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A superfast algorithm for multi-dimensional Padé systems

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Abstract

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n 0,...,n k ) inO(‖n‖ · log2‖n‖) block-matrix operations, where ‖n‖=n 0+...+n k . Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(‖n‖2). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.

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References

  1. G.A. Baker and P. Graves-Morris,Padé Approximants Part II: Extensions and Applications (Addison-Wesley, Reading, 1981).

    Google Scholar 

  2. M. Van Barel and A. Bultheel, The computation of non-perfect Padé-Hermite approximants, Preprint, to appear in Num. Algorithms (1991).

  3. B. Beckermann, A reliable method for computingM-Padé approximants on arbitrary staircases, J. Comp. Appl. Math. Preprint (1990).

  4. B. Beckermann, The structure of the singular solution table of theM-Padé approximation problem, J. Comp. Appl. Math. 32 (1990) 3–15.

    Article  Google Scholar 

  5. R. Brent, F.G. Gustavson and D.Y.Y. Yun, Fast solution of Toeplitz systems of equations and computation of Padé approximants, J. Algorithms 1 (1980) 259–295.

    Article  Google Scholar 

  6. M.G. de Bruin, The interruption phenomenon for generalized continued fractions, Bull. Austral. Math. Soc. 19 (1978) 245–272.

    Google Scholar 

  7. M.G. de Bruin, Some aspects of simultaneous rational approximation, in:Proc. PAN (1987).

  8. A. Bultheel and M. Van Barel, Minimal vector Padé approximation, J. Comp. Appl. Math. 32 (1990) 27–37.

    Article  Google Scholar 

  9. S. Cabay and D.K. Choi, Algebraic computations of scaled Padé fractions, SIAM J. Comput. (Feb. 1986) 243–270.

  10. S. Cabay, G. Labahn and B. Beckermann. On the theory and computation of non-perfect Padé-Hermite approximants, to appear in J. Comp. Appl. Math. (1991).

  11. J. Coates, On the algebraic approximation of functions, I–III, K. Nederl. Ak. Wetenschappen 69 (1966) 421–461.

    Google Scholar 

  12. J. Della Dora and C. Direscenzo, Approximants de Padé-Hermite, Numer. Math. 43 (1984) 23–57.

    Article  Google Scholar 

  13. W.B. Gragg, The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev. 14 (1972) 1–61.

    Article  Google Scholar 

  14. P. Graves-Morris, Vector-valued rational interpolants II, IMA J. Numer. Anal. 4 (1984) 209–224.

    Google Scholar 

  15. M.Ch. Hermite, Sur l'ExpressionUsinx+Vcosx+W, Extrait d'une lettre a M. Paul Gordan,Oeuvres Tome III (1873) pp. 135–145.

    Google Scholar 

  16. M.Ch. Hermite, Sur quelques approximations algebriques, Extrait d'une lettre a M. Borchardt,Oeuvres Tome III (1873) pp. 146–149.

    Google Scholar 

  17. H. Jager, A multidimensional generalization of the Padé table I–VI, K. Nederl. Ak. Wetenschappen 67 (1964) 192–249.

    Google Scholar 

  18. G.A. Baker Jr. and P.R. Graves-Morris, Definition and uniqueness of integral approximants, J. Comput. Appl. Math. 31 (1990) 357–372.

    Article  Google Scholar 

  19. G. Labahn and S. Cabay, Matrix Padé fractions and their computation, SIAM J. Comput. 18 (1989) 639–657.

    Article  Google Scholar 

  20. G. Labahn, D.K. Choi and S. Cabay, Inverses of block Hankel and block Toeplitz matrices, SIAM J. Comput. 19 (1990) 98–123.

    Article  Google Scholar 

  21. G. Labahn, Inversion components for block Hankel-like matrices, to appear in Lin. Algebra Appl. (1991).

  22. L. Lerer and M. Tismenetsky, Generalized Bezoutians and the inversion problem for block matrices, Integral Eq. Oper. Th. 9 (1986) 790–819.

    Article  Google Scholar 

  23. K. Mahler, Zur Approximation der Exponentialfunktion und des Logarithmus, Teil I, Journ. f.d.r.u.a. 166 (1932) 118–136.

    Google Scholar 

  24. K. Mahler, Perfect systems, Compositio Math. 19 (1968) 95–166.

    Google Scholar 

  25. R.J. Meleshko, A stable algorithm for the computation of Padé approximants, Ph.D. Thesis, Dept. of Computing Science, University of Alberta, Edmonton, Canada (1990).

    Google Scholar 

  26. H. Padé, Sur la représentation approchée d'une fonction par des fractions rationelles, Ann. Ecole Nor., supplement Thesis (1892) pp. 1–93.

  27. B.N. Parlett and J.R. Bunch Direct methods for solving symmetric indefinite systems of linear equations, SIAM J. Numer Anal. 8 (1971) 639–655.

    Article  Google Scholar 

  28. B.N. Parlett, D.R. Taylor and Z.A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comput. 44 (1985) 105–124.

    Google Scholar 

  29. S. Paszkowski, Recurrence relations in Padé-Hermite approximation, J. Comp. Appl. Math. 19 (1987) 99–107.

    Google Scholar 

  30. Y. Sugiyama, An algorithm for solving discrete-time Wiener-Hopf equations based on Euclid's algorithm, IEEE Trans. Inform. Theory IF-32 (1986) 394–409.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computing Science, University of Alberta, T6G 2H1, Edmonton, Alberta, Canada

    Stan Cabay

  2. Department of Computer Science, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    George Labahn

Authors
  1. Stan Cabay
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  2. George Labahn
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Additional information

Communicated by C. Brezinski

Supported in part by NSERC grant No. A8035.

Partially supported by NSERC operating grant No. 6194.

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Cabay, S., Labahn, G. A superfast algorithm for multi-dimensional Padé systems. Numer Algor 2, 201–224 (1992). https://doi.org/10.1007/BF02145386

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  • DOI: https://doi.org/10.1007/BF02145386

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