Skip to main content
Log in

Characterization of linear stationary iterative processes for solving a singular system of linear equations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

There are many iterative methods for solving a consistent singular system of linear equations [2, 3, 5, 6, 8, 10–13]. In this paper we characterize the class of all singular system of linear equations using generalized inverses.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Afriat, S.N.: Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Cambridge Philosophi. Soc.53, 800–816 (1957)

    Google Scholar 

  2. Douglas, J., Jr., Pearcy, C. M.: On convergence of alternating direction procedures in the presence of singular operators. Numer. Math.5, 175–184 (1963)

    Google Scholar 

  3. Joshi, V. N.: A note on the solution of rectangular linear systems by iteration. SIAM Review12, 463–466 (1970)

    Google Scholar 

  4. Forsythe, G. E., Moler, C. B.: Computer solution of linear algebraic systems. Prentice-Hall (1967)

  5. Keller, H. B.: On the solution of singular and semidefinite linear systems by iteration. J. SIAM Numer. Anal. B-2, 289–290 (1965)

    Google Scholar 

  6. Pavel-Parvu, M., Korganoff, A.: Iteration functions for solving polynomial matrix equations. Constructive aspects of the fundamental theorem of algebra. (eds. Dejon, B., Henrici, P.) London: Wiley (1969)

    Google Scholar 

  7. Oldenburger, R.: Infinite powers of matrices and characteristic roots. Duke Math. J.6, 357–361 (1940)

    Google Scholar 

  8. Pyle, L. D.: A generalized inverseε-algorithm for constructing intersection projection matrices, with applications. Numer. Math.10, 86–102 (1967)

    Google Scholar 

  9. Rao, C. R., Mitra, S. K.: Generalized inverse of matrices and its applications. New York: Wiley 1971

    Google Scholar 

  10. Tanabe, K.: Projection method for solving a singular system of linear equations and its applications. Numer. Math.17, 203–214 (1971)

    Google Scholar 

  11. Tewarson, R. P.: On some representations of generalized inverses, SIAM Rev.11, 272–276 (1969)

    Google Scholar 

  12. Whitney, T. M., Meany, R. K.: Two algorithms related to the method of steepest descent. J. SIAM Numer. Anal.4, 109–118 (1967)

    Google Scholar 

  13. Young, D. M.: On the consistency of linear stationary iterative methods. J. SIAM Numer. Anal.9, 89–96 (1972)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tanabe, K. Characterization of linear stationary iterative processes for solving a singular system of linear equations. Numer. Math. 22, 349–359 (1974). https://doi.org/10.1007/BF01436918

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01436918

Keywords

Navigation