Skip to main content

Templates for linear algebra problems

  • Chapter
  • First Online:
Computer Science Today

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1000))

  • 199 Accesses

Abstract

The increasing availability of advanced-architecture computers is having a very significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra -in particular, the solution of linear systems of equations and eigenvalue problems — lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to help the user on advanced-architecture computers.

Much of the work in developing linear algebra software for advancedarchitecture computers is motivated by the need to solve large problems on the fastest computers available. In this paper, we focus on four basic issues: (1) the motivation for the work; (2) the development of standards for use in linear algebra and the building blocks for a library; (3) aspects of templates for the solution of large sparse systems of linear algorithm; and (4) templates for the solution of large sparse eigenvalue problems. This last project is under development and we will pay more attention to it in this paper.

This work was made possible in part by grants from the Defense Advanced Research Projects Agency under contract DAAL03-91-C-0047 administered by the Army Research Office, the Office of Scientific Computing U.S. Department of Energy under Contract DE-AC05-84OR21400, the National Science Foundation Science and Technology Center Cooperative Agreement No. CCR-8809615, and National Science Foundation Grant No. ASC-9005933.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen. LAPACK Users' Guide, Release 2.0. SIAM, Philadelphia, 1995. 324 pages. URL http://www.netlib.org/lapack/lug/lapack_lug.html.

    Google Scholar 

  2. E. Anderson and J. Dongarra. Results from the initial release of LAPACK. Techn. Rep. LAPACK working note 16, Computer Science Department, University of Tennessee, Knoxville, TN, 1989.

    Google Scholar 

  3. E. Anderson and J. Dongarra. Evaluating block algorithm variants in LAPACK. Techn. Rep. LAPACK working note 19, Computer Science Department, University of Tennessee, Knoxville, TN, 1990.

    Google Scholar 

  4. Z. Bai. Progress in the numerical solution of the nonsymmetric eigenvalue problem, 1993. To appear in J. Num. Lin. Alg. Appl.

    Google Scholar 

  5. Z. Bai and J. Demmel. Design of a parallel nonsymmetric eigenroutine toolbox, Part I. In Proc. 6th SIAM Conference on Parallel Processing for Scientific Computing. SIAM, 1993. Long version: UC Berkeley Comp. Sc. Rep. all.ps.Z via anonymous ftp from tr-ftp.cs.berkeley.edu, directory pub/tech-reports/csd/csd-92-718.

    Google Scholar 

  6. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, V. Pozo, Romime C., and H. van der Vorst. Templates for the solution of linear systems: Building blocks for iterative methods. SIAM, 1994. URL http://www.netlib.org/templates/templates.ps.

    Google Scholar 

  7. R. Boisvert. The architecture of an intelligent virtual mathematical software repository system. Mathematics and Computers in Simulation, 36:269–279, 1994.

    Article  Google Scholar 

  8. J. Choi, J. J. Dongarra, R. Pozo, and D. W. Walker. ScaLAPACK: A scalable linear algebra library for distributed memory concurrent computers. In Proc. 4th Symposium on the Frontiers of Massively Parallel Computation, pages 120–127. IEEE Computer Society Press, 1992.

    Google Scholar 

  9. J. Choi, J. J. Dongarra, and D. W. Walker. The design of scalable software libraries for distributed memory concurrent computers. In J. J. Dongarra and B. Tourancheau, editors, Environments and Tools for Parallel Scientific Computing. Elsevier Science Publishers, 1993.

    Google Scholar 

  10. D. E. Culler, A. Dusseau, S. C. Goldstein, A. Krishnamurthy, S. Lumetta, T. von Eicken, and K. Yelick. Introduction to Split-C: Version 0.9. Techn. Rep. Computer Science Division — EECS, Univ. of California, Berkeley, CA, February 1993.

    Google Scholar 

  11. J. Cullum and R. A. Willoughby. Lanczos algorithms for large symmetric eigenvalue computations. Birkhaüser, Basel, 1985. Vol.1, Theory, Vol.2. Program.

    Google Scholar 

  12. J. Demmel. Berkeley Lecture Notes in Numerical Linear Algebra. Mathematics Department, University of California, 1993. 215 pages.

    Google Scholar 

  13. J. Demmel and B. Kågström. The generalized Schur decomposition of an arbitrary pencil A — λB: Robust software with error bounds and applications. Parts I and II. ACM Trans. Math. Soft., 19(2), June 1993.

    Google Scholar 

  14. J. Demmel. LAPACK: A portable linear algebra library for supercomputers. In Proc. 1989 IEEE Control Systems Society Workshop on Computer-Aided Control System Design, December 1989.

    Google Scholar 

  15. J. J. Dongarra. Increasing the performance of mathematical software through high-level modularity. In Proc. Sixth Int. Symp. Comp. Methods in Eng. & Applied Sciences, Versailles, France, pages 239–248. North-Holland, 1984.

    Google Scholar 

  16. J. J. Dongarra, J. R. Bunch, C. B. Moler and G. W. Stewart. LINPACK Users' Guide. SIAM Press, 1979.

    Google Scholar 

  17. J. J. Dongarra. LAPACK Working Note 34: Workshop on the BLACS. Computer Science Dept. Technical Report CS-91-134, University of Tennessee, Knoxville, TN, May 1991.

    Google Scholar 

  18. J. J. Dongarra, J. Du Croz, S. Hammarling, and I. Duff. A set of level 3 basic linear algebra subprograms. ACM Trans. Math. Softw., 16(1):1–17, 1990.

    Article  Google Scholar 

  19. J. J. Dongarra, J. Du Croz, S. Hammarling, and R. Hanson. An extended set of Fortran basic linear algebra subroutines. ACM Trans. Math. Softw., 14(1):1–17, March 1988.

    Article  Google Scholar 

  20. J. J. Dongarra and R. A. van de Geijn. LAPACK Working Note 37: Twodimensional basic linear algebra communication subprograms. Computer Science Department, University of Tennessee, Knoxville, TN, October 1991.

    Google Scholar 

  21. J. Dongarra and E. Grosse. Distribution of mathematical software via electronic mail. Communications of the ACM, 30(5):403–407, July 1987. URL http://www.netlib.org/.

    Article  Google Scholar 

  22. E. W. Felten and S. W. Otto. Coherent parallel C. In G. C. Fox, editor, Proc. 3rd Conference on Hypercube Concurrent Computers and Applications, pages 440–450. ACM Press, 1988.

    Google Scholar 

  23. F. Gantmacher. The Theory of Matrices, Vol. II (transl.). Chelsea, New York, 1959.

    Google Scholar 

  24. B. S. Garbow, J. M. Boyle, J. J. Dongarra, and C. B. Moler. Matrix Eigensystem Routines — EISPACK Guide Extension, Lecture Notes in Computer Science, Vol. 51. Springer-Verlag, Berlin, 1977.

    Google Scholar 

  25. G. Golub and C. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, MD, 2nd edition, 1989.

    Google Scholar 

  26. R. W. Hockney and C. R. Jesshope. Parallel Computers. Adam Hilger Ltd., Bristol, UK, 1981.

    Google Scholar 

  27. T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, 2 edition, 1980.

    Google Scholar 

  28. C. Lawson, R. Hanson, D. Kincaid, and F. Krogh. Basic Linear Algebra Subprograms for Fortran usage. ACM Trans. Math. Soft., 5:308–323, 1979.

    Article  Google Scholar 

  29. A. Packard, M. Fan, and J. Doyle. A power method for the structured singular value. In IEEE Conf. on Decision and Control, pages 2132–2137, 1988.

    Google Scholar 

  30. B. Parlett. The Symmetric Eigenvalue Problem. Prentice Hall, Englewood Cliffs, NJ, 1980.

    Google Scholar 

  31. B. Parlett. The software scene in the extraction of eigenvalues from sparse matrices. SIAM J. Sci. Stat. Comp., 5:590–604, 1984.

    Article  Google Scholar 

  32. W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Numerical Recipes in C. Cambridge University Press, Cambridge, UK, 1991.

    Google Scholar 

  33. Y. Saad. Numerical methods for large eigenvalue problems. Manchester University Press, 1992.

    Google Scholar 

  34. G. Sleijpen and H. van der Vorst. Reliable updated residuals in hybrid Bi-CG methods. Preprint 886, Utrecht University, the Netherlands, 1994. to appear in Computing.

    Google Scholar 

  35. G. Sleijpen and H. van der Vorst. A Jacobi-Davidson iteration method for linear eigenvalue problems. Preprint 856 (revised), Utrecht University, the Netherlands, 1995. to appear in SIMAX.

    Google Scholar 

  36. B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler. Matrix Eigensystem Routines — EISPACK Guide, Lecture Notes in Computer Science, Vol. 6. Springer-Verlag, Berlin, 1976.

    Google Scholar 

  37. G. W. Stewart. Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Review, 15(4):727–764, Oct 1973.

    Article  Google Scholar 

  38. G. W. Stewart and J.-G. Sun. Matrix Perturbation Theory. Academic Press, New York, 1990.

    Google Scholar 

  39. C. überhuber. Computer-Numerik, Part 1 and 2, Springer-Verlag, Berlin, 1995.

    Google Scholar 

  40. A.J. van der Steen (ed). Aspects of Computational Science. NCF, Den Haag, the Netherlands, 1995.

    Google Scholar 

  41. P. Van Dooren. The computation of Kronecker's canonical form of a singular pencil. Lin. Alg. Appl., 27:103–141, 1979.

    Article  Google Scholar 

  42. P. Van Dooren. Reducing subspaces: Definitions, properties and algorithms. In B. Kågström and A. Ruhe, editors, Matrix Pencils, pages 58–73. Lecture Notes in Mathematics, Vol. 973, Proceedings, Springer-Verlag, Berlin, 1983.

    Google Scholar 

  43. J. H. Wilkinson. The Algebraic Eigenvalue Problem. Oxford University Press, Oxford, 1965.

    Google Scholar 

  44. J. Wilkinson and C. Reinsch. Handbook for Automatic Computation: Volume II — Linear Algebra. Springer-Verlag, New York, 1971.

    Google Scholar 

  45. P. Young, M. Newlin, and J. Doyle. Practical computation of the mixed Μ problem. In Proc. American Control Conference, pages 2190–2194, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Jan van Leeuwen

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bai, Z. et al. (1995). Templates for linear algebra problems. In: van Leeuwen, J. (eds) Computer Science Today. Lecture Notes in Computer Science, vol 1000. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015240

Download citation

  • DOI: https://doi.org/10.1007/BFb0015240

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60105-0

  • Online ISBN: 978-3-540-49435-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics