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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Explicit, nearly optimal, linear rational approximation with preassigned poles
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by Frank Stenger PDF
Math. Comp. 47 (1986), 225-252 Request permission

Abstract:

This paper gives explicit rational functions for interpolating and approximating functions on the intervals $[ - 1,1]$, $[0,\infty ]$, and $[ - \infty ,\infty ]$. The rational functions are linear in the functions to be approximated, and they have preassigned poles. The error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees. Regions of analyticity are described, which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation.
References
  • Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
  • Jan-Erik Andersson, Optimal quadrature of $H^{p}$ functions, Math. Z. 172 (1980), no. 1, 55–62. MR 576296, DOI 10.1007/BF01182779
    • D. W. Barclay, J. B. Moodie & J. B. Haddow, "Analysis of unloading waves from suddenly punched hole in an axially loaded elastic plate," Wave Motion, v. 3, 1981, pp. 105-113.
  • Peter B. Borwein, Approximations with negative roots and poles, J. Approx. Theory 35 (1982), no. 2, 132–141. MR 662161, DOI 10.1016/0021-9045(82)90031-4
  • H. G. Burchard and K. Höllig, $N$-width and entropy of $H_p$-classes in $L_q(-1,1)$, SIAM J. Math. Anal. 16 (1985), no. 2, 405–421. MR 777476, DOI 10.1137/0516029
  • W. Cauer, Bemerkung über eine Extremalaufgabe von E. Zolotareff, Z. Angew. Math. Mech. 20 (1940), 358 (German). MR 8747, DOI 10.1002/zamm.19400200606
  • Tord Ganelius, Rational approximation in the complex plane and on the line, Ann. Acad. Sci. Fenn. Ser. A I Math. 2 (1976), 129–145. MR 0499176, DOI 10.5186/aasfm.1976.0211
  • Walter Gautschi, Computational methods in special functions—a survey, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 1–98. MR 0391476
  • P. R. Graves-Morris, Efficient reliable rational interpolation, Padé approximation and its applications, Amsterdam 1980 (Amsterdam, 1980), Lecture Notes in Math., vol. 888, Springer, Berlin-New York, 1981, pp. 28–63. MR 649084
  • Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
  • Peter Henrici, Essentials of numerical analysis with pocket calculator demonstrations, John Wiley & Sons, Inc., New York, 1982. MR 655251
    • J. R. Lund, Numerical Evaluation of Integral Transforms, Ph.D. Thesis, University of Utah, 1978.
  • Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968, DOI 10.1007/978-3-662-11761-3
    • National Bureau of Standards, Handbook of Mathematical Functions, N.B.S. Applied Math. Series, vol. 55, 1964.
  • D. J. Newman, Rational approximation to $| x|$, Michigan Math. J. 11 (1964), 11–14. MR 171113, DOI 10.1307/mmj/1028999029
  • Donald J. Newman, Approximation with rational functions, CBMS Regional Conference Series in Mathematics, vol. 41, Conference Board of the Mathematical Sciences, Washington, D.C., 1979. Expository lectures from the CBMS Regional Conference held at the University of Rhode Island, Providence, R.I., June 12–16, 1978. MR 539314, DOI 10.1090/cbms/041
    • H. Padé, "Sur la représentation approchée d’une fonction par des fractions rationelles," Ann. Fac. Sci. École Norm. Sup., v. 9, 1892, pp. 1-93.
  • D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41. MR 71874, DOI 10.1137/0103003
  • M. J. D. Powell, On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria, Comput. J. 9 (1967), 404–407. MR 208807, DOI 10.1093/comjnl/9.4.404
  • E. B. Saff and Richard S. Varga (eds.), Padé and rational approximation, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Theory and applications. MR 0458010
  • Daniel Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. and Phys. 34 (1955), 1–42. MR 68901, DOI 10.1002/sapm19553411
  • Oved Shisha (ed.), Inequalities, Academic Press, New York-London, 1967. MR 0218162
  • Frank Stenger, Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), no. 2, 165–224. MR 618638, DOI 10.1137/1023037
  • Frank Stenger, Optimal convergence of minimum norm approximations in $H_{p}$, Numer. Math. 29 (1977/78), no. 4, 345–362. MR 483329, DOI 10.1007/BF01432874
  • Frank Stenger, Integration formulae based on the trapezoidal formula, J. Inst. Math. Appl. 12 (1973), 103–114. MR 381261, DOI 10.1093/imamat/12.1.103
    • F. Stenger, M. R. Berggren, S. A. Johnson & C. H. Wilcox, "Rational function frequency extrapolation for diffraction correction in ultrasonic tomography," in Wave Phenomena: Modern Theory and Applications (C. Rogers and T. B. Moodie, eds.), North-Holland, Amsterdam, 1984, pp. 19-34.
  • T. J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Math. (6) 4 (1995), no. 1, Ji–Jiv, J1–J35 (French). Reprint of the 1894 original; With an introduction by Jean Cassinet. MR 1344720, DOI 10.5802/afst.789
    • J. Szabados, Uniform Approximation of Continuous Functions by Rational Functions, Ph.D. Thesis, Hungarian Academy of Sciences, 1968. (Hungarian) T. N. Thiele, "Differences reciproques," Overs, danske Vids, Selsk. Forhadl. (Sitzber., Akad. Kopenhagen), 1906.
  • Richard S. Varga, Topics in polynomial and rational interpolation and approximation, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 81, Presses de l’Université de Montréal, Montreal, Que., 1982. MR 654329
  • P. Wynn, On a device for computing the $e_m(S_n)$ tranformation, Math. Tables Aids Comput. 10 (1956), 91–96. MR 84056, DOI 10.1090/S0025-5718-1956-0084056-6
  • P. Wynn, Upon some continuous prediction algorithms. I, Calcolo 9 (1972), 197–234. MR 362820, DOI 10.1007/BF02576490
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Math. Comp. 47 (1986), 225-252
  • MSC: Primary 41A20; Secondary 41A25, 65D15
  • DOI: https://doi.org/10.1090/S0025-5718-1986-0842132-0
  • MathSciNet review: 842132