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Numerical Algorithms

, Volume 59, Issue 1, pp 1–27 | Cite as

Novel modifications of parallel Jacobi algorithms

  • Sanja SingerEmail author
  • Saša Singer
  • Vedran Novaković
  • Aleksandar Ušćumlić
  • Vedran Dunjko
Original Paper

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.

Keywords

Hermitian matrices Eigenvalues Jacobi algorithm Parallelization 

Mathematics Subject Classifications (2010)

65F15 65Y05 65Y20 68W10 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  • Sanja Singer
    • 1
    Email author
  • Saša Singer
    • 2
  • Vedran Novaković
    • 1
  • Aleksandar Ušćumlić
    • 3
  • Vedran Dunjko
    • 4
  1. 1.Faculty of Mechanical Engineering and Naval ArchitectureUniversity of ZagrebZagrebCroatia
  2. 2.Faculty of Science, Department of MathematicsUniversity of ZagrebZagrebCroatia
  3. 3.MSV sustavi d.o.o.SamoborCroatia
  4. 4.School of EPS – Physics Department, David Brewster BuildingHeriot-Watt UniversityEdinburghUK

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