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

TL;DR

This paper presents a novel mixed-precision projection method for eigenvalue and SVD problems that eliminates orthogonalization, maintaining accuracy and stability while leveraging computational efficiency.

Contribution

It introduces an orthogonalization-free approach using non-orthogonal bases in mixed-precision, enhancing efficiency without sacrificing accuracy in large-scale matrix computations.

Findings

Achieves accurate eigenvalues and singular values with reduced precision bases.

Maintains linear independence of basis vectors in mixed-precision setting.

Demonstrates effectiveness on synthetic and real-world data.

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 (SVD) remains challenging. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal 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.