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International Conference on High-Performance Computing

HiPC 2002: High Performance Computing — HiPC 2002 pp 545–554Cite as

Fast Stable Solver for Sequentially Semi-separable Linear Systems of Equations

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2552))

Abstract

In this paper we will present a fast backward stable algorithm for the solution of certain structured matrices which can be either sparse or dense. It essentially combines the fast solution techniques for banded plus semi-separable linear systems of equations of Chandrasekaran and Gu [4] with similar techniques of Dewilde and van der Veen for time-varying systems [12].

Partially supported by grant B521537 from Lawrence Livermore national Laboratory and NSF Career Award CCR-9734290.

This research was supported in part by NSF Career Award CCR-9702866 and by Alfred Sloan Research Fellowship BR-3720.

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References

  1. J. J. Dongarra, J. Du Croz, S. Hammarling, and I. Du., Algorithm 679: A Set of Level 3 Basic Linear Algebra Subprograms: Model Implementation and Test Programs, ACM Transactions on Mathematical Software, 16 (1990), pp. 18–28.

    Article  MATH  Google Scholar 

  2. F.X. Canning and K. Rogovin, Fast direct solution of moment-method matrices, IEEE Antennas and Propagation Magazine, 40 (1998).

    Google Scholar 

  3. J. Carrier, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 669–686.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Chandrasekaran and M. Gu, Fast and stable algorithms for banded plus semiseparable matrices, submitted to SIAM J. Matrix Anal. Appl., 2000. 545, 546, 549, 552

    Google Scholar 

  5. S. Chandrasekaran and M. Gu, A fast and stable solver for recursively semiseparable systems of equations, in Structured matrices in mathematics, computer science and engineering, II, edited by Vadim Olshevsky, in the Contemporary Mathematics series, AMS publications, 2001.

    Google Scholar 

  6. S. Chandrasekaran and M. Gu, Fast and Stable Eigendecomposition of Symmetric Banded plus Semi-separable Matrices, 1999, Linear Algebra and its Applications, Volume 313, Issues 1-3, 1 July 2000, pages 107–114.

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Chandrasekaran and M. Gu, A Divide-and-Conquer Algorithm for the Eigendecomposition of Symmetric Block-Diagonal Plus Semiseparable Matrices, 1999, accepted for publication in Numerische Mathematik.

    Google Scholar 

  8. S. Chandrasekaran, M. Gu, and T. Pals, A fast and stable solver for smooth recursively semi-separable systems. Paper presented at the SIAM Annual Conference, San Diego, CA, 2001, and SIAM Conference of Linear Algebra in Controls, Signals and Systems, Boston, MA, 2001.

    Google Scholar 

  9. S. Chandrasekaran, P. Dewilde, M. Gu, T. Pals, and A.-J. van der Veen, Fast and stable solvers for sequentially recursively semi-separable linear systems of equations. Submitted to SIAM Journal on Matrix Analysis and its Applications, 2002. 549, 552

    Google Scholar 

  10. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, 1983.

    Google Scholar 

  11. D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, vol. 93, Springer-Verlag, 1992. 545, 552

    Google Scholar 

  12. P. Dewilde and A. van der Veen, Time-varying systems and computations. Kluwer Academic Publishers, 1998. 545, 546, 550

    Google Scholar 

  13. Y. Eidelman and I. Gohberg, Inversion formulas and linear complexity algorithm for diagonal plus semiseparable matrices, Computers and Mathematics with Applications, 33 (1997), Elsevier, pp. 69–79.

    Article  MATH  MathSciNet  Google Scholar 

  14. Y. Eidelman and I. Gohberg, A look-ahead block Schur algorithm for diagonal plus semiseparable matrices, Computers and Mathematics with Applications, 35 1998), pp. 25–34.

    Article  MATH  MathSciNet  Google Scholar 

  15. Y. Eidelman and I. Gohberg, A modification of the Dewilde van der Veen method for inversion of finite structured matrices, Linear Algebra and its Applications, volumes 343-344, 1 march 2001, pages 419–450.

    Google Scholar 

  16. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comp. Phys., 73 (1987), pp. 325–348.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Gurel and W. C. Chew, Fast direct (non-iterative) solvers for integral-equation formulations of scattering problems, IEEE Antennas and Propagation Society International Symposium, 1998 Digest, Antennas: Gateways to the Global Network, pp. 298–301, vol. 1.

    Google Scholar 

  18. W. Hackbusch, A sparse arithmetic based on H-matrices. Part-I: Introduction to H-matrices, Computing 62, pp. 89–108, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  19. W. Hackbusch and B.N. Khoromskij, A sparse H-matrix arithmetic. Part-II: application to multi-dimensional problems, Computing 64, pp. 21–47, 2000.

    MATH  MathSciNet  Google Scholar 

  20. W. Hackbusch and B.N. Khoromskij, A sparse H-matrix arithmetic: general complexity estimates, Journal of Computational and Applied Mathematics, volume 125, pp. 79–501, 2000.

    Article  MathSciNet  Google Scholar 

  21. D. Gope and V. Jandhyala, An iteration-free fast multilevel solver for dense method of moment systems, Electrical Performance of Electronic Packaging, 2001, pp. 177–180.

    Google Scholar 

  22. R. Kussmaul, DeEin numerisches Verfahren zur Lösung des Neumannschen Aussenraumproblems für die Helmholtzsche Schwingungsgleichung, Computing 4, 246–273, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  23. June-Yub Lee and L. Greengard, A fast adaptive numerical method for stiff twopoint boundary value problems, SIAM Journal on Scientific Computing, vol.18, (no.2), SIAM, March 1997, pp. 403–29. 552

    Article  MATH  MathSciNet  Google Scholar 

  24. E. Martensen, DeÜber eine Methode zum räumlichen Neumannschen Problem mit einer Anwendung für torusartige Berandungen, Acta Math. 109, 75–135, 1963.

    Article  MATH  MathSciNet  Google Scholar 

  25. N. Mastronardi, S. Chandrasekaran and S. van Huffel, Fast and stable two-way chasing algorithm for diagonal plus semi-separable systems of linear equations, Numerical Linear Algebra with Applications, Volume 38, Issue 1, January 2000, pages 7–12.

    Google Scholar 

  26. N. Mastronardi, S. Chandrasekaran and S. van Huffel, Fast and stable algorithms for reducing diagonal plus semi-separable matrices to tridiagonal and bidiagonal form, BIT, volume 41, number 1, March 2001, pages 149–157.

    Article  MATH  MathSciNet  Google Scholar 

  27. T. Pals. Ph. D. Thesis, Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, 2002.

    Google Scholar 

  28. V. Rokhlin, Applications of volume integrals to the solution of PDEs, J. Comp. Phys., 86 (1990), pp. 414–439.

    Article  MATH  MathSciNet  Google Scholar 

  29. P. Starr, On the numerical solution of one-dimensional integral and differential equations, Thesis advisor: V. Rokhlin, Research Report YALEU/DCS/RR-888, Department of Computer Science, Yale University, New Haven, CT, December 1991. 552

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, Santa Barbara

    S. Chandrasekaran & T. Pals

  2. University of California, Berkeley

    M. Gu

  3. DIMES, Delft

    P. Dewilde & A. J. van der Veen

Authors
  1. S. Chandrasekaran
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  2. P. Dewilde
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  3. M. Gu
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  4. T. Pals
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  5. A. J. van der Veen
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Editor information

Editors and Affiliations

  1. CISE Department, University of Florida, 32611, FL, Gainesville, USA

    Sartaj Sahni

  2. Department of Electrical Engineering, EEB 200C, University of Southern California, 3740 McClintok Ave., 90089-2562, CA, Los Angeles, USA

    Viktor K. Prasanna

  3. India Software Lab, IBM Global Services India Exports, Golden Enclave, TISL Tower, Airport Road, 560 017, Bangalore, India

    Uday Shukla

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© 2002 Springer-Verlag Berlin Heidelberg

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Chandrasekaran, S., Dewilde, P., Gu, M., Pals, T., van der Veen, A.J. (2002). Fast Stable Solver for Sequentially Semi-separable Linear Systems of Equations. In: Sahni, S., Prasanna, V.K., Shukla, U. (eds) High Performance Computing — HiPC 2002. HiPC 2002. Lecture Notes in Computer Science, vol 2552. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36265-7_51

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  • DOI: https://doi.org/10.1007/3-540-36265-7_51

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00303-8

  • Online ISBN: 978-3-540-36265-4

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