A mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem

TL;DR

This paper introduces a mixed precision preconditioned Jacobi method for symmetric eigenvalue problems, combining low and high precision computations to improve efficiency and accuracy, with theoretical error analysis and numerical validation.

Contribution

The paper proposes a novel mixed precision approach for the Jacobi method, integrating low and high precision steps with error analysis and GPU-accelerated experiments.

Findings

The mixed precision method improves computational efficiency.

Error bounds are established for the proposed approach.

Numerical experiments demonstrate superior performance over traditional Jacobi methods.

Abstract

The eigenvalue problem is a fundamental problem in scientific computing. In this paper, we first give the error analysis for a single step or sweep of Jacobi's method in floating point arithmetic. Then we propose a mixed precision preconditioned Jacobi method for the symmetric eigenvalue problem: We first compute the eigenvalue decomposition of a real symmetric matrix by an eigensolver at low precision and we obtain a low-precision matrix of eigenvectors; Then by using the high-precision modified Gram-Schmidt orthogonalization process, a high-precision orthogonal matrix is obtained, which is used as an initial guess for Jacobi's method. The rounding error analysis of the proposed method is established under some conditions. We also present a mixed precision preconditioned one-sided Jacobi method for the singular value problem and the corresponding rounding error analysis is discussed. Numerical experiments on CPUs and GPUs are reported to illustrate the efficiency of the proposed method over the original Jacobi method.