Iteris: Agentic Research Loops for Computational Mathematics

2026-06-01

AI Summary

This work addresses the lack of systematic intelligent research tools for open problems in computational mathematics by introducing Iteris—the first agent-based research system tailored to this domain. Iteris integrates large language models with numerical computation, algorithm generation, formal reasoning, and iterative verification to establish an autonomous research loop spanning numerical experimentation, counterexample construction, and proof sketching. The system achieves breakthroughs on two frontier problems: it produces the first asymptotic performance phase diagram comparing the conjugate gradient method and randomized coordinate descent under power-law spectra, and constructs a rigorous counterexample demonstrating the failure of QR factorization with column pivoting. Both results have been validated by domain experts.

Abstract

Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.

Problem

• computational mathematics
• open problems
• numerical experimentation
• algorithm design
• adversarial constructions

Innovation

• agentic AI
• computational mathematics
• numerical experimentation
• algorithm design
• open mathematical problems