Journal of Computational and Applied Mathematics ( IF 2.1 ) Pub Date : 2020-06-03 , DOI: 10.1016/j.cam.2020.113027 Carl Jagels , Khalide Jbilou , Lothar Reichel
The need to evaluate expressions of the form trace , where the matrix is symmetric, with , and is a function defined on the convex hull of the spectrum of , arises in many applications including network analysis and machine learning. When the matrix is large, the evaluation of by first computing may be prohibitively expensive. In this situation it is attractive to compute an approximation of by first applying a few steps of a global Lanczos-type method to reduce to a small matrix and then evaluating at this reduced matrix. The computed approximation can be interpreted as a quadrature rule. The present paper generalizes the extended global Lanczos method introduced in Bentbib et al. (2018) and discusses the computation of error-bounds and error estimates. Numerical examples illustrate the performance of the techniques described.
中文翻译:
扩展的全局Lanczos方法,Gauss-Radau正交和矩阵函数逼近
需要评估表格的表达式 跟踪 ,其中矩阵 是对称的 与 和 是定义在频谱的凸包上的函数 出现在许多应用程序中,包括网络分析和机器学习。当矩阵 很大,对 通过首先计算 可能太贵了。在这种情况下,计算 首先应用全局Lanczos型方法的一些步骤来减少 到一个小的矩阵,然后求值 在这个简化的矩阵上。计算出的近似值可以解释为正交规则。本文概括了Bentbib等人引入的扩展全局Lanczos方法。(2018),并讨论了误差范围和误差估计的计算。数值示例说明了所描述技术的性能。