# A robust and efficient implementation of LOBPCG

## TL;DR

This paper presents a robust and efficient implementation of the LOBPCG algorithm, improving stability and performance for computing eigenvalues of large sparse symmetric matrices through new basis selection, convergence criteria, and optimizations.

## Contribution

It introduces an improved basis selection strategy, a robust convergence criterion, and several optimizations, enhancing the stability and efficiency of LOBPCG implementations.

## Key findings

- Outperforms previous methods in stability.
- Achieves faster convergence in numerical tests.
- Demonstrates robustness across various large sparse matrices.

## Abstract

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. We also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.