Estimates in quadratic formulas
Estimates in quadratic formulas
复制标题
二次公式中的估计
DOI：
10.1007/bf02142693
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发表时间：
1994
影响因子：
2.1
通讯作者：
Z. Strakoš
中科院分区：
文献类型：
--
作者：
G. Golub;Z. Strakoš
LETA是一个真实的对称阳性矩阵。 1B”，以及矩阵inversea -1的条目。 resp.f（a）= a-2是一个实际矢量。回想上述问题的确切算术解决方案，然后分析正交计算中的圆形误差的效果。相应的共轭梯度过程的收敛性甚至有限的精度计算也能够解释基于持续分数回报的正交计算以及物理化学和固态物理学计算中观察到的实验结果。
LetA be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systemsAx=b, minimizing quadratic functional minx(xTAx−2bTx) subject to the constraint ‖x‖=α, α<‖A−1b‖, and estimates for the entries of the matrix inverseA−1. All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound onuTF(A)u, whereF(A)=A−1 resp.F(A)=A−2,u is a real vector. This problem can be considered in terms of estimates in the Gauss-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauss quadrature forf(λ)=λ−1 and the rate of convergence of the corresponding conjugate gradient process holds true even for finite precision computation. This allows us to explain experimental results observed in quadrature calculations and in physical chemistry and solid state physics computations which are based on continued fraction recurrences.