Work

Mixed-Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems

Year: 2026

Type: article

Abstract: . Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh–Ritz projection methods. The proposed method employs nonorthogonal bases computed at reduced precision, resulting in bases computed without inner products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic.

Source: SIAM Journal on Scientific Computing

DOI: 10.1137/25m1756922

Authors: Tianshi Xu, Zechen Zhang, Jie Chen, Yousef Saad, Yuanzhe Xi

Institutions: Emory University, University of Minnesota, IBM (United States), Massachusetts Institute of Technology

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Collaborative Research: Robust Acceleration and Preconditioning Methods for Data-Related Applications: Theory and Practice, Collaborative Research: Acceleration and Preconditioning Methods for Deep Learning, Collaborative Research: Robust Acceleration and Preconditioning Methods for Data-Related Applications: Theory and Practice, Collaborative Research: Acceleration and Preconditioning Methods for Deep Learning