Skip to main content
SpringerLink
Log in

PCR algorithm for parallel computing the solution of the general restricted linear equations

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper we present a parallel algorithm for parallel computing the solution of the general restricted linear equations Ax=b,x∈T, where T is a subspace of ℂn and b∈AT. By this algorithm the solution x=A (2) T,S b is obtained in n(log 2 m+log 2(n−s+1)+7)+log 2 m+1 steps with P=mn processors when m≥2(n−1) and with P=2n(n−1) processors otherwise.

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

Access this article

Log in via an institution

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. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  2. Chen, Y.: A Cramer rule for solution of the general restricted linear equation. Linear Multilinear Algebra 34, 177–186 (1993)

    Article  MATH  Google Scholar 

  3. Chen, Y.: Finite algorithms for the (2)-generalized inverse A (2) T,S . Linear Multilinear Algebra 40, 61–68 (1995)

    Article  MATH  Google Scholar 

  4. Chipman, J.S.: On least-squares with insufficient observations. J. Am. Stat. Assoc. 54, 1078–1111 (1964)

    Article  MathSciNet  Google Scholar 

  5. Cline, R.E., Greville, T.N.E.: A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 53–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gu, C., Wang, G., Xu, Z.: PCR algorithm for the parallel computation of the solution of a class of singular linear systems. Appl. Math. Comput. 176, 237–244 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Penrose, R.: On best approximate solutions of linear matrix equations. Proc. Camb. Philos. Soc. 52, 17–19 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  8. Sridhar, M.K.: A new algorithm for parallel solution of linear equations. Inf. Process. Lett. 24, 407–412 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. Srinivas, M.A.: Optimal parallel scheduling of Gaussian eliminations DAG’s. IEEE Trans. Comput. C-32, 109–117 (1983)

    Article  Google Scholar 

  10. Wing, O., Huang, J.W.: A computational model for parallel solution of linear equations. IEEE Trans. Comput. C-29, 632–638 (1980)

    Article  MathSciNet  Google Scholar 

  11. Wang, G.: PCR algorithm for parallel computing minimum-norm least-squares solution of inconsistent linear equations. Numer. Math., J. Chin. Univ., Engl. Ser. 2, 1–10 (1993)

    Google Scholar 

  12. Wang, G., Lu, S.: Fast parallel algorithm for computing the generalized inverse A † and A † MN . J. Comput. Math. 6, 348–354 (1988)

    MATH  MathSciNet  Google Scholar 

  13. Wei, Y.: A characterization and representation of the generalized inverse A (2) T,S and its applications. Linear Algebra Appl. 280, 87–96 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wei, Y., Wang, G.: PCR algorithm for parallel computing minimum-norm (T) least-squares (S) solution of inconsistent linear equations. Appl. Math. Comput. 133, 547–557 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Yu, Y., Wang, G.: On the generalized inverse A (2) T,S over integral domains. Aust. J. Math. Appl. 4(1), 1–20 (2007). Article 16

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Education, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Yaoming Yu

Authors
  1. Yaoming Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaoming Yu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yu, Y. PCR algorithm for parallel computing the solution of the general restricted linear equations. J. Appl. Math. Comput. 27, 125–136 (2008). https://doi.org/10.1007/s12190-008-0062-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-008-0062-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation