收敛速度,作者把Chebyshev多项式和预处理技术同时应用到子空间迭代法中,对预处理过的残余矩阵用Chebyshev多项式加速.即讨论了Chebyshev迭代法对预处理子空间迭代法的应用.这样既缩小了矩阵特征值的分布范围,又改善了每次循环的初始矩阵.从而给出了用Chebyshev多项式加速的预处理子空间迭代法.最后给出了数值例子,结果表明加速后的预处理子空间迭代法比原来的预处理子空间迭代法更优越,进一步加速了迭代法的收敛速度,减少了计算量和计算时间.

The paper deals with the subspace iteration method for computing a few of the largest （or smallest） eigenvalues of a large sparse symmetric matrix. Firstly the Chebyshev iteration and the preconditioning techniques are considered for computing approximation of the large sparse matrix A, which accelerated the convergence rate of the subspace iteration method. In order to faster accelerate the convergence rate of the subspace iteration method. The Chebyshev polynomial and the preconditioning techniques are simultaneously used to the subspace iteration method, improving the preconditioned residual matrix with the Chebyshev polynomial. That is to discuss application of the Chebyshev iteration to the preconditioning subspace iteration method. This reduces the distribution range of eignvaluces and improves the starting matrix obtained from every iteration procedure. Then the preconditioning subspace iteration accelerated by using Chebyshev polynomial is presented. The numerical experiments show that the accelerated preconditioning subspace iteration method is more effective in convergence of algorithm than the original preconditioning subspace iteration method. And it decreases the computation cost and computation time.