Numerical Algorithms

, Volume 76, Issue 1, pp 211–235 | Cite as

An asynchronous direct solver for banded linear systems

  • Michael A. Jandron
  • Anthony A. Ruffa
  • James Baglama
Original Paper


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.


LAPACK Banded Linear systems Parallel Direct solver Asynchronous 

Mathematics Subject Classification (2010)

15A06 65F05 


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  1. 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)CrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)Google Scholar
  6. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Duff, I., Reid, J.: The Multifrontal Solution of Indefinite Sparse Symmetric Linear Equations. ACM Trans. Math. Softw. 9(3), 302–325 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duff, I.: A review of frontal methods for solving linear systems. Comput. Phys. Commun. 97, 4552 (1996)Google Scholar
  10. 10.
    Gavel, D.T.: Solution to the Problem of Instability in Banded Toeplitz Solvers. IEEE Trans. Signal Process. 40, 464 (1992)Google Scholar
  11. 11.
    Golub, G., Van Loan, C.: Matrix Computations, 4th Edn. John Hopkins University Press, Baltimore, MD (2013)Google Scholar
  12. 12.
    Fortran ISML Numerical Library. User’s Guide, Version 7.0, Rogue Wave Software, 1315 West Century Drive, Suite 150, Louisville, CO 80027 (2010)Google Scholar
  13. 13.
    Irons, B.: A Frontal Solution Program for Finite Element Analysisss. Int. J. Numer. Methods Eng. 2, 5–32 (1970)CrossRefzbMATHGoogle Scholar
  14. 14.
    Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications NY (2009)Google Scholar
  15. 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)Google Scholar
  16. 16.
    Liu, J.: The Multifrontal Method for Sparse Matrix Solution. Theory and Practice, SIAM Review 34(1), 82–109 (1992)MathSciNetzbMATHGoogle Scholar
  17. 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)Google Scholar
  18. 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)Google Scholar
  19. 19.
    MacLeod, A.J.: Instability in the solution of banded Toeplitz systems. IEEE Trans. Acoust. Speech Signal Process. 37, 1449 (1989)Google Scholar
  20. 20.
    Mandel, J.: Balancing domain decomposition. Comm. Numer. Methods Engrg. 9, 233–241 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Meek, D.S.: The Inverses of Toeplitz Band Matrices. Linear Algebra Appl. 49, 117 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Intel Math Kernel Library, Reference Manual, Intel Corporation, 2200 Mission College Blvd., Santa Clara, CA 95052-8119, USA (2010)Google Scholar
  23. 23.
    Ruffa, A.: A solution approach for lower Hessenberg linear systems. ISRN Applied Mathematics 2011, 236727 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)Google Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Thomas, L.H.: Elliptic Problems in Linear Differential Equations over a Network, Watson Sci. Comput. Lab Report. Columbia University, New York (1949)Google Scholar
  29. 29.
    Trench, W.F.: An algorithm for the inversion of finite Toeplitz smatrices. J. Soc. Ind. Appl. Math. 12, 515 (1964)Google Scholar
  30. 30.
    Van der Vorst, H.A.: Analysis of a parallel solution method for tridiagonal linear systems 5(3), 303311 (1987)Google Scholar
  31. 31.
    Zohar, S.: The solution of a Toeplitz set of linear equations. J. ACM 21, 272 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York (outside the USA) 2017

Authors and Affiliations

  • Michael A. Jandron
    • 1
  • Anthony A. Ruffa
    • 1
  • James Baglama
    • 2
  1. 1.Naval Undersea Warfare CenterNewportUSA
  2. 2.Department of MathematicsUniversity of Rhode IslandKingstonUSA

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