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Towards a Lanczos’ \(\tau \)-Method Toolkit for Differential Problems

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Abstract

The aim of this work is to build a numerical software library based on the \(\tau \)-method to solve differential problems using MATLAB. The \(\tau \)-method can be very effective in the solution of certain type of these problems, and therefore, the existence of a numerical library for its dissemination is of major importance. Furthermore, the method has been used for the solution of particular problems but has not yet been explored as a general technique. Focus will be on stability issues, namely those issued from the solution of algebraic linear systems required for the process. Additionally, preconditioners for the solution with the \(\tau \)-method will be tackled, with emphasizes on incomplete LU factorizations and (block) Jacobi preconditioners. We also propose an iterative approach, build upon an LU factorization over a moderate initial size, generating better approximations and providing a priori error estimate at each iteration. Numerical results enlightening the efficiency of the proposed methods will be presented.

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References

  1. Crisci, M.R., Russo, E.: An extension of Ortiz recursive formulation of the Tau method to certain linear systems of ordinary differential equations. Math. Comput. 41(163), 27–42 (1983)

  2. Lanczos, C.: Trigonometric interpolation of empirical and analytical functions. J. Math. Phys. 17(1), 123–199 (1938)

    Article  MATH  Google Scholar 

  3. Liu, K., Pan, C.: The automatic solution to systems of ordinary differential equations by the Tau method. Comput. Math. Appl. 38(9), 197–210 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, Boca Raton, Florida (2002)

    Book  MATH  Google Scholar 

  5. Matos, J., Rodrigues, M.J., de Matos, J.C.: Avoiding similarity transformations in the operational Tau method (submitted)

  6. Matos, J., Rodrigues, M.J., Vasconcelos, P.B.: New implementation of the Tau method for PDEs. J. Comput. Math. Appl. 164, 555–567 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Namasivayam, S., Ortiz, E.L.: Best approximation and the numerical solution of partial differential equations with the Tau method. Port. Math. 40(1), 97–119 (1981)

    MathSciNet  MATH  Google Scholar 

  8. Olver, S., Townsend, A.: A fast and well-conditioned spectral method. SIAM Rev. 55(3), 462–489 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ortiz, E.: On the numerical solution of nonlinear and functional differential equations with the Tau method. In: Numerical Treatment of Differential Equations in Applications: Proceedings, Oberwolfach, Germany, December 1977. Lecture Notes in Mathematics, vol. 679, pp. 127–139. Springer, Berlin, Heidelberg (1978)

  10. Ortiz, E., Dinh, A.P.N.: Linear recursive schemes associated with some nonlinear partial differential equations in one dimension and the Tau method. SIAM J. Math. Anal. 18(2), 452–464 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ortiz, E.L., Samara, H.: An operational approach to the Tau method for the numerical solution of non-linear differential equations. Computing 27(1), 15–25 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pour-Mahmoud, J., Rahimi-Ardabili, M., Shahmorad, S.: Numerical solution of the system of fredholm integro-differential equations by the Tau method. Appl. Math. Comput. 168(1), 465–478 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Quintana-Ortí, E.S., Van De Geijn, R.A.: Updating an lu factorization with pivoting. ACM Trans. Math. Softw. (TOMS) 35(2), 11 (2008)

    Article  MathSciNet  Google Scholar 

  14. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM (2003)

  15. Trefethen, L.N.: Spectral Methods in MATLAB, vol. 10. SIAM (2000)

  16. Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM (2013)

  17. Van der Vorst, H.A.: Iterative Krylov methods for large linear systems, vol. 13. Cambridge University Press (2003)

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Authors and Affiliations

  1. Faculty of Sciences of the University of Porto, Porto, Portugal

    M. Trindade

  2. Laboratório Engenharia Matemática, Instituto Superior Engenharia Porto, Porto, Portugal

    J. Matos

  3. Center of Mathematics at the University of Porto, Porto, Portugal

    J. Matos & P. B. Vasconcelos

  4. Faculty of Economics of the University of Porto, rua Dr. Roberto Frias s/n, 4200-465, Porto, Portugal

    P. B. Vasconcelos

Authors
  1. M. Trindade
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  2. J. Matos
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  3. P. B. Vasconcelos
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Corresponding author

Correspondence to P. B. Vasconcelos.

Additional information

The first author was supported by CAPES, Coordination of Superior Level Staff Improvement—Brazil. The second and third authors were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.

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Trindade, M., Matos, J. & Vasconcelos, P.B. Towards a Lanczos’ \(\tau \)-Method Toolkit for Differential Problems. Math.Comput.Sci. 10, 313–329 (2016). https://doi.org/10.1007/s11786-016-0269-x

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  • DOI: https://doi.org/10.1007/s11786-016-0269-x

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