Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

2026-02-05

AI Summary

This work addresses fundamental open problems in solving linear systems and computing eigenvalues by synthesizing consensus reached among experts in theoretical computer science and numerical analysis during a workshop at the Simons Institute. It presents the first systematic survey of interdisciplinary challenges, covering key techniques such as iterative solvers, low-rank approximations, and randomized sketching, while also extending to emerging areas including tensors, quantum systems, and matrix functions. The study distills these challenges into five major categories of critical questions, offering a clear research roadmap that advances the theory of computational complexity in linear algebra and guides the design of efficient algorithms, thereby fostering collaborative innovation between the two disciplines.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Solvers and Tools
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop \emph{Linear Systems and Eigenvalue Problems}, which was organized at the Simons Institute for the Theory of Computing program on \emph{Complexity and Linear Algebra} in Fall 2025. The complexity and numerical solution of linear algebra problems %in matrix computations and related fields is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.

Problem

• linear systems
• eigenvalue problems
• matrix computations
• numerical linear algebra
• open questions

Innovation

• iterative solvers
• eigenvalue computation
• low-rank approximation
• randomized sketching
• matrix computations