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Local fill reduction techniques for sparse symmetric linear systems

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Abstract

Local algorithms for obtaining pivot orderings for sparse symmetric coefficient matrices are reviewed together with their mathematical background, appropriate data structures and details of efficient implementation. Heuristics that go beyond the classical Minimum Degree and Minimum Local Fill scoring functions are discussed, illustrated, improved and extensively tested on a test suite of matrices from various applications. Our tests indicate that the presented techniques have the potential of accelerating circuit simulation significantly.

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References

  1. George A (1973) Nested dissection of a regular finite element mesh. SIAM J Numer Anal 10:345–363

    Article  MATH  MathSciNet  Google Scholar 

  2. Cavers IA (1989) Using deficiency measure for tiebreaking the minimum degree algorithm. Technical Report 89-2, Department Computer Science, The University of British Columbia, Vancouver, B. C., Canada V6T 1W5, 10 Jan 1989

  3. Lustig IJ, Marsten RE, Shanno DF (1992) The interaction of algorithms and architectures for interior point methods. In: Pardalos PM (ed) Advances in optimization and parallel computing, North-Holland, pp. 190–204

  4. Mészáros C (1998) Ordering heuristics in interior point LP methods. In: Giannessi F, Komlósi S, Rapcśak T (eds), New trends in mathematical programming. Kluwer, The Netherlands, pp. 203–221

    Google Scholar 

  5. Chua LO, Desoer CA, Kuh ES (1987) Linear and nonlinear circuits. McGraw–Hill

  6. Feldmann U, Wever UA, Zheng Q, Schultz R, Wriedt H (1992) Algorithms for modern circuit simulation. Int J Electron Commun (AEÜ) 46(4):274–285

    Google Scholar 

  7. Nagel LW (1975) SPICE 2: A computer program to simulate semiconductor circuits. Technical report ERL-M520, University of California Berkeley, Electronic Res Lab, Berkeley, CA

  8. Golub GH, Van Loan CF (1996) Matrix computations, The John Hopkins Univ Press, 3rd edn. Baltimore

  9. Ortega JM (1988) The ijk forms of factorization methods. I.Vector computers. Parallel Comput 7(2):135–147

    Article  MATH  MathSciNet  Google Scholar 

  10. Duff IS, Erisman AM, Reid JK (1986) Direct methods for sparse matrices. Oxford University Press, Oxford

    MATH  Google Scholar 

  11. Yannakakis M (1981) Computing the minimum fill-in is NP-complete. SIAM J Algebraic Discrete Methods 2(1):77–79

    MATH  MathSciNet  Google Scholar 

  12. Markowitz HM (1957) The elimination form of the inverse and its application to linear programming. Manage Sci 3:255–269

    Article  MATH  MathSciNet  Google Scholar 

  13. Tinney WF, Walker JW (1967) Direct solution of sparse network equations by optimally ordered triangular factorization. Proc IEEE 55:1801–1809

    Article  Google Scholar 

  14. Parter SV (1961) The use of linear graphs in Gauss elimination. SIAM Rev 3:119–130

    Article  MATH  MathSciNet  Google Scholar 

  15. Rose DJ (1972) A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations. In: Read RC (eds) Graph theory and computing. Academic, New York, pp 183–217

    Google Scholar 

  16. Ng EG, Raghavan P (1999) Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM J Matrix Anal Appl 20(4):902–914

    Article  MATH  MathSciNet  Google Scholar 

  17. Rothberg E, Eisenstat SC (1998) Node selection strategies for bottom-up sparse matrix ordering. SIAM J Matrix Anal Appl 19(3):682–695

    Article  MATH  MathSciNet  Google Scholar 

  18. Schenk O (2000) Scalable parallel sparse LU factorization methods on shared memory multiprocessors. Dissertationsschrift ETH no. 13515, ETH Zürich, Integrated Systems Lab.

  19. Reißig G (2001) On methods for ordering sparse matrices in circuit simulation. In: Proceedings 2001 IEEE int symp on circuits and systems (ISCAS), vol 5, Sydney, Australia, May 6–9, pp. 315–318

  20. George A, Liu JW (1989) The evolution of the minimum degree ordering algorithm. SIAM Rev 31(1):1–19

    Article  MATH  MathSciNet  Google Scholar 

  21. Knuth DE (1973) The Art of Computer Programming, vol 3, Sorting and Searching. Addison Wesley, Reading MA

    Google Scholar 

  22. Harary F (1969) Graph theory. Addison-Wesley, Reading

    Google Scholar 

  23. Speelpenning B (1973) The generalized element method. Preliminary report. Notices Amer Math Soc, vol 20, no. 2, p. A-280, 73T-C18

  24. Speelpenning B (1973) The generalized element method. Technical Report UIUCDCS-R-78-946, Univ Illinois at Urbana-Champaign, Dept Comp Sci, Urbana, IL, Nov 1978. Reprint of 1973.

  25. Duff IS, Reid JK (1983) The multifrontal solution of indefinite sparse symmetric linear equations. ACM Trans Math Software 9(3):302–325

    Article  MATH  MathSciNet  Google Scholar 

  26. George JA, Liu JW (1981) Computer solution of large sparse positive definite systems. Prentice-Hall, Englewood Cliffs NJ

    MATH  Google Scholar 

  27. George A, McIntyre DR (1978) On the application of the minimum degree algorithm to finite element systems. SIAM J Numer Anal 15(1):90–112

    Article  MATH  MathSciNet  Google Scholar 

  28. Sherman AH (1975) Yale sparse matrix package - User’s guide. Technical Report UCID-30114, Lawrence Livermore Lab, Univ of California, Livermore, CA, Aug 1975

  29. Liu JWH (1985) Modification of the minimum-degree algorithm by multiple elimination. ACM Trans Math Software 11(2):141–153

    Article  MATH  MathSciNet  Google Scholar 

  30. SPOOLES. An object oriented software library for solving sparse linear systems of equations. netlib.bell-labs.com/netlib/linalg/spooles/, 1999

  31. Ashcraft C (1995) Compressed graphs and the minimum degree algorithm. SIAM J Sci Comput 16(6):1404–1411

    Article  MATH  MathSciNet  Google Scholar 

  32. Amestoy PR, Davis TA, Duff IS (1996) An approximate minimum degree ordering algorithm. SIAM J Matrix Anal Appl 17(4):886–905

    Article  MATH  MathSciNet  Google Scholar 

  33. Wing O, Huang J (1975) SCAP - a sparse matrix circuit analysis program. In: Proceedings of 1975 IEEE International Symposium on circuits and systems (ISCAS), pp 213–215

  34. Vlach J, Singhal K (1983) Computer methods for circuit analysis and design. Van Nostrand Rheinhold

  35. Duff IS, Erisman AM, Reid JK (1976) On George’s nested dissection method. SIAM J Numer Anal 13(5):686–695

    Article  MATH  MathSciNet  Google Scholar 

  36. Reißig G, Klimpel T (2000) Fill-In Minimierung in der Schaltkreissimulation. Internal rept, Infineon Technologies, MP PTS, München, 28 Mar.

  37. Reißig G, Klimpel T (2001) Verfahren zum computergestützten Vorhersagen des Verhaltens eines durch Differentialgleichungen beschreibbaren Systems. Patent appl. DE 101 03 793 A 1 (pend.), 28 Jan 2001. (“Method for computer-aided prediction of the behavior of a system described by differential equations”, in German)

  38. Duff IS, Grimes RG, Lewis JG (1992) Users’ guide for the Harwell–Boeing sparse matrix collection (Release I). Technical Report TR/PA/92/86, Rutherford Appleton Lab., Oxon, England, Boeing Computer Services, Res. and Techn. Div., Seattle, WA, USA, Oct 1992

  39. Davis T (1995) University of florida sparse matrix collection. http://www.cise.ufl.edu/research/sparse/matrices/, NA Digest, vol 92, no. 42, Oct. 16, 1994, vol 96, no. 28, July 23, 1996, vol 97, no. 23, June 7, 1997, 2 June 1995

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  1. Fakultät IV – Fachgebiet Regelungssysteme, TU Berlin, Sekretariat EN 11, Einsteinufer 17, 10587, Berlin, Germany

    Gunther Reißig

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  1. Gunther Reißig
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Reißig, G. Local fill reduction techniques for sparse symmetric linear systems. Electr Eng 89, 639–652 (2007). https://doi.org/10.1007/s00202-006-0042-2

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