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The Additive Peaceman–Rachford Method

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A new version of the parallel Alternating Direction Implicit (ADI) method by Peaceman and Rachford for solving systems of linear algebraic equations with positive-definite coefficient matrices represented as sums of two commuting terms is suggested. The algorithms considered are suited for solving two-dimensional grid boundary-value problems with separable variables, as well as the Sylvester and Lyapunov matrix equations. The approach to rising parallel efficiency proposed in the paper is based on representing rational functions as sums of partial fractions. An additive version of the factorized ADI method for solving Sylvester’s equation is described. Estimates of the speedup resulting from increasing the number of computer units are presented. These estimates demonstrate a potential advantage of using the additive algorithms when implemented on a supercomputer with large number of processors or cores.

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References

  1. A. A. Samarskiĭ and E. S. Nikolaev, Methods for Solving Grid Equations [in Russian], Nauka, Moscow (1978).

  2. V. P. Il’in, Finite-Difference and Finite-Volume Methods for Elliptic Equations [in Russian], IVM SO RAN, Novosibirsk (2000).

  3. V. P. Il’in, Methods and Tehnologies of Finite Elements [in Russian], IVM SO RAN, Novosibirsk (2007).

  4. V. P. Il’in and D. V. Knysh, “Parallel algorithms for solving separable boundary value problems,” in: Int. Sci. Conf. “Parallel Computer Tehnologies,” SPbPGU, St.Petersburg (2008), pp. 107–118.

  5. P. Benner, Ren-Cang Li, and Nin Truhar, “On the ADI method for Sylvester equations,” J. Comput. Appl. Math., 233, 1035–1045 (2009).

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Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, SB RAS, Novosibirsk, Russia

    N. I. Gorbenko & V. P. Il’in

  2. Novosibirsk State University, Novosibirsk, Russia

    N. I. Gorbenko & V. P. Il’in

Authors
  1. N. I. Gorbenko
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  2. V. P. Il’in
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Correspondence to V. P. Il’in.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 439, 2015, pp. 47–58.

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Gorbenko, N.I., Il’in, V.P. The Additive Peaceman–Rachford Method. J Math Sci 216, 753–760 (2016). https://doi.org/10.1007/s10958-016-2939-2

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  • DOI: https://doi.org/10.1007/s10958-016-2939-2

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