We consider approximating $B^T(A+sI)^{-1}B$, where $A\in\mathbb{R}^{n\times n}$ is large, symmetric positive definite, and $B\in\mathbb{R}^{n\times p}$ with $p\ll n$. We focus on the case where $A$ has an almost continuous (dense) spectrum: its eigenvalues fill one or more intervals so densely that Krylov methods cannot resolve them individually. Our target is computing multiple-input multiple-output transfer functions arising from large-scale discretizations of problems with continuous spectral measures, such as linear time-invariant PDEs on unbounded domains. Traditional Krylov methods, such as Lanczos or conjugate gradients, resolve individual eigenvalues of a dense discretization while ignoring the underlying continuous spectral measure these points approximate. We argue it is more efficient to model the operator's inherent branch cut than to exhaustively resolve the artificial point spectrum induced by discretization. We adapt the framework of Kreĭn-Nudelman semi-infinite strings to the block Lanczos algorithm, with parameters chosen adaptively by maximizing the energy absorbed at the string termination relative to the energy stored in the string. This yields a low-rank modification to the block Lanczos matrix, dependent on $\sqrt{s}$, at an additional $O(n)$ cost. We show significant error reductions for large-scale self-adjoint PDE discretizations in unbounded domains, including two- and three-dimensional Maxwell's equations in diffusive regimes. The method is especially advantageous for computing state-space solutions for wave propagation, specifically for 2D wave and 3D Maxwell operators. Replacing the conventional Lanczos spectral decomposition with the continuous Kreĭn-Nudelman spectrum yields a qualitative improvement in finite-difference approximations, transforming standing-wave artifacts into outgoing propagating waves.
A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices
Tsinghua University（清华大学）；Dalian University of Technology（大连理工大学）
A Range-Deflated Stochastic Lanczos Quadrature Method for Large-Scale Log-Determinant Estimation
The University of Dodoma（多多马大学）；African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Western Norway University of Applied Sciences（挪威西部应用科学大学）；University of Cape Town（开普敦大学）
Fast Botnet Detection From Streaming Logs Using Online Lanczos Method
CA Technologies, Inc.（CA Technologies公司）；College of Computing & Informatics, Drexel University（德雷塞尔大学计算与信息学院）；Department of Mathematics, Temple University（Temple大学数学系）；Department of Computer Science, Maryland University at Baltimore County（马里兰大学巴尔的摩县计算机科学系）
