Skip to main content
SpringerLink
Log in

Fast Rational Lanczos Method for the Toeplitz Symmetric Positive Semidefinite Matrix Functions

  • Conference paper
  • First Online:
Simulation Tools and Techniques (SIMUtools 2020)

Included in the following conference series:

Abstract

In this paper, we use the rational Lanczos method to approximate Toeplitz matrix functions, in which the matrices are symmetric positive semidefinite (SPSD). In order to reduce the computational cost, we use the inverse of the Toeplitz matrix and the fast Fourier transform (FFT). Then, we apply this method to solve a heat equation. Numerical examples are given to show the effectiveness of the rational Lanczos method.

Supported by the “Peiyu” Project from Xuzhou University of Technology (Grant Number XKY2019104).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Al-Mohy, A., Higham, N.J.: Computing the action of matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)

    Google Scholar 

  2. Berland, H., Skaflestad, B., Wright, W.: Expint-A matlab package for exponential integrators. ACM Tran. Math. Soft. 33(4), 4-es (2007)

    Google Scholar 

  3. Botchev, M., Grimm, V., Hochbruck, M.: Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35, A1376–A1397 (2013)

    Article  MathSciNet  Google Scholar 

  4. Caliari, M., Kandolf, P., Zivcovich, F.: Backward error analysis of polynomial approximations for computing the action of the matrix exponential. BIT Numer. Math. 58, 907–935 (2018)

    Article  MathSciNet  Google Scholar 

  5. Chan, R., Jin, X.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  6. Chan, R., Ng, M.: Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38, 427–482 (1996)

    Article  MathSciNet  Google Scholar 

  7. Eiermann, M., Ernst, O.: A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal. 44, 2481–2504 (2006)

    Article  MathSciNet  Google Scholar 

  8. Frommer, A., Gttel, S., Schweitzer, M.: Efficient and stable Arnoldi restarts for matrix functions based on quadrature. SIAM J. Matrix Anal. Appl. 35, 661–683 (2014)

    Article  MathSciNet  Google Scholar 

  9. Frommer, A., Simoncini, V.: Stopping criteria for rational matrix functions of hermitian and symmetric matrices. SIAM J. Sci. Comput. 30, 1387–1412 (2008)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  11. Grimm, V.: Resolvent Krylov subspace approximation to operator functions. BIT Numer. Math. 52, 639–659 (2012)

    Article  MathSciNet  Google Scholar 

  12. Gockenbach, M.: Partial Differential Equations-Analytical and Numerical Methods. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

  13. Gohberg, I., Semencul, A.: On the inversion of finite Toeplitz matrices and their continuous analogs. Matem. Issled. 2, 201–233 (1972)

    MATH  Google Scholar 

  14. Heinig, G., Rost, L.: Algebraic Methods for Toeplitz-like Matrices and Operators. Birkhäuser, Basel (1984). https://doi.org/10.1007/978-3-0348-6241-7

    Book  MATH  Google Scholar 

  15. Higham, N.J., Kandolf, P.: Computing the action of trigonometric and hyperbolic matrix functions. SIAM J. Sci. Comput. 36, A613–A627 (2017)

    Article  MathSciNet  Google Scholar 

  16. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  Google Scholar 

  17. Huo, L., Jiang, D., Lv, Z., et al.: An intelligent optimization-based traffic information acquirement approach to software-defined networking. Comput. Intell. 36, 1–21 (2019)

    Google Scholar 

  18. Huo, L., Jiang, D., Qi, S., Song, H., Miao, L.: An AI-based adaptive cognitive modeling and measurement method of network traffic for EIS. Mob. Net. Appl., 1–11 (2019). https://doi.org/10.1007/s11036-019-01419-z

  19. Jiang, D., Huo, L., Song, H.: Rethinking behaviors and activities of base stations in mobile cellular networks based on big data analysis. IEEE Trans. Netw. Sci. Eng. 1(1), 1–12 (2018)

    MathSciNet  Google Scholar 

  20. Jiang, D., Wang, Y., Lv, Z., et al.: Big data analysis-based network behavior insight of cellular networks for industry 4.0 applications. IEEE Trans. Ind. Inf. 16(2), 1310–1320 (2020)

    Google Scholar 

  21. Jiang, D., Huo, L., Li, Y.: Fine-granularity inference and estimations to network traffic for SDN. PLoS One 13(5), 1–23 (2018)

    Google Scholar 

  22. Jiang, D., Huo, L., Lv, Z., et al.: A joint multi-criteria utility-based network selection approach for vehicle-to-infrastructure networking. IEEE Trans. Intell. Transp. Syst. 19(10), 3305–3319 (2018)

    Article  Google Scholar 

  23. Jiang, D., Li, W., Lv, H.: An energy-efficient cooperative multicast routing in multi-hop wireless networks for smart medical applications. Neurocomputing 220, 160–169 (2017)

    Article  Google Scholar 

  24. Jiang, D., Wang, Y., Lv, Z., et al.: Intelligent Optimization-based reliable energy-efficient networking in cloud services for IIoT networks. IEEE J. Sel. Areas Commun. Online available (2019)

    Google Scholar 

  25. Jiang, D., Wang, W., Shi, L., et al.: A compressive sensing-based approach to end-to-end network traffic reconstruction. IEEE Trans. Netw. Sci. Eng. 5(3), 1–12 (2018)

    Google Scholar 

  26. Jiang, D., Zhang, P., Lv, Z., et al.: Energy-efficient multi-constraint routing algorithm with load balancing for smart city applications. IEEE Internet Things J. 3(6), 1437–1447 (2016)

    Article  Google Scholar 

  27. Kooij, G.L., Botchev, M.A., Geurts, B.J.: An exponential time integrator for the incompressible Navier-stokes equation. SIAM J. Sci. Comput. 40, B684–B705 (2018)

    Article  MathSciNet  Google Scholar 

  28. Lee, S., Pang, H., Sun, H.: Shift-invert Arnoldi approximation to the Toeplitz matrix exponential. SIAM J. Sci. Comput. 32, 774–792 (2010)

    Article  MathSciNet  Google Scholar 

  29. Lopez, L., Simoncini, V.: Analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. 44, 613–635 (2006)

    Article  MathSciNet  Google Scholar 

  30. Moret, I.: On RD-rational Krylov approximations to the core-functions of exponential integrators. Numer. Linear Algebra Appl. 14, 445–457 (2007)

    Article  MathSciNet  Google Scholar 

  31. Ng, M., Sun, H., Jin, X.: Recursive-based PCG methods for Toeplitz systems with nonnegative generating functions. SIAM J. Sci. Comput. 24, 1507–1529 (2003)

    Article  MathSciNet  Google Scholar 

  32. Noferini, V.: A formula for the Fréchet derivative of a generalized matrix function. SIAM J. Matrix Anal. Appl. 38, 434–457 (2017)

    Article  MathSciNet  Google Scholar 

  33. Pang, H., Sun, H.: Shift-invert Lanczos method for the symmetric positive semidefinite Toeplitz matrix exponential. Numer. Linear Algebra Appl. 18, 603–614 (2011)

    Article  MathSciNet  Google Scholar 

  34. Qi S., Jiang, D., Huo, L.: A prediction approach to end-to-end traffic in space information networks. Mob. Netw. Appl. Online available (2019)

    Google Scholar 

  35. Saad, Y.: Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29, 209–228 (1992)

    Article  MathSciNet  Google Scholar 

  36. Tangman, D.Y., Gopaul, A., Bhuruth, M.: Exponential time integration and Chebychev discretisation schemes for fast pricing of options. Appl. Numer. Math. 58, 1309–1319 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  38. Wang, Y., Jiang, D., Huo, L., et al.: A new traffic prediction algorithm to software defined networking. Mob. Netw. Appl. Online available (2019)

    Google Scholar 

  39. Wang, F., Jiang, D., Qi, S.: An adaptive routing algorithm for integrated information networks. China Commun. 7(1), 196–207 (2019)

    Google Scholar 

  40. Wu, G., Feng, T., Wei, Y.: An inexact shift-and-invert Arnoldi algorithm for Toeplitz matrix exponential. Numer. Linear Algebra Appl. 22, 777–792 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Engineering, Xuzhou University of Technology, Xuzhou, 221018, Jiangsu, China

    Lei Chen

  2. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou, 221018, Jiangsu, China

    Lu Zhang, Mengjia Wu & Jianqiang Zhao

Authors
  1. Lei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mengjia Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jianqiang Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Embry-Riddle Aeronautical University, Daytona Beach, FL, USA

    Houbing Song

  2. School of Astronautics and Aeronautic, UESTC, Chengdu, China

    Dingde Jiang

Rights and permissions

Reprints and permissions

Copyright information

© 2021 ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chen, L., Zhang, L., Wu, M., Zhao, J. (2021). Fast Rational Lanczos Method for the Toeplitz Symmetric Positive Semidefinite Matrix Functions. In: Song, H., Jiang, D. (eds) Simulation Tools and Techniques. SIMUtools 2020. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 369. Springer, Cham. https://doi.org/10.1007/978-3-030-72792-5_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-72792-5_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-72791-8

  • Online ISBN: 978-3-030-72792-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation