计算矩阵函数双线性形式的Krylov子空间算法的误差分析

矩阵函数的双线性形式u^Tf（A）v出现在很多应用问题中，其中u，v ∈ Rⁿ，A ∈ R^n×n，f（z）为给定的解析函数.开发其有效可靠的数值算法一直是近年来学术界所关注的问题，其中关于其数值算法的停机准则多种多样，但欠缺理论支持，可靠性存疑.本文将对矩阵函数的双线性形式u^Tf（A）v的数值算法和后验误差估计进行研究，给出其基于Krylov子空间算法的误差分析，导出相应的误差展开式，证明误差展开式的首项是一个可靠的后验误差估计，据此可以为算法设计出可靠的停机准则.

THE ERROR ANALYSIS OF THE KRYLOV SUBSPACE METHODS FOR COMPUTING THE BILINEAR FORM OF MATRIX FUNCTIONS

The bilinear form u^Tf(A)v of matrix functions is of wide interest in many applications, where u, v ∈ Rⁿ, A ∈ R^n×n, f(z) is a given analytic function. In recent years, the efficient and reliable numerical algorithms for the bilinear form has been a research focus. Although there are numerous stopping criteria, they lack solid theoretical supports, and the reliability is unknown. In this paper, we consider the posteriori error estimates for the errors of approximate solutions of the matrix functions u^Tf(A)v. We derive an error expansion and prove that the first term of the error expansion can used as a reliable stopping criterion.