Skip to main content
SpringerLink
Log in

An asynchronous direct solver for banded linear systems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

Banded linear systems occur frequently in mathematics and physics. However, direct solvers for large systems cannot be performed in parallel without communication. The aim of this paper is to develop a general asymmetric banded solver with a direct approach that scales across many processors efficiently. The key mechanism behind this is that reduction to a row-echelon form is not required by the solver. The method requires more floating point calculations than a standard solver such as LU decomposition, but by leveraging multiple processors the overall solution time is reduced. We present a solver using a superposition approach that decomposes the original linear system into q subsystems, where q is the number of superdiagonals. These methods show optimal computational cost when q processors are available because each system can be solved in parallel asynchronously. This is followed by a q×q dense constraint matrix problem that is solved before a final vectorized superposition is performed. Reduction to row echelon form is not required by the solver, and hence the method avoids fill-in. The algorithm is first developed for tridiagonal systems followed by an extension to arbitrary banded systems. Accuracy and performance is compared with existing solvers and software is provided in the supplementary material.

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. Amestoy, P.R., Duff, I.S., l’Excellent, J.-Y.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184(2–4), 501–520 (2000)

    Article  MATH  Google Scholar 

  2. Amestoy, P.R., Duff, I.S., l’Excellent, J.-Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amestoy, P.R., Guermouche, A., l’Excellent, J.-Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)

    Article  MathSciNet  Google Scholar 

  4. Bartels, R.H., Beatty, J.C., Barsky, B.A.: Hermite and Cubic Spline Interpolation. In: Ch. 3 in An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. San Francisco, CA: Morgan Kaufmann, pp 9–17 (1998)

  5. Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)

  6. Demmel, J.W., Gilbert, J.R., Li, X.S.: An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination. SIAM J. Matrix Anal. Appl. 20(4), 915952 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donfack, S., Dongarra, J., Faverge, M., Gates, M., Kurzak, J., Luszczek, P., Yamazaki, I.: A survey of recent developments in parallel implementations of Gaussian elimination, Concurrency and Computation: Practice and Experience. DOI: 10.1002/cpe.3306 (2014)

  8. Duff, I., Reid, J.: The Multifrontal Solution of Indefinite Sparse Symmetric Linear Equations. ACM Trans. Math. Softw. 9(3), 302–325 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duff, I.: A review of frontal methods for solving linear systems. Comput. Phys. Commun. 97, 4552 (1996)

  10. Gavel, D.T.: Solution to the Problem of Instability in Banded Toeplitz Solvers. IEEE Trans. Signal Process. 40, 464 (1992)

  11. Golub, G., Van Loan, C.: Matrix Computations, 4th Edn. John Hopkins University Press, Baltimore, MD (2013)

  12. Fortran ISML Numerical Library. User’s Guide, Version 7.0, Rogue Wave Software, 1315 West Century Drive, Suite 150, Louisville, CO 80027 (2010)

  13. Irons, B.: A Frontal Solution Program for Finite Element Analysisss. Int. J. Numer. Methods Eng. 2, 5–32 (1970)

    Article  MATH  Google Scholar 

  14. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications NY (2009)

  15. Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, Third. In: Society for Industrial and Applied Mathematics. (paperback), pp 0–89871-447-8, Philadelphia, PA (1999)

  16. Liu, J.: The Multifrontal Method for Sparse Matrix Solution. Theory and Practice, SIAM Review 34(1), 82–109 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Levinson, N.: The Wiener RMS (root mean square) error criterion in filter design and prediction. In: Wiener, N. (ed.) Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications, pp 129–148. Wiley, Appendix B, New York (1949)

  18. Luisier, M., Schenk, O., et al: Fast Methods for Computing Selected Elements of the Green’s Function. In: Wolf, F., Mohr, B., an Ney, D. (eds.) Massively Parallel Nanoelectronic Device Simulations, Euro-Par 2013, LNCS 8097, pp 533–544. Springer-Verlag, Berlin Heidelberg (2013)

  19. MacLeod, A.J.: Instability in the solution of banded Toeplitz systems. IEEE Trans. Acoust. Speech Signal Process. 37, 1449 (1989)

  20. Mandel, J.: Balancing domain decomposition. Comm. Numer. Methods Engrg. 9, 233–241 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Meek, D.S.: The Inverses of Toeplitz Band Matrices. Linear Algebra Appl. 49, 117 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Intel Math Kernel Library, Reference Manual, Intel Corporation, 2200 Mission College Blvd., Santa Clara, CA 95052-8119, USA (2010)

  23. Ruffa, A.: A solution approach for lower Hessenberg linear systems. ISRN Applied Mathematics 2011, 236727 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ruffa, A., Jandron, M., Toni, B.: Parallelized Solution of Banded Linear Systems with an Introduction to p-adic Computation. In: Toni, B. (ed.) Mathematical Sciences with Multidisciplinary Applications Springer Proceedings in Mathematics & Statistics, vol. 157 (2016)

  25. Schenk, O., Bollhoefer, M., Roemer, R.: On large-scale diagonalization techniques for the Anderson model of localization. Featured SIGEST paper in the SIAM Review selected on the basis of its exceptional interest to the entire SIAM community. SIAM Rev. 50, 91–112 (2008)

    Article  MathSciNet  Google Scholar 

  26. Schenk, O., Waechter, A., Hagemann, M.: Matching-based Preprocessing Algorithms to the Solution of Saddle-Point Problems in Large-Scale Nonconvex Interior-Point Optimization. Journal of Computational Optimization and Applications 36(2-3), 321–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stone, H. S.: An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations. J. Assoc. Comput. Mach. 20(1), 27–38 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  28. Thomas, L.H.: Elliptic Problems in Linear Differential Equations over a Network, Watson Sci. Comput. Lab Report. Columbia University, New York (1949)

  29. Trench, W.F.: An algorithm for the inversion of finite Toeplitz smatrices. J. Soc. Ind. Appl. Math. 12, 515 (1964)

  30. Van der Vorst, H.A.: Analysis of a parallel solution method for tridiagonal linear systems 5(3), 303311 (1987)

  31. Zohar, S.: The solution of a Toeplitz set of linear equations. J. ACM 21, 272 (1974)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Naval Undersea Warfare Center, Newport, RI, 02841, USA

    Michael A. Jandron & Anthony A. Ruffa

  2. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881, USA

    James Baglama

Authors
  1. Michael A. Jandron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anthony A. Ruffa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James Baglama
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael A. Jandron.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jandron, M.A., Ruffa, A.A. & Baglama, J. An asynchronous direct solver for banded linear systems. Numer Algor 76, 211–235 (2017). https://doi.org/10.1007/s11075-016-0251-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0251-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation