The computation of bounds for the norm of the error in the conjugate gradient algorithm
The computation of bounds for the norm of the error in the conjugate gradient algorithm
复制标题
共轭梯度算法中误差范数界限的计算
作者：
G. Meurant
在本文中，我们考虑了结合梯度（CG）算法中误差规范的计算估计值。在Golub and Meurant（1997）的论文中给出了公式。在这里，我们首先证明这些表达式确实是误差的A-norm的上限和下限。此外，从这些公式开始，我们研究了误差的L2-norm的计算。最后，我们定义了一种自适应算法，其中在运行CG时计算获得上限所需的极端特征值的近似值，从而改善了误差范围的上限。数值实验显示了该算法的有效性。
In this paper we consider computing estimates of the norm of the error in the conjugate gradient (CG) algorithm. Formulas were given in a paper by Golub and Meurant (1997). Here, we first prove that these expressions are indeed upper and lower bounds for the A-norm of the error. Moreover, starting from these formulas, we investigate the computation of the l2-norm of the error. Finally, we define an adaptive algorithm where the approximations of the extreme eigenvalues that are needed to obtain upper bounds are computed when running CG leading to an improvement of the upper bounds for the norm of the error. Numerical experiments show the effectiveness of this algorithm.