2024/05/13 by Richard B. Lehoucq, Lehoucq, Richard B., Michael Weylandt +3
Mathematics · #Advanced Optimization Algorithms Research #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA) #Social and Information Networks (cs.SI)
paper · pdf · doi:10.48550/arxiv.2405.07877
openalex publication_date 2024/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We show that certain Graph Laplacian linear sets of equations exhibit optimal accuracy, guaranteeing that the relative error is no larger than the norm of the relative residual and that optimality occurs for carefully chosen right-hand sides. Such sets of equations arise in PageRank and Markov chain theory. We establish new relationships among the PageRank teleportation parameter, the Markov chain discount, and approximations to linear sets of equations. The set of optimally accurate systems can be separated into two groups for an undirected graph -- those that achieve optimality asymptotically with the graph size and those that do not -- determined by the angle between the right-hand side of the linear system and the vector of all ones. We provide supporting numerical experiments.