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Iterative methods for the solution of elliptic difference equations on multiprocessors

  • Nonnumerical Parallel Algorithms
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Conpar 81 (CONPAR 1981)

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References

  1. Axelsson, O., On preconditioning and convergence acceleration in sparse matrix problems, Report CERN 74-10 of the CERN European Organization for Nuclear Research, Data Handling Division, Laboratory I, May 1974.

    Google Scholar 

  2. Boley, D., Buzbee, B. and Parter, S., On block relaxation techniques, University of Wisconsin, Computer Sciences Technical Report #318, June 1978.

    Google Scholar 

  3. Brezinski, C., Accélération de la convergence en analyse numerique, Lecture Notes in Mathematics 584, Springer-Verlag, 1977.

    Google Scholar 

  4. Brezinski, C., Padé-Type Approximation and General Orthogonal Polynomials, Birkhäuser-Verlag, 1980.

    Google Scholar 

  5. Buzbee, B., Golub, G. and Howell, J., Vectorization for the CRAY-1 of some methods for solving elliptic difference equations, High Speed Computer and Algorithm Organization, Ed. D. Kuck, D. Lawrie, and A. Sameh, 255–272, Academic Press, 1977.

    Google Scholar 

  6. Concus, P., Golub, G., and O'Leary, D., A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, Sparse Matrix Computations, Ed. J. Bunch and D. Rose, 309–332, Academic Press, 1976.

    Google Scholar 

  7. Ericksen, J., Iterative and direct methods for solving the Poisson's equation and their adaptability to Illiac IV, CAC document No. 60, University of Illinois at Urbana-Champaign, December 1972.

    Google Scholar 

  8. George, A., Poole, W., and Voigt, R., Analysis of dissection algorithms for vector computers, ICASE Report No. 76-17, June 1976.

    Google Scholar 

  9. Golub, G., and Varga, R., Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods, Parts I and II, Numer. Math. 3, 147–168, 1961.

    Article  MathSciNet  Google Scholar 

  10. Hageman, L., The estimation of acceleration parameters for the Chebyshev polynomial and successive overrelaxation iterative methods, WAPD-TM-1038, Bettis Atomic Power Laboratory, 1972.

    Google Scholar 

  11. Hageman, L., Luk, F., and Young, D., On the equivalence of certain iterative acceleration methods, SIAM J. Numer. Anal. 17 (6), 852–873, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hayes, L., Comparative analysis of iterative techniques for solving Laplace's equation on the unit square on a parallel processor, M.S. Thesis, Department of Mathematics, University of Texas-Austin, 1974.

    Google Scholar 

  13. Hestenes, M., The conjugate gradient method for solving linear equations, Proceedings of the Sixth Symposium in Applied Mathematics, Ed. J. Curtiss, 83-102, A.M.S., Providence, R.I., 1956.

    Google Scholar 

  14. Householder, A., The Theory of Matrices in Numerical Analysis, Blaisdell, 1964.

    Google Scholar 

  15. Householder, A., and Bauer, F., On certain iterative methods for solving linear systems, Numer. Math. 2, 55–59, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  16. Lambiotte, J., The solution of linear systems of equations on a vector computer, Ph.D. dissertation, Department of Applied Mathematics and Computer Science, University of Virginia, 1975.

    Google Scholar 

  17. Liu, J., The solution of mesh equations on a parallel computer, Department of Computer Science, Report CS-74-19, University of Waterloo, October 1974.

    Google Scholar 

  18. Mitchell, A., and Wait, R., The Finite Element Method for Partial Differential Equations, Wiley, 1976.

    Google Scholar 

  19. Miura, K., The block-iterative method for Illiac IV, CAC document No. 41, University of Illinois at Urbana-Champaign, July 1971.

    Google Scholar 

  20. Ortega, J. and Voigt, R., Solution of partial differential equations on vector computers, 1977 Army Numerical Analysis and Computers Conference.

    Google Scholar 

  21. Parter, S., and Steuerwalt, M., On k-line and k x k block iterative schemes for a problem arising in three-dimensional elliptic difference equations, SIAM J. Numer. Anal. 17 (6), 823–839, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  22. Reid, J., The use of conjugate gradients for systems of linear equations possessing "property A", SIAM J. Numer. Anal. 9 (2), 325–332, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  23. Saad, Y., and Sameh, A., A parallel block Stiefel method for solving positive definite systems, Conference on Elliptic Problem Solvers, Los Alamos Scientific Laboratory, Santa Fe, N.M., 1980.

    Google Scholar 

  24. Stewart, G., Introduction to Matrix Computations, Academic Press, 1973.

    Google Scholar 

  25. Stiefel, E., Kernel polynomials in linear algebra and their numerical applications, National Bureau of Standards, Applied Math. Series 49, 1–22, 1958.

    MathSciNet  Google Scholar 

  26. Varga, R., Matrix Iterative Analysis, Prentice-Hall, 1962.

    Google Scholar 

  27. Wachspress, E., Iterative Solution of Elliptic Systems, Prentice-Hall, 1966.

    Google Scholar 

  28. Wynn, P., Acceleration techniques for iterated vector and matrix problems, Math. Comp. 16, 301–322, 1962.

    Article  MATH  MathSciNet  Google Scholar 

  29. Young, D., Iterative Solution of Large Linear Systems, Academic Press, 1971.

    Google Scholar 

  30. Zienkiewicz, O., The Finite Element Method in Engineering Science, McGraw-Hill, 1971.

    Google Scholar 

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W. Brauer P. Brinch Hansen D. Gries C. Moler G. Seegmüller J. Stoer N. Wirth Wolfgang Händler

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© 1981 Springer-Verlag Berlin Heidelberg

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Saad, Y., Sameh, A. (1981). Iterative methods for the solution of elliptic difference equations on multiprocessors. In: Brauer, W., et al. Conpar 81. CONPAR 1981. Lecture Notes in Computer Science, vol 111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0105132

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  • DOI: https://doi.org/10.1007/BFb0105132

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  • Print ISBN: 978-3-540-10827-6

  • Online ISBN: 978-3-540-38715-2

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