Novel Modifications of Parallel Jacobi Algorithms

TL;DR

This paper introduces two novel classes of parallel Jacobi algorithms for eigenvalue problems of Hermitian matrices, highlighting their high accuracy and efficiency, especially with new parallelization and pivoting techniques applicable to modern computing architectures.

Contribution

It presents new parallelization methods and pivoting strategies for trigonometric and hyperbolic Jacobi algorithms, improving their speed and applicability on distributed and shared-memory systems.

Findings

Hyperbolic algorithms may outperform trigonometric ones in practice

Both algorithm types achieve high relative accuracy when applicable

Parallelization techniques enhance computational efficiency

Abstract

We describe two main classes of one-sided trigonometric and hyperbolic Jacobi-type algorithms for computing eigenvalues and eigenvectors of Hermitian matrices. These types of algorithms exhibit significant advantages over many other eigenvalue algorithms. If the matrices permit, both types of algorithms compute the eigenvalues and eigenvectors with high relative accuracy. We present novel parallelization techniques for both trigonometric and hyperbolic classes of algorithms, as well as some new ideas on how pivoting in each cycle of the algorithm can improve the speed of the parallel one-sided algorithms. These parallelization approaches are applicable to both distributed-memory and shared-memory machines. The numerical testing performed indicates that the hyperbolic algorithms may be superior to the trigonometric ones, although, in theory, the latter seem more natural.