Novel modifications of parallel Jacobi algorithms

Authors:
Sanja Singer;Saša Singer;Vedran Novaković;Aleksandar Ušćumlić;Vedran Dunjko
Affiliations:
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia 10000;Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia 10002;Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia 10000;MSV sustavi d.o.o., Samobor, Croatia 10430;School of EPS --- Physics Department, David Brewster Building, Heriot-Watt University, Edinburgh, UK EH14 4AS
Venue:
Numerical Algorithms
Year:
2012

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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.