The need to compute matrix functions of the form f(A)v, where \(A\in {\mathbb {R}}^{N\times N}\) is a large symmetric matrix, f is a function such that f(A) is well defined, and v≠ 0 is a vector, arises in many applications. This paper is concerned with the situation when A is so large that the evaluation of f(A) is prohibitively expensive. Then, an approximation of f(A)v often is computed by applying a few, say 1 ≤ n ≪ N, steps of the symmetric Lanczos process to A with initial vector v to determine a symmetric tridiagonal matrix \(T_{n}\in {\mathbb {R}}^{n\times n}\) and a matrix \(V_{n}\in {\mathbb {R}}^{N\times n}\), whose orthonormal columns span a Krylov subspace. The expression V_nf(T_n)e₁∥v∥ furnishes an approximation of f(A)v. The evaluation of f(T_n) is inexpensive, because the matrix T_n is small. It is important to be able to estimate the error in the computed approximation. This paper describes a novel approach that is based on a technique proposed by Spalević for estimating the error in Gauss quadrature rules.

需要计算形式为f ( A ) v 的矩阵函数，其中\(A\in {\mathbb {R}}^{N\times N}\)是一个大的对称矩阵，f是一个函数，使得f ( A ) 是明确定义的，并且v ≠ 0 是一个向量，出现在许多应用中。本文关注的是当A太大以至于f ( A )的评估成本过高时的情况。然后，通常通过应用一些来计算f ( A ) v的近似值，比如 1 ≤ n ≪ N，使用初始向量v对A进行对称 Lanczos 过程的步骤，以确定对称三对角矩阵\(T_{n}\in {\mathbb {R}}^{n\times n}\)和矩阵\(V_{ n}\in {\mathbb {R}}^{N\times n}\)，其正交列跨越 Krylov 子空间。表达式V _n f ( T _n ) e ₁ ∥ v ∥ 提供了f ( A ) v的近似值。对f ( T _n )的评估并不昂贵，因为矩阵T _n是小。能够估计计算出的近似值的误差很重要。本文描述了一种新方法，该方法基于 Spalević 提出的用于估计高斯正交规则中的误差的技术。