Skip to main content
Springer Nature Link
Log in

A multicore solution to Block–Toeplitz linear systems of equations

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

There exist algorithms, also called “fast” algorithms, which exploit the special structure of Toeplitz matrices so that, e.g., allow to solve a linear system of equations in O(n 2) flops. However, some implementations of classical algorithms that do not use this structure (O(n 3) flops) highly reduce the time to solution when several cores are available. That is why it is necessary to work on “fast” algorithms so that they do not lose track of the benefits of new hardware/software. In this work, we propose a new approach to the Generalized Schur Algorithm, a very known algorithm for the solution of Toeplitz systems, to work on a Block–Toeplitz matrix. Our algorithm is based on matrix–matrix multiplications, thus allowing to exploit an efficient implementation of this operation if it exists. Our algorithm also makes use of the thread level parallelism featured by multicores to decrease execution time.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Algorithm 2
Algorithm 3
Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Alonso P, Badía JM, Vidal AM (2005) An efficient parallel algorithm to solve block–Toeplitz systems. J Supercomput 32:251–278

    Article  Google Scholar 

  2. Alonso P, Argüeso F, Cortina R, Ranilla J, Vidal AM Non-linear parallel solver for detecting point sources in CMB maps using Bayesian techniques. J Math Chem. doi:10.1007/s10910-012-0078-7

  3. Anderson E et al (1999) LAPACK users’ guide, 3rd edn. SIAM, Philadelphia

    Book  Google Scholar 

  4. Bischof C, van Loan C (1987) The WY representation for products of householder matrices. SIAM J Sci Stat Comput 8(1):2–13

    Article  Google Scholar 

  5. Chun J, Kailath T, Lev-Ari H (1987) Fast parallel algorithms for QR and triangular factorization. SIAM J Sci Stat Comput 8(6):899–913

    Article  MathSciNet  MATH  Google Scholar 

  6. Cybenko G, Berry M (1990) Hyperbolic householder algorithms for factoring structured matrices. SIAM J Matrix Anal Appl 11(4):499–520

    Article  MathSciNet  MATH  Google Scholar 

  7. Gallivan K, Thirumalai S, Van Dooren P (1994) On solving block Toeplitz systems using a block Schur algorithm. In: Proceedings of the 23rd international conference on parallel processing, vol 3. CRC Press, Boca Raton, pp 274–281

    Google Scholar 

  8. Gustavson FG (1997) Recursion leads to automatic variable blocking for dense linear-algebra algorithms. IBM J Res Dev 41(6):737–755

    Article  Google Scholar 

  9. Intel MKL (2012) http://software.intel.com/en-us/articles/intel-mkl

  10. Jin XQ (2002) Developments and applications of Block Toeplitz iterative solvers. Combinatorics and computer science. Science Press, Beijing

    Google Scholar 

  11. Kailath T, Sayed AH (1995) Displacement structure: theory and applications. SIAM Rev 37(3):297–386

    Article  MathSciNet  MATH  Google Scholar 

  12. PLASMA Project (2012) The parallel linear algebra for scalable multi-core architectures. http://icl.cs.utk.edu/plasma

  13. StructPack (2012) A high performance computing library for structured matrices. http://www.inco2.upv.es/structpack.html

Download references

Acknowledgements

PROMETEO/2009/013, Generalitat Valenciana. Projects TEC2009-13741, TIN2010-14971 and TIN2011-15734-E of the Ministerio Español de Ciencia e Innovación, and TEC2012-38142-C04 of the Ministerio Español de Economía y Competitividad.

Author information

Authors and Affiliations

  1. Departamento de Sistemas Informáticos y Computación, Universitat Politècnica de València, València, Spain

    Pedro Alonso & Antonio M. Vidal

  2. Departamento de Informática, Universidad de Oviedo, Gijón, Spain

    Daniel Argüelles & José Ranilla

Authors
  1. Pedro Alonso
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Daniel Argüelles
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. José Ranilla
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Antonio M. Vidal
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Pedro Alonso.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alonso, P., Argüelles, D., Ranilla, J. et al. A multicore solution to Block–Toeplitz linear systems of equations. J Supercomput 65, 999–1009 (2013). https://doi.org/10.1007/s11227-012-0824-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-012-0824-4

Keywords