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Double-shift-invert Arnoldi method for computing the matrix exponential

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Abstract

Computing the matrix exponential for large sparse matrices is essential for solving evolution equations numerically. Conventionally, the shift-invert Arnoldi method is used to compute a matrix exponential and a vector product. This method transforms the original matrix using a shift, and generates the Krylov subspace of the transformed matrix for approximation. The transformation makes the approximation converge faster. In this paper, a new method called the double-shift-invert Arnoldi method is explored. This method uses two shifts for transforming the matrix, and this transformation results in a faster convergence than the shift-invert Arnoldi method.

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References

  1. Andersson, J.E.: Approximation of \(e^{-x}\) by rational functions with concentrated negative poles. J. Approx. Theory 32, 85–95 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beckermann, B., Reichel, L.: Error estimates and evaluation of matrix functions via the faber transform. SIAM J. Numer. Anal. 47, 3849–3883 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berljafa, M., Güttel, S.: Parallelization of the rational Arnoldi algorithm. SIAM J. Sci. Comput. 39, S197–S221 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gallopoulos, E., Saad, Y.: Efficient Solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. 13, 1236–1264 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Göckler, T., Grimm, V.: Uniform approximation of \(\varphi \)-functions in exponential integrators by a rational Krylov subspace method with simple poles. SIAM J. Matrix Anal. Appl. 35, 1467–1489 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Güttel, S.: Rational Krylov Methods for Operator Functions. Ph.D. thesis. Technischen Universität Bergakademie Freiberg. (2010)

  7. Hashimoto, Y., Nodera, T.: Inexact shift-invert Arnoldi method for evolution equations. ANZIAM J. 58, E1–E27 (2016)

    Article  MathSciNet  Google Scholar 

  8. Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26, 1179–1193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Isaacson, S.A.: The reaction-diffusion master equation as an asymptotic approximation of diffusion to a small target. SIAM J. Appl. Math. 70, 77–111 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Moler, C., Van Loan, C.F.: Nineteen dubious ways to compute the exponential of a matrix. Twenty-five years later. SIAM Rev. 4, 49–53 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Moret, I., Novati, P.: RD-rational approximations of the matrix exponential. BIT Numer. Math. 44, 595–615 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ruhe, A.: Rational Krylov sequence methods for eigenvalue computation. Linear Algebra Appl. 58, 391–405 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  14. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Spijker, M.N.: Numerical ranges and stability estimates. Appl. Numer. Math. 13, 241–249 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Svoboda, Z.: The convective-diffusion equation and its use in building physics. Int. J. Arch. Sci. 1, 68–79 (2000)

    Google Scholar 

  17. Van den Eshof, J., Hochbruck, M.: Preconditioning lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27, 1438–1457 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Fundamental Science and Technology, Graduate School of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan

    Yuka Hashimoto

  2. Center for Advanced Intelligence Project, RIKEN, 1-4-1 Nihonbashi, Chuo, Tokyo, 103-0027, Japan

    Yuka Hashimoto

  3. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku, Yokohama, Kanagawa, 223-8522, Japan

    Takashi Nodera

Authors
  1. Yuka Hashimoto
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  2. Takashi Nodera
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Correspondence to Yuka Hashimoto.

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Hashimoto, Y., Nodera, T. Double-shift-invert Arnoldi method for computing the matrix exponential. Japan J. Indust. Appl. Math. 35, 727–738 (2018). https://doi.org/10.1007/s13160-018-0309-9

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  • DOI: https://doi.org/10.1007/s13160-018-0309-9

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