A note on the relative merits of Padé and Maehly’s diagonal convergents in computing $e^{x}$
HTML articles powered by AMS MathViewer
 by R. Sankar and V. Malini PDF
 Math. Comp. 17 (1963), 414418 Request permission
Abstract:
Methods for calculating functions to a high degree of accuracy have assumed increased importance following the advent of the computers. It has been found that rational approximations require fewer operations on a computer than the older polynomial approximations. Among the known methods those due to Padé [1] and Maehly [2] are perhaps the most important. In this paper we have analyzed these methods as applied to the exponential function. It is observed that Maehly’s method is superior to the Padé method in the sense of yielding better accuracy over a given range on the real axis for a given order of approximation. Maehly’s formulas for computing ${e^x}$ correct to eight decimal places have been worked out.References

H. Padé, “Sur la répresentation approchee dúne function par des fractions rationnelles,” Ann. Sci. École. Norm. Sup., Paris, v. 9, 1892, p. 193; v. 16, 1899, p. 315426.
H. Maehly, First Interim Progress Report on Rational Approximations, Project NR 44196, Princeton University, June 23, 1958.
 Modern computing methods, National Physical Laboratory, Teddington, England; Her Majesty’s Stationery Office, London, 1957. Notes on applied science, no. 16. MR 0088783
 E. G. Kogbetliantz, Computation of $e^N$ for $\infty <N<+\infty$ using an electronic computer, IBM J. Res. Develop. 1 (1957), 110–115. MR 90146, DOI 10.1147/rd.12.0110
 Kurt Spielberg, Efficient continued fraction approximations to elementary functions, Math. Comp. 15 (1961), 409–417. MR 134842, DOI 10.1090/S00255718196101348420 Nat. Bur. Standards, “Tables of the exponential function ${e^x}$,” Appl. Math. Ser. 14, Department of Commerce, Washington, D. C., 1951. British Association for the Advancement of Science, Comittee on Mathematical Tables, Vol. X, Bessel Functions, Part II, Cambridge University Press, 1952, p. 220237.
Additional Information
 © Copyright 1963 American Mathematical Society
 Journal: Math. Comp. 17 (1963), 414418
 MSC: Primary 65.20
 DOI: https://doi.org/10.1090/S00255718196301564493
 MathSciNet review: 0156449