BIT Numerical Mathematics

, Volume 44, Issue 1, pp 189–207

Semi-Conjugate Direction Methods for Real Positive Definite Systems

  • J. Y. Yuan
  • G. H. Golub
  • R. J. Plemmons
  • W. A. G. Cecílio

DOI: 10.1023/B:BITN.0000025092.92213.da

Cite this article as:
Yuan, J.Y., Golub, G.H., Plemmons, R.J. et al. BIT Numerical Mathematics (2004) 44: 189. doi:10.1023/B:BITN.0000025092.92213.da


In this preliminary work, left and right conjugate direction vectors are defined for nonsymmetric, nonsingular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method has no breakdown for real positive definite systems. The method reduces to the usual conjugate gradient method when A is symmetric positive definite. A finite termination property of the semi-conjugate direction method is shown, providing a new simple proof of the finite termination property of conjugate gradient methods. The new method is well defined for all nonsingular M-matrices. Some techniques for overcoming breakdown are suggested for general nonsymmetric A. The connection between the semi-conjugate direction method and LU decomposition is established. The semi-conjugate direction method is successfully applied to solve some sample linear systems arising from linear partial differential equations, with attractive convergence rates. Some numerical experiments show the benefits of this method in comparison to well-known methods.

left conjugate direction vectors right conjugate direction vectors left conjugate direction method semi-conjugate direction method LU decomposition conjugate gradient method Gaussian elimination solution of nonsymmetric linear systems 

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • J. Y. Yuan
    • 1
  • G. H. Golub
    • 2
  • R. J. Plemmons
    • 3
  • W. A. G. Cecílio
    • 4
  1. 1.Departamento de MatemáticaUniversidade Federal do ParanáCuritiba, PRBrazil
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA
  3. 3.Department of Computer ScienceWake Forest UniversityWinston-SalemUSA
  4. 4.Departamento de MatemáticaPontifícia Universidade Católica do ParanáCuritiba, PRBrazil

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