Skip to main content

Jacobi-Type Algorithms for Eigenvalues on Vector and Parallel Computers

  • Chapter
Scientific Computing on Supercomputers
  • 75 Accesses

Abstract

After a short introduction to Jacobi-like algorithms a review is given of a vector and a parallel implementation of the Jacobi method for symmetric matrices. In the last section a modification of Sameh’s parallel eigen-algorithm is presented based on a problem formulation with so-called Euclidean parameters of non-orthogonal shears.

This research is part of the VF-program “Parallele Algoritmiek”, THD-WI-08/85-25 which has been approved by the Netherlands Ministry of Education and Sciences.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  • Brent, R.P., and Luk, F.L., 1985, The solution of singular-value and symmetric eigenvalue problems on multiprocessor arrays,SIAM J. Sci. Stat. Comput, 6: 69 –84.

    Article  MathSciNet  MATH  Google Scholar 

  • Dollinger, E., 1981, Ein linear konvergentes zyklisches Jacobi ähnliches Verfahren für beliebige reele Matrizen,Num. Math, 38: 245 –253.

    MathSciNet  MATH  Google Scholar 

  • Eberlein, P.J., 1962, A Jacobi-like method for the automatic computation of eigenvalues and eigenvectors of an arbitrary matrix,J. Soc. Industr. Appl. Math, 10: 74 –88.

    Article  MathSciNet  Google Scholar 

  • Elsner, L., and Paardekooper, M.H.C., On measures of non-normality of matrices,Lin Alg. Appl., 92:107–24.

    Google Scholar 

  • Forsyth, G.E., and Henrici, P., 1960, The cyclic Jacobi method for computing the principal values of a complex matrix,Trans. Am. M.S, 94: 1 –23.

    Article  Google Scholar 

  • Goldstine, H.H., and Horowitz, L.P., 1959, A procedure for the diagonalization of normal matrices,J. Assoc. Comp. Mach, 6: 176 –195.

    MATH  Google Scholar 

  • Hari, V., 1982, On the quadratic convergence of the paardekooper method I,Glasnik Matem, 17: 183 –195.

    MathSciNet  Google Scholar 

  • Hari, V., 1982, On the global convergence of the Eberlein method for real matrices,Num. Math., 39: 361 –369.

    Article  MathSciNet  MATH  Google Scholar 

  • Hestenes, M.R., 1958, Inversion of matrices by bio-orthogonalization and related results,J. Soc. Indust. Appl. Math, 6: 51 –90.

    Article  MathSciNet  MATH  Google Scholar 

  • Jacobi, C.G.J., Uber ein leichtes Verfahren die in der Theorie der Secul’r st’rungen vorkommenden Gleichungen numerisch aufzul’sen,Crelle’s J., 30:51–94.

    Google Scholar 

  • Kempen, H.P.M. van, 1966, On the convergence of the classical Jacobi method for real symmetric matrices with non- distinct eigenvalues,Num. Math, 9: 11 –18.

    MATH  Google Scholar 

  • Kempen, H.P.M. van, 1966, On the quadratic convergence of the serial cyclic Jacobi method,Num. Math, 9: 19 –22.

    MATH  Google Scholar 

  • Luk, F.T., 1380, Computing the singular-value decomposition on the ILLIAC IV,ACMTOMS, 6: 524: 539.

    Google Scholar 

  • Modi, J.J.,and Pryce, J.D., 1985, Efficient implementation of Jacobi1s diagonalization method on the DAP,Num. Math, 46: 443 –454.

    Google Scholar 

  • Paardekooper, M.H.C., 1969, An eigenvalue algorithm based on norm-reducing transformation, Technical University Eindhoven, Thesis.

    Google Scholar 

  • Paardekooper, M.H.C., 1971, An eigenvalue algorithm for skew symmetric matrices,Num. Math, 17: 189 –202.

    MathSciNet  MATH  Google Scholar 

  • Paardekooper M.H.C., 1986, Sameh’s parallel eigenvalue algorithm revisited, Research Memorandum, Tilburg University.

    Google Scholar 

  • Ruhe, A., 1968, On the quadratic convergence of a generalization of the Jacobi method for arbitrary matrices,BIT, 8: 210 –231.

    Article  MathSciNet  MATH  Google Scholar 

  • Rutishauser, H., 1966, The Jacobi method for real symmetric matrices,Num. Math, 9: 1 –10.

    MathSciNet  MATH  Google Scholar 

  • Sameh, A.H., 1971, On Jacobi and Jacobi-like algorithms for a parallel computer,Math. Comput, 25: 579 –590.

    MathSciNet  MATH  Google Scholar 

  • Schönhage, A., 1964, On the quadratic convergence of the Jacobi process,Num. Math, 6: 410 –412.

    Article  MATH  Google Scholar 

  • Stewart, G.W., 1985, A Jacobi-like algorithm for computing the Schur decomposition of a non-hermitian matrix,SIAM J. Sci. Stat. Comput, 6: 853 –864.

    Article  MATH  Google Scholar 

  • Veselic, K., 1976, A convergent Jacobi method for solving the eigenproblem of arbitrary real matrices,Num. Math25: 179 –184.

    Article  MathSciNet  MATH  Google Scholar 

  • Wilkinson, J.H., 1962, Note on the quadratic convergence of the cyclic Jacobi process,Num. Math4: 296 –300.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Plenum Press, New York

About this chapter

Cite this chapter

Paardekooper, M.H.C. (1989). Jacobi-Type Algorithms for Eigenvalues on Vector and Parallel Computers. In: Devreese, J.T., Van Camp, P.E. (eds) Scientific Computing on Supercomputers. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0819-5_11

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-0819-5_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4612-8098-9

  • Online ISBN: 978-1-4613-0819-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics