Skip to main content
SpringerLink
Log in

Banded Linear Systems

  • Chapter
  • First Online:
Parallelism in Matrix Computations

Part of the book series: Scientific Computation ((SCIENTCOMP))

  • 3029 Accesses

Abstract

We encounter banded linear systems in many areas of computational science and engineering, including computational mechanics and nanoelectronics, to name but a few.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arbenz, P., Hegland, M.: On the stable parallel solution of general narrow banded linear systems. High Perform. Algorithms Struct. Matrix Probl. 47–73 (1998)

    Google Scholar 

  2. Arbenz, P., Cleary, A., Dongarra, J., Hegland, M.: A comparison of parallel solvers for general narrow banded linear systems. Parallel Distrib. Comput. Pract. 2(4), 385–400 (1999)

    MATH  Google Scholar 

  3. Blackford, L., Choi, J., Cleary, A., D’Azevedo, E., Demmel, J., Dhillon, I., Dongarra, J., Hammarling, S., Henry, G., Petitet, A., Stanley, K., Walker, D., Whaley, R.: ScaLAPACK User’s Guide. SIAM, Philadelphia (1997). URL http://www.netlib.org/scalapack

  4. Conroy, J.: Parallel algorithms for the solution of narrow banded systems. Appl. Numer. Math. 5, 409–421 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dongarra, J., Johnsson, L.: Solving banded systems on a parallel processor. Parallel Comput. 5(1–2), 219–246 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. George, A.: Numerical experiments using dissection methods to solve \(n\) by \(n\) grid problems. SIAM J. Numer. Anal. 14, 161–179 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golub, G., Sameh, A., Sarin, V.: A parallel balance scheme for banded linear systems. Numer. Linear Algebra Appl. 8, 297–316 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Johnsson, S.: Solving narrow banded systems on ensemble architectures. ACM Trans. Math. Softw. 11, 271–288 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Meier, U.: A parallel partition method for solving banded systems of linear equations. Parallel Comput. 2, 33–43 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Tang, W.: Generalized Schwarz splittings. SIAM J. Sci. Stat. Comput. 13, 573–595 (1992)

    Article  MATH  Google Scholar 

  11. Wright, S.: Parallel algorithms for banded linear systems. SIAM J. Sci. Stat. Comput. 12, 824–842 (1991)

    Article  MATH  Google Scholar 

  12. Sameh, A., Kuck, D.: On stable parallel linear system solvers. J. Assoc. Comput. Mach. 25(1), 81–91 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dongarra, J.J., Sameh, A.: On some parallel banded system solvers. Technical Report ANL/MCS-TM-27, Mathematics Computer Science Division at Argonne National Laboratory (1984)

    Google Scholar 

  14. Gallivan, K., Gallopoulos, E., Sameh, A.: CEDAR—an experiment in parallel computing. Comput. Math. Appl. 1(1), 77–98 (1994)

    MATH  Google Scholar 

  15. Lawrie, D.H., Sameh, A.: The computation and communication complexity of a parallel banded system solver. ACM TOMS 10(2), 185–195 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Polizzi, E., Sameh, A.: A parallel hybrid banded system solver: the SPIKE algorithm. Parallel Comput. 32, 177–194 (2006)

    Article  MathSciNet  Google Scholar 

  17. Polizzi, E., Sameh, A.: SPIKE: a parallel environment for solving banded linear systems. Compon. Fluids 36, 113–120 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sameh, A., Kuck, D.: A parallel QR algorithm for symmetric tridiagonal matrices. IEEE Trans. Comput. 26(2), 147–153 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia (1999)

    Book  MATH  Google Scholar 

  20. Demko, S., Moss, W., Smith, P.: Decay rates for inverses of band matrices. Math. Comput. 43(168), 491–499 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  21. Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

    Book  MATH  Google Scholar 

  22. Golub, G., Van Loan, C.: Matrix Computations, 4th edn. Johns Hopkins. University Press, Baltimore (2013)

    Google Scholar 

  23. Davis, T.: Algorithm 915, SuiteSparseQR: multifrontal multithreaded rank-revealing sparse QR factorization. ACM Trans. Math. Softw. 38(1), 8:1–8:22 (2011). doi:10.1145/2049662.2049670, URL http://doi.acm.org/10.1145/2049662.2049670

    Google Scholar 

  24. Lou, G.: Parallel methods for solving linear systems via overlapping decompositions. Ph.D. thesis, University of Illinois at Urbana-Champaign (1989)

    Google Scholar 

  25. Naumov, M., Sameh, A.: A tearing-based hybrid parallel banded linear system solver. J. Comput. Appl. Math. 226, 306–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle-point problems. Acta Numer. 1–137 (2005)

    Google Scholar 

  27. Hockney, R., Jesshope, C.: Parallel Computers. Adam Hilger (1983)

    Google Scholar 

  28. Ortega, J.M.: Introduction to Parallel and Vector Solution of Linear Systems. Plenum Press, New York (1988)

    Book  MATH  Google Scholar 

  29. Golub, G., Ortega, J.: Scientific Computing: An Introduction with Parallel Computing. Academic Press Inc., San Diego (1993)

    Google Scholar 

  30. Davidson, A., Zhang, Y., Owens, J.: An auto-tuned method for solving large tridiagonal systems on the GPU. In: Proceedings of IEEE IPDPS, pp. 956–965 (2011)

    Google Scholar 

  31. Lopez, J., Zapata, E.: Unified architecture for divide and conquer based tridiagonal system solvers. IEEE Trans. Comput. 43(12), 1413–1425 (1994). doi:10.1109/12.338101

    Article  Google Scholar 

  32. Santos, E.: Optimal and efficient parallel tridiagonal solvers using direct methods. J. Supercomput. 30(2), 97–115 (2004). doi:10.1023/B:SUPE.0000040615.60545.c6, URL http://dx.doi.org/10.1023/B:SUPE.0000040615.60545.c6

    Google Scholar 

  33. Chang, L.W., Stratton, J., Kim, H., Hwu, W.M.: A scalable, numerically stable, high-performance tridiagonal solver using GPUs. In: Proceedings International Conference High Performance Computing, Networking Storage and Analysis, SC’12, pp. 27:1–27:11. IEEE Computer Society Press, Los Alamitos (2012). URL http://dl.acm.org/citation.cfm?id=2388996.2389033

  34. Goeddeke, D., Strzodka, R.: Cyclic reduction tridiagonal solvers on GPUs applied to mixed-precision multigrid. IEEE Trans. Parallel Distrib. Syst. 22(1), 22–32 (2011)

    Article  Google Scholar 

  35. Codenotti, B., Leoncini, M.: Parallel Complexity of Linear System Solution. World Scientific, Singapore (1991)

    Book  Google Scholar 

  36. Ascher, U., Mattheij, R., Russell, R.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  37. Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1966)

    MATH  Google Scholar 

  38. Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Dover Publications, New York (1992)

    Google Scholar 

  39. Bank, R.E.: Marching algorithms and block Gaussian elimination. In: Bunch, J.R., Rose, D. (eds.) Sparse Matrix Computations, pp. 293–307. Academic Press, New York (1976)

    Google Scholar 

  40. Bank, R.E., Rose, D.: Marching algorithms for elliptic boundary value problems. I: the constant coefficient case. SIAM J. Numer. Anal. 14(5), 792–829 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  41. Roache, P.: Elliptic Marching Methods and Domain Decomposition. CRC Press Inc., Boca Raton (1995)

    Google Scholar 

  42. Richardson, L.F.: Weather Prediction by Numerical Process. Cambridge University Press. Reprinted by Dover Publications, 1965 (1922)

    Google Scholar 

  43. Arbenz, P., Hegland, M.: The stable parallel solution of narrow banded linear systems. In: Heath, M., et al. (eds.) Proceedings of Eighth SIAM Conference Parallel Processing and Scientific Computing SIAM, Philadelphia (1997)

    Google Scholar 

  44. Bank, R.E., Rose, D.: Marching algorithms for elliptic boundary value problems. II: the variable coefficient case. SIAM J. Numer. Anal. 14(5), 950–969 (1977)

    Article  MathSciNet  Google Scholar 

  45. Higham, N.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)

    Book  MATH  Google Scholar 

  46. Higham, N.: Stability of parallel triangular system solvers. SIAM J. Sci. Comput. 16(2), 400–413 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  47. Viswanath, D., Trefethen, L.: Condition numbers of random triangular matrices. SIAM J. Matrix Anal. Appl. 19(2), 564–581 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hockney, R.: A fast direct solution of Poisson’s equation using Fourier analysis. J. Assoc. Comput. Mach. 12, 95–113 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  49. Gander, W., Golub, G.H.: Cyclic reduction: history and applications. In: Luk, F., Plemmons, R. (eds.) Proceedings of the Workshop on Scientific Computing, pp. 73–85. Springer, New York (1997). URL http://people.inf.ethz.ch/gander/papers/cyclic.pdf

  50. Amodio, P., Brugnano, L.: Parallel factorizations and parallel solvers for tridiagonal linear systems. Linear Algebra Appl. 172, 347–364 (1992). doi:10.1016/0024-3795(92)90034-8, URL http://www.sciencedirect.com/science/article/pii/0024379592900348

    Google Scholar 

  51. Heller, D.: Some aspects of the cyclic reduction algorithm for block tridiagonal linear systems. SIAM J. Numer. Anal. 13(4), 484–496 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lambiotte Jr, J., Voigt, R.: The solution of tridiagonal linear systems on the CDC STAR 100 computer. ACM Trans. Math. Softw. 1(4), 308–329 (1975). doi:10.1145/355656.355658, URL http://doi.acm.org/10.1145/355656.355658

    Google Scholar 

  53. Nassimi, D., Sahni, S.: An optimal routing algorithm for mesh-connected parallel computers. J. Assoc. Comput. Mach. 27(1), 6–29 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  54. Nassimi, D., Sahni, S.: Parallel permutation and sorting algorithms and a new generalized connection network. J. Assoc. Comput. Mach. 29(3), 642–667 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  55. George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973). URL http://www.jstor.org/stable/2156361

    Google Scholar 

  56. Amodio, P., Brugnano, L., Politi, T.: Parallel factorization for tridiagonal matrices. SIAM J. Numer. Anal. 30(3), 813–823 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  57. Johnsson, S.: Solving tridiagonal systems on ensemble architectures. SIAM J. Sci. Stat. Comput. 8, 354–392 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, Y., Cohen, J., Owens, J.: Fast tridiagonal solvers on the GPU. ACM SIGPLAN Not. 45(5), 127–136 (2010)

    Article  Google Scholar 

  59. Amodio, P., Mazzia, F.: Backward error analysis of cyclic reduction for the solution of tridiagonal systems. Math. Comput. 62(206), 601–617 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  60. Higham, N.: Bounding the error in Gaussian elimination for tridiagonal systems. SIAM J. Matrix Anal. Appl. 11(4), 521–530 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhang, Y., Owens, J.: A quantitative performance analysis model for GPU architectures. In: Proceedings of the 17th IEEE International Symposium on High-Performance Computer Architecture (HPCA 17) (2011)

    Google Scholar 

  62. El-Mikkawy, M., Sogabe, T.: A new family of k-Fibonacci numbers. Appl. Math. Comput. 215(12), 4456–4461 (2010). URL http://www.sciencedirect.com/science/article/pii/S009630031000007X

    Google Scholar 

  63. Fang, H.R., O’Leary, D.: Stable factorizations of symmetric tridiagonal and triadic matrices. SIAM J. Math. Anal. Appl. 28(2), 576–595 (2006)

    Article  MathSciNet  Google Scholar 

  64. Mikkelsen, C., Kågström, B.: Parallel solution of narrow banded diagonally dominant linear systems. In: Jónasson, L. (ed.) PARA 2010. LNCS, vol. 7134, pp. 280–290. Springer (2012). doi:10.1007/978-3-642-28145-7_28, URL http://dx.doi.org/10.1007/978-3-642-28145-7_28

    Google Scholar 

  65. Mikkelsen, C., Kågström, B.: Approximate incomplete cyclic reduction for systems which are tridiagonal and strictly diagonally dominant by rows. In: Manninen, P., Öster, P. (eds.) PARA 2012. LNCS, vol. 7782, pp. 250–264. Springer (2013). doi:10.1007/978-3-642-36803-5_18, URL http://dx.doi.org/10.1007/978-3-642-36803-5_18

    Google Scholar 

  66. Bini, D., Meini, B.: The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond. Numer. Algorithms 51(1), 23–60 (2008). doi:10.1007/s11075-008-9253-0, URL http://www.springerlink.com/index/10.1007/s11075-008-9253-0; http://www.springerlink.com/content/m40t072h273w8841/fulltext.pdf

    Google Scholar 

  67. Sameh, A.: Numerical parallel algorithms—a survey. In: Kuck, D., Lawrie, D., Sameh, A. (eds.) High Speed Computer and Algorithm Optimization, pp. 207–228. Academic Press, Sans Diego (1977)

    Google Scholar 

  68. Mathias, R.: The instability of parallel prefix matrix multiplication. SIAM J. Sci. Comput. 16(4) (1995), to appear

    Google Scholar 

  69. Eğecioğlu, O., Koç, C., Laub, A.: A recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors. J. Comput. Appl. Math. 27, 95–108 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  70. Dubois, P., Rodrigue, G.: An analysis of the recursive doubling algorithm. In: Kuck, D., Lawrie, D., Sameh, A. (eds.) High Speed Computer and Algorithm Organization, pp. 299–305. Academic Press, San Diego (1977)

    Google Scholar 

  71. Hammarling, S.: A survey of numerical aspects of plane rotations. Report Maths. 1, Middlesex Polytechnic (1977). URL http://eprints.ma.man.ac.uk/1122/. Available as Manchester Institute for Mathematical Sciences MIMS EPrint 2008.69

  72. Bar-On, I., Codenotti, B.: A fast and stable parallel QR algorithm for symmetric tridiagonal matrices. Linear Algebra Appl. 220, 63–95 (1995). doi:10.1016/0024-3795(93)00360-C, URL http://www.sciencedirect.com/science/article/pii/002437959300360C

    Google Scholar 

  73. Gill, P.E., Golub, G., Murray, W., Saunders, M.: Methods for modifying matrix factorizations. Math. Comput. 28, 505–535 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  74. Lakshmivarahan, S., Dhall, S.: Parallelism in the Prefix Problem. Oxford University Press, New York (1994)

    Google Scholar 

  75. Cleary, A., Dongarra, J.: Implementation in ScaLAPACK of divide and conquer algorithms for banded and tridiagonal linear systems. Technical Report UT-CS-97-358, University of Tennessee Computer Science Technical Report (1997)

    Google Scholar 

  76. Bar-On, I., Codenotti, B., Leoncini, M.: Checking robust nonsingularity of tridiagonal matrices in linear time. BIT Numer. Math. 36(2), 206–220 (1996). doi:10.1007/BF01731979, URL http://dx.doi.org/10.1007/BF01731979

    Google Scholar 

  77. Bar-On, I.: Checking non-singularity of tridiagonal matrices. Electron. J. Linear Algebra 6, 11–19 (1999). URL http://math.technion.ac.il/iic/ela

  78. Bondeli, S.: Divide and conquer: a parallel algorithm for the solution of a tridiagonal system of equations. Parallel Comput. 17, 419–434 (1991)

    Article  MATH  Google Scholar 

  79. Wang, H.: A parallel method for tridiagonal equations. ACM Trans. Math. Softw. 7, 170–183 (1981)

    Article  MATH  Google Scholar 

  80. Wright, S.: Parallel algorithms for banded linear systems. SIAM J. Sci. Stat. Comput. 12(4), 824–842 (1991)

    Article  MATH  Google Scholar 

  81. Stewart, G.: Modifying pivot elements in Gaussian elimination. Math. Comput. 28(126), 537–542 (1974)

    Article  MATH  Google Scholar 

  82. Li, X., Demmel, J.: SuperLU-DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM TOMS 29(2), 110–140 (2003). URL http://doi.acm.org/10.1145/779359.779361

  83. Venetis, I.E., Kouris, A., Sobczyk, A., Gallopoulos, E., Sameh, A.: A direct tridiagonal solver based on Givens rotations for GPU-based architectures. Technical Report HPCLAB-SCG-06/11-14, CEID, University of Patras (2014)

    Google Scholar 

  84. Bunch, J.: Partial pivoting strategies for symmetric matrices. SIAM J. Numer. Anal. 11(3), 521–528 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  85. Bunch, J., Kaufman, K.: Some stable methods for calculating inertia and solving symmetric linear systems. Math. Comput. 31, 162–179 (1977)

    Article  MathSciNet  Google Scholar 

  86. Erway, J., Marcia, R.: A backward stability analysis of diagonal pivoting methods for solving unsymmetric tridiagonal systems without interchanges. Numer. Linear Algebra Appl. 18, 41–54 (2011). doi:10.1002/nla.674, URL http://dx.doi.org/10.1002/nla.674

    Google Scholar 

  87. Erway, J.B., Marcia, R.F., Tyson, J.: Generalized diagonal pivoting methods for tridiagonal systems without interchanges. IAENG Int. J. Appl. Math. 4(40), 269–275 (2010)

    MathSciNet  Google Scholar 

  88. Golub, G.H., Meurant, G.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton (2009)

    Book  Google Scholar 

  89. Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices. Volume I: Linear Systems. Johns Hopkins University Press (2008)

    Google Scholar 

  90. Gantmacher, F., Krein, M.: Sur les matrices oscillatoires et complèments non négatives. Composition Mathematica 4, 445–476 (1937)

    Google Scholar 

  91. Bukhberger, B., Emelyneko, G.: Methods of inverting tridiagonal matrices. USSR Comput. Math. Math. Phys. 13, 10–20 (1973)

    Article  Google Scholar 

  92. Swarztrauber, P.N.: A parallel algorithm for solving general tridiagonal equations. Math. Comput. 33, 185–199 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  93. Yamamoto, T., Ikebe, Y.: Inversion of band matrices. Linear Algebra Appl. 24, 105–111 (1979). doi:10.1016/0024-3795(79)90151-4, URL http://www.sciencedirect.com/science/article/pii/0024379579901514

    Google Scholar 

  94. Strang, G., Nguyen, T.: The interplay of ranks of submatrices. SIAM Rev. 46(4), 637–646 (2004). URL http://www.jstor.org/stable/20453569

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Engineering and Informatics Department, University of Patras, Patras, Greece

    Efstratios Gallopoulos

  2. Campus de Beaulieu, INRIA/IRISA, Rennes Cedex, France

    Bernard Philippe

  3. Department of Computer Science, Purdue University, West Lafayette, IN, USA

    Ahmed H. Sameh

Authors
  1. Efstratios Gallopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard Philippe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ahmed H. Sameh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Efstratios Gallopoulos .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Gallopoulos, E., Philippe, B., Sameh, A.H. (2016). Banded Linear Systems. In: Parallelism in Matrix Computations. Scientific Computation. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7188-7_5

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-7188-7_5

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-017-7187-0

  • Online ISBN: 978-94-017-7188-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation