Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 


Generalization of Padé approximation from rational functions to arbitrary analytic functions -- Theory

Authors: Can Evren Yarman and Garret M. Flagg
Journal: Math. Comp. 84 (2015), 1835-1860
MSC (2010): Primary 30E05, 32A17, 32A26, 65D15, 41A21; Secondary 94A12, 94A11, 32A27, 33C10, 65T99
Published electronically: January 13, 2015
MathSciNet review: 3335894
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Padé approximation has a long and rich history of theory and application and is known to produce excellent local approximations. We present a method for extending the basic idea of Padé approximation, that of matching a prescribed number of terms in the Taylor series expansion of a given function using rational functions, to any arbitrary function holormorphic in a neighborhood of the Taylor series expansion point. We demonstrate that providing the flexibility of using other functions in a Padé-type approximation yields highly accurate approximations having additional desirable properties. Additional properties that can be preserved in our method include comparable asymptotic behavior to the function to be approximated, or the preservation of band-limitedness in the approximation.

References [Enhancements On Off] (What's this?)

  • [1] Milton Abramowitz and Irene A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables, Dover Publications Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604 (94b:00012)
  • [2] Robert F Abramson, The sinc and cosinc transform, Electromagnetic Compatibility, IEEE Transactions on (1977), no. 2, 88-94.
  • [3] Lars V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 510197 (80c:30001)
  • [4] D.E. Amos, A subroutine package for bessel functions of a complex argument and nonnegative order, Sandia National Laboratory Report SAND85-1018, Sandia National Laboratory, May 1985.
  • [5] D. E. Amos, Algorithm 644: a portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Software 12 (1986), no. 3, 265-273. MR 889069,
  • [6] Athanasios C. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. With a foreword by Jan C. Willems. MR 2155615 (2006c:93001)
  • [7] George A. Baker Jr., Essentials of Padé Approximants, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0454459 (56 #12710)
  • [8] George A. Baker Jr. and Peter Graves-Morris, Padé Approximants, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 59, Cambridge University Press, Cambridge, 1996. MR 1383091 (97h:41001)
  • [9] Gregory Beylkin and Lucas Monzón, On approximation of functions by exponential sums, Appl. Comput. Harmon. Anal. 19 (2005), no. 1, 17-48. MR 2147060 (2006h:65012),
  • [10] Gregory Beylkin and Lucas Monzón, Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal. 28 (2010), no. 2, 131-149. MR 2595881 (2011e:41032),
  • [11] C. F. Gauss, Methodus nova integralium valores per approximationem inveniendi, Comment. Soc. Reg. Scient. Gotting. Recent. (1814).
  • [12] Walter Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968), 251-270. MR 0228171 (37 #3755)
  • [13] R. E. Kalman, On partial realizations, transfer functions, and canonical forms, Acta Polytech. Scand. Math. Comput. Sci. Ser. 31 (1979), 9-32. Topics in systems theory. MR 557691 (80k:93022)
  • [14] Rudolf Emil Kalman, Mathematical description of linear dynamical systems, Journal of the Society for Industrial & Applied Mathematics, Series A: Control 1 (1963), no. 2, 152-192.
  • [15] R. E. Kalman, Advanced theory of linear systems, Topics in Mathematical System Theory, McGraw-Hill, New York, 1969, pp. 237-339. MR 0286517 (44 #3726)
  • [16] Sun-Yuan Kung, A new identification and model reduction algorithm via singular value decomposition, Proc. 12th Asilomar Conf. Circuits, Syst. Comput., Pacific Grove, CA, 1978, pp. 705-714.
  • [17] Sun Yuan Kung and David W. Lin, A state-space formulation for optimal Hankel-norm approximations, IEEE Trans. Automat. Control 26 (1981), no. 4, 942-946. MR 635856 (84a:93017b),
  • [18] A. J. Mayo and A. C. Antoulas, A framework for the solution of the generalized realization problem, Linear Algebra Appl. 425 (2007), no. 2-3, 634-662. MR 2343060 (2008e:93028),
  • [19] I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831 (55 #8786a)
  • [20] Leonard M. Silverman, Realization of linear dynamical systems, IEEE Trans. Automatic Control AC-16 (1971), 554-567. MR 0307749 (46 #6869)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 30E05, 32A17, 32A26, 65D15, 41A21, 94A12, 94A11, 32A27, 33C10, 65T99

Retrieve articles in all journals with MSC (2010): 30E05, 32A17, 32A26, 65D15, 41A21, 94A12, 94A11, 32A27, 33C10, 65T99

Additional Information

Can Evren Yarman
Affiliation: Schlumberger, 3750 Briar Park Dr., Houston, Texas 77042

Garret M. Flagg
Affiliation: Schlumberger, 3750 Briar Park Dr., Houston, Texas 77042

Received by editor(s): June 13, 2013
Received by editor(s) in revised form: November 18, 2013
Published electronically: January 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society