On computing quadrature-based bounds for the A-norm of the error in conjugate gradients
On computing quadrature-based bounds for the A-norm of the error in conjugate gradients
复制标题
计算共轭梯度误差 A 范数的基于正交的界限
作者：
G. Meurant;Petr Tichý
Golub和Meurant(Bit 37：687-705,1997)在他们的原文中建议使用Gauss、Gauss-Radau和Gauss-Lobatto求积来计算共轭梯度法中误差的A范数的界。使用相应雅可比矩阵的逆的(或其一阶或二阶修正)的(1，1)项来计算求积。由此产生的被称为CGQL的算法显式地计算雅可比矩阵的条目及其根据CG系数的修改。在本文中，我们利用CG计算隐式给出的Jacobi矩阵的Cholesky分解的事实。对于Gauss-Radau和Gauss-Lobatto求积，我们不计算修正的Jacobi矩阵的项，而是直接计算(修正的)Jacobi矩阵的Cholesky分解的项。与CGQL中使用的公式相比，这导致了更简单的公式。
In their original paper, Golub and Meurant (BIT 37:687–705, 1997) suggest to compute bounds for the A-norm of the error in the conjugate gradient (CG) method using Gauss, Gauss-Radau and Gauss-Lobatto quadratures. The quadratures are computed using the (1,1)-entry of the inverse of the corresponding Jacobi matrix (or its rank-one or rank-two modifications). The resulting algorithm called CGQL computes explicitly the entries of the Jacobi matrix and its modifications from the CG coefficients. In this paper, we use the fact that CG computes the Cholesky decomposition of the Jacobi matrix which is given implicitly. For Gauss-Radau and Gauss-Lobatto quadratures, instead of computing the entries of the modified Jacobi matrices, we directly compute the entries of the Cholesky decompositions of the (modified) Jacobi matrices. This leads to simpler formulas in comparison to those used in CGQL.