Reliable Computing

, Volume 6, Issue 4, pp 365–390 | Cite as

Effective Computation of Rational Approximants and Interpolants

  • Bernhard Beckermann
  • George Labahn
Article

Abstract

This paper considers the problem of effective algorithms for some problems having structured coefficient matrices. Examples of such problems include rational approximation and rational interpolation. The corresponding coefficient matrices include Hankel, Toeplitz and Vandermonde-like matrices. Effective implies that the algorithms studied are suitable for implementation in either a numeric environment or else a symbolic environment.

The paper includes two algorithms for the computation of rational interpolants which are both effective in symbolic environments. The algorithms use arithmetic that is free of fractions but at the same time control the growth of coefficients during intermediate computations. One algorithm is a look-around procedure which computes along a path of closest normal points to an offdiagonal path while the second computes along an arbitrary path using a look-ahead strategy. Along an antidiagonal path the look-ahead recurrence is closely related to the Subresultant PRS algorithm for polynomial GCD computation. Both algorithms are an order of magnitude faster than alternative methods which are effective in symbolic environments.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akritas, A. G.: Elements of Computer Algebra with Applications, Wiley-Interscience, 1989.Google Scholar
  2. 2.
    Baker, G. A., and Graves-Morris, P. R.: Padé Approximants, second edition, Cambridge Univ. Press, Cambridge, 1995.Google Scholar
  3. 3.
    Bareiss, E.: Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination, Math. Comp. 22(103) (1968), pp. 565-578.Google Scholar
  4. 4.
    Beckermann, B.: A Reliable Method for Computing M-Padé Approximants on Arbitrary Staircases, J. Comput. Appl. Math. 40 (1992), pp. 19-42.Google Scholar
  5. 5.
    Beckermann, B.: The Stable Computation of Formal Orthogonal Polynomials, Numerical Algorithms 11 (1996), pp. 1-23.Google Scholar
  6. 6.
    Beckermann, B., Cabay, S., and Labahn, G.: Fraction-Free Computation of Matrix Padé Systems, in: Proceedings of ISSAC'97, Maui, ACM Press, 1997, pp. 125-132.Google Scholar
  7. 7.
    Beckermann, B. and Carstensen, C.: QD-Type Algorithms for the Non-Normal Newton-Padé Approximation Table, Constructive Approximation 12 (1996), pp. 307-330.Google Scholar
  8. 8.
    Beckermann, B. and Labahn, G.: A Fast and Numerically Stable Euclidean-Like Algorithm for Detecting Relatively Prime Numerical Polynomials, J. of Symbolic Computation 26 (1998), pp. 691-714.Google Scholar
  9. 9.
    Beckermann, B. and Labahn, G.: Fraction-Free Computation of Matrix GCD's and Rational Interpolants, University of Waterloo, Tech Report, 1997.Google Scholar
  10. 10.
    Beckermann, B. and Labahn, G.: Recursiveness in Matrix Rational Interpolation Problems, J. Comput. Appl. Math. 77 (1997), pp. 5-34.Google Scholar
  11. 11.
    Beckermann, B. and Labahn, G.: When Are Two Numerical Polynomials Relatively Prime? J. of Symbolic Computation 26 (1998), pp. 677-689.Google Scholar
  12. 12.
    Beckermann, B., Labahn, G., and Villard, G.: Shifted Normal Forms of Polynomial Matrices, in: Proceedings of ISSAC'99, Vancouver, 1999, pp. 189-196.Google Scholar
  13. 13.
    Bojanczyk, A. W., Brent, R. P., and de Hoog, F. R.: Stability Analysis of a General Toeplitz Systems Solver, Numerical Algorithms 10 (1995), pp. 225-244.Google Scholar
  14. 14.
    Bojanczyk, A. W., Brent, R. P., de Hoog, F. R., and Sweet, D. R.: On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms, SIAM J. Matrix Anal. Appl. 16 (1995), pp. 40-57.Google Scholar
  15. 15.
    Brent, R., Gustavson, F. G., and Yun, D. Y. Y.: Fast Solution of Toeplitz Systems of Equations and Computation of Padé Approximants, J. of Algorithms 1 (1980), pp. 259-295.Google Scholar
  16. 16.
    Brown, W. and Traub, J. F.: On Euclid's Algorithm and the Theory of Subresultants, J. ACM 18 (1971), pp. 505-514.Google Scholar
  17. 17.
    Cabay, S., Jones, A. R., and Labahn, G.: Computation of Numerical Padé-Hermite and Simultaneous Padé Systems II: A Weakly Stable Algorithm, SIAM J. Matrix Anal. Appl. 17 (1996), pp. 268-297.Google Scholar
  18. 18.
    Cabay, S. and Kossowski, P.: Power Series Remainder Sequences and Padé Fractions over an Integral Domain, J. Symbolic Computation 10 (1990), pp. 139-163.Google Scholar
  19. 19.
    Cabay, S. and Meleshko, R.: A Weakly Stable Algorithm for Padé Approximants and the Inversion of Hankel Matrices, SIAM J. Matrix Anal. Appl. 14 (1993), pp. 735-765.Google Scholar
  20. 20.
    Chandrasekaran, S. and Sayed, A. H.: Stabilizing the Generalized Schur Algorithm, SIAM J. Matrix Anal. Appl. 14 (1996), pp. 950-983.Google Scholar
  21. 21.
    Claessens, G.: On the Newton-Padé Approximation Problem, J. Approx. Th. 22 (1978), pp. 150-160.Google Scholar
  22. 22.
    Claessens, G.: On the Structure of the Newton-Padé Table, J. Approx. Th. 22 (1978), pp. 304-319.Google Scholar
  23. 23.
    Claessens, G.: Some Aspects of the Rational Hermite Interpolation Table and Its Applications, Ph.D. Thesis, University of Antwerp, 1979.Google Scholar
  24. 24.
    Collins, G.: Subresultant and Reduced Polynomial Remainder Sequences, J. ACM 14 (1967), pp. 128-142.Google Scholar
  25. 25.
    Czapor, S. R. and Geddes, K. O.: A Comparison of Algorithms for the Symbolic Computation of Padé Approximants, in: Fitch, J. (ed.), Proceedings of EUROSAM'84 (Lecture Notes in Computer Science 174), Springer-Verlag, Berlin, 1984, pp. 248-259.Google Scholar
  26. 26.
    Freund, R. W. and Zha, H.: A Look-Ahead Algorithm for the Solution of General Hankel Systems, Numer. Math. 64 (1993), pp. 295-321.Google Scholar
  27. 27.
    Freund, R. W. and Zha, H.: Formally Biorthogonal Polynomials and a Look-Ahead Levinson Algorithm for General Toeplitz Systems, Linear Algebra Appl. 188/89 (1993), pp. 255-303.Google Scholar
  28. 28.
    Geddes, K. O., Czapor, S. R., and Labahn, G.: Algorithms for Computer Algebra, Kluwer Academic Publishers, Boston, 1992.Google Scholar
  29. 29.
    Gohberg, I., Kailath, T., and Olshevski, V.: Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure, Math. Comp. 64 (1995), pp. 1557-1567.Google Scholar
  30. 30.
    Golub, G. and Olshevski, V.: Pivoting for Structured Matrices, with Applications, Manuscript, 1997, http://www-isl.stanford.edu/~olshevsk.Google Scholar
  31. 31.
    Graves-Morris, P.: Efficient Reliable Rational Interpolation, in: Padé Approximation and Its Applications 1980, Springer-Verlag, 1980, pp. 28-63.Google Scholar
  32. 32.
    Gu, M.: Stable and Efficient Algorithms for Structured Systems of Linear Equations, SIAM J. Matrix Anal. Appl. 19(2) (1998), pp. 279-306.Google Scholar
  33. 33.
    Gutknecht, M. H.: Stable Row Recurrences for the Padé Table and Generically Superfast Look-Ahead Solvers for Non-Hermitian Toeplitz Systems, Linear Algebra Appl. 188/89 (1993), pp. 351-421.Google Scholar
  34. 34.
    Gutknecht, M. H.: The Multipoint Padé Table and General Recurrences for Rational Interpolation, in: Cuyt, A. (ed.), Nonlinear Numerical Methods and Rational Approximation, Kluwer Academic Publishers, 1994, pp. 109-136.Google Scholar
  35. 35.
    Gutknecht, M. H. and Hochbruck, M.: Look-Ahead Levinson and Schur Algorithms for Non-Hermitian Toeplitz Systems, Numer. Math. 70 (1995), pp. 181-227.Google Scholar
  36. 36.
    Habicht, W.: Eine Verallgemeinerung des Sturmschen Wurzelzählverfahrens, Commentarii Mathematici Helvetici 21 (1948), pp. 99-116.Google Scholar
  37. 37.
    Heinig, G. and Rost, K.: Algebraic Methods for Toeplitz-Like Matrices and Operators, Operator Theory 13, Basel, 1984.Google Scholar
  38. 38.
    Knuth, D.: The Art of Computer Programming Vol. 2, Addison-Wesley, 1981.Google Scholar
  39. 39.
    Kravanja, P. and Van Barel, M.: A Fast Block Hankel Solver Based on an Inversion Formula for Block Loewner Matrices, CALCOLO 33 (1996), pp. 147-164.Google Scholar
  40. 40.
    Kravanja, P. and Van Barel, M.: A Fast Hankel Solver Based on an Inversion Formula for Loewner Matrices, Linear Algebr. Appl. 282 (1998), pp. 275-295.Google Scholar
  41. 41.
    Kravanja, P. and Van Barel, M.: Coupled Vandermonde Matrices and the Superfast Computation of Toeplitz Determinants, Submitted to Numerical Algorithms, in: Proceedings of the International Conference on Rational Approximation (ICRA99), Antwerp, Belgium.Google Scholar
  42. 42.
    Labahn, G.: Inversion Components of Block Hankel-Like Matrices, Linear Algebra Appl. 177 (1992), pp. 7-48.Google Scholar
  43. 43.
    Li, Z.: A Subresultant Theory for Linear Differential, Linear Difference and Ore Polynomials, with Applications, PhD Thesis, Univ. Linz, 1996.Google Scholar
  44. 44.
    Mahler, K.: Perfect Systems, Compos. Math. 19 (1968), pp. 95-166; theorems: J. Comput. Appl. Math. 32 (1990), pp. 229–236.Google Scholar
  45. 45.
    Mishra, B.: Algorithmic Algebra, Springer Verlag, 1993.Google Scholar
  46. 46.
    Salvy, B. and Zimmermann, P.: Gfun: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable, ACM Transactions on Mathematical Software (TOMS) 20(2) (1994), pp. 163-177.Google Scholar
  47. 47.
    Sylvester, J. J.: On a Theory of the Syzgetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and That of the Greatest Algebraic Common Measure, Philosophical Transactions 143 (1853), pp. 407-548.Google Scholar
  48. 48.
    Van Barel, M. and Bultheel, A.: A Look-Ahead Algorithm for the Solution of Block Toeplitz Systems, Linear Algebra Appl. 266 (1997), pp. 291-335.Google Scholar
  49. 49.
    Van Barel, M. and Bultheel, A.: A New Formal Approach to the Rational Interpolation Problem, Numerische Mathematik 62 (1992), pp. 87-122.Google Scholar
  50. 50.
    Van Hoeij, M.: Factorization of Differential Operators with Rational Function Coefficients, Journal of Symbolic Computation (1998).Google Scholar
  51. 51.
    Warner, D. D.: Hermite Interpolation with Rational Functions, Ph.D. Thesis, Univ. of California, San Diego, 1974.Google Scholar
  52. 52.
    Werner, H.: A Reliable Method for Rational Interpolation, in: Wuytack, L. (ed.), Padé Approximation and Its Applications, Antwerp 1979 (Lecture Notes in Math. 765), Springer, Berlin, 1979, pp. 257-277.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Bernhard Beckermann
    • 1
  • George Labahn
    • 2
  1. 1.Laboratoire d'Analyse Numérique et d'OptimisationUniversité des Sciences et Technologies de LilleVilleneuve d'Ascq CedexFrance
  2. 2.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations