Estimating the condition number of Chebyshev filtered vectors with application to the ChASE library

2026-03-11

AI Summary

This work addresses the challenge posed by the unknown condition number of Chebyshev-filtered vector sets, which hinders the efficient selection of QR decomposition algorithms. We propose a lightweight algorithm to compute a tight upper bound on the condition number of the filtered subspace, enabling accurate and efficient prediction. Leveraging this estimate, our method dynamically selects the optimal QR decomposition strategy. Integrated into the ChASE library for the first time, this approach establishes an adaptive QR decomposition mechanism that significantly enhances computational performance without compromising numerical accuracy. The key innovation lies in tightly coupling condition number estimation with algorithmic choice, thereby providing intelligent scheduling support for linear algebra operations within subspace iteration methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Chebyshev filtered subspace iteration is a well-known algorithm for the solution of (symmetric/Hermitian) algebraic eigenproblems which has been implemented in several application codes~\cite{Kronik:2006ff, abinit:2020} or in stand alone libraries~\cite{ChASE}. An essential part of the algorithm is the QR-factorization of the array of vectors spanning the active subspace that have been filtered by the Chebyshev filter. Typically such an array has an a-priori unknown high condition number that directly influences the choice of QR-factorization algorithm. In this work we show how such condition number can be bound from above with precise and inexpensive estimates. We then proceed to use these estimates to implement a mechanism for the choice of QR-factorization in the ChASE library. We show how such mechanism enhance the performance of the library without compromising on its accuracy.

Problem

• condition number
• Chebyshev filtering
• subspace iteration
• QR factorization
• eigenproblems

Innovation

• condition number estimation
• Chebyshev filtering
• subspace iteration
• QR factorization
• ChASE library