In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.

翻译：在本说明中,我们利用“共振梯度”方法的多元先决条件来解决平行环境中的大型对称正确定线性系统。我们结合了一种专门的牛顿方法来解决矩阵方程式X ⁇ -1}=A和用于先决条件的Chebyshev多面体。我们建议简单修改一个参数,避免将极端电子元值组合起来,以加快趋同速度。我们提供了在平行环境中的非常大的矩阵(高达80亿未知数)的结果,显示拟议预设物类别的效率。