Computing accurate singular values using a mixed-precision one-sided Jacobi algorithm

TL;DR

This paper introduces a mixed-precision one-sided Jacobi algorithm for computing singular values with higher accuracy and efficiency, especially for ill-conditioned matrices, outperforming standard routines in accuracy and speed.

Contribution

The paper develops a novel mixed-precision preconditioned one-sided Jacobi algorithm with improved error bounds and demonstrates its superior accuracy and potential speed compared to existing methods.

Findings

Achieves smaller relative forward errors than LAPACK and MATLAB routines.

Supports faster convergence through effective preconditioning.

Offers potential speed advantages with hardware/software improvements.

Abstract

We present a relative forward error analysis of a mixed-precision preconditioned one-sided Jacobi algorithm, analogous to a two-sided version introduced in [N. J. Higham, F. Tisseur, M. Webb and Z. Zhou, SIAM J. Matrix Anal. Appl. 46 (2025), pp. 2423-2448], which uses low precision to compute the preconditioner, applies it in high precision, and computes the singular value decomposition using the one-sided Jacobi algorithm at working precision. Our analysis yields smaller relative forward error bounds for the computed singular values than those of standard SVD algorithms. We present and analyse two approaches for constructing effective preconditioners. Our numerical experiments support the theoretical results and demonstrate that our algorithm achieves smaller relative forward errors than the LAPACK routines $\texttt{DGESVJ}$ and $\texttt{DGEJSV}$, as well as the MATLAB function $\texttt{svd}$, particularly for ill-conditioned matrices. Timing tests show that our approach accelerates the convergence of the Jacobi iterations and that the dominant cost arises from a single high-precision matrix-matrix multiplication. With improved software or hardware support for this bottleneck, our algorithm would be faster than the LAPACK one-sided Jacobi algorithm $\texttt{DGESVJ}$ and comparable in speed to the state-of-the-art preconditioned one-sided Jacobi algorithm $\texttt{DGEJSV}$, but much more accurate.