The need to estimate upper and lower bounds for matrix functions of the form \({\mathrm{trace}}(W^Tf(A)V)\), where the matrix \(A\in {{\mathbb {R}}}^{n\times n}\) is large and sparse, \(V,W\in {{\mathbb {R}}}^{n\times s}\) are block vectors with \(1\le s\ll n\) columns, and f is a function arises in many applications, including network analysis and machine learning. This paper describes the shifted extended global symmetric and nonsymmetric Lanczos processes and how they can be applied to approximate the trace. These processes compute approximations in the union of Krylov subspaces determined by positive powers of A and negative powers of \(A-\sigma I_n\), where the shift \(\sigma\) is a user-chosen parameter. When A is nonsymmetric, transposes of these powers also are used. When A is symmetric and \(W=V\), we describe how error bounds or estimates of bounds for the trace can be computed by pairs of Gauss and Gauss-Radau quadrature rules, or by pairs of Gauss and anti-Gauss quadrature rules. These Gauss-type quadrature rules are defined by recursion coefficients for the shifted extended global Lanczos processes. Gauss and anti-Gauss quadrature rules also can be applied to give estimates of error bounds for the trace when A is nonsymmetric and \(W\ne V\). Applications to the computation of the Estrada index for networks and to the nuclear norm of a large matrix are presented. Computed examples show the shifted extended symmetric and nonsymmetric Lanczos processes to produce accurate approximations in fewer steps than the standard symmetric and nonsymmetric global Lanczos processes, respectively.

需要估计形式为\（{\ mathrm {trace}}（W ^ Tf（A）V）\）的矩阵函数的上下界，其中{{\ mathbb {R} }} ^ {n \ times n} \）大而稀疏，\（V，W \ in {{\ mathbb {R}}} ^ {n \ times s \\）是\（1 \ le s \ ll n \）列，而f是许多应用程序中出现的函数，包括网络分析和机器学习。本文介绍了移位的扩展全局对称和非对称Lanczos过程以及如何将其应用于近似轨迹。这些过程计算由A的正幂和\（A- \ sigma I_n \）的负幂确定的Krylov子空间的并集中的近似值，其中移位\（\ sigma \）是用户选择的参数。当A不对称时，也会使用这些幂的转置。当A是对称且\（W = V \）时，我们描述如何通过成对的Gauss和Gauss-Radau正交规则或成对的Gauss和反Gauss正交规则来计算迹线的误差边界或边界估计。这些高斯类型的正交规则由移位扩展的全局Lanczos过程的递归系数定义。当A为非对称且\（W \ ne V \）时，也可以应用高斯和反高斯正交规则来给出轨迹的误差范围的估计。。介绍了在网络的Estrada指数的计算以及大型矩阵的核范数中的应用。计算的例子表明，与标准的对称和非对称全局Lanczos过程相比，扩展后的扩展对称和非对称Lanczos过程分别以更少的步长产生了精确的近似值。