A note on the relative merits of Padé and Maehly's diagonal convergents in computing

Authors:
R. Sankar and V. Malini

Journal:
Math. Comp. **17** (1963), 414-418

MSC:
Primary 65.20

MathSciNet review:
0156449

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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 correct to eight decimal places have been worked out.

**[1]**H. Padé, ``Sur la répresentation approchee dúne function par des fractions rationnelles,''*Ann. Sci. École. Norm. Sup.*, Paris, v. 9, 1892, p. 1-93; v. 16, 1899, p. 315-426.**[2]**H. Maehly,*First Interim Progress Report on Rational Approximations*, Project NR 44-196, Princeton University, June 23, 1958.**[3]***Modern computing methods*, Notes on applied science, no. 16, National Physical Laboratory, Teddington, England. Her Majesty’s Stationery Office, London, 1957. MR**0088783****[4]**E. G. Kogbetliantz,*Computation of 𝑒^{𝑁} for -∞<𝑁<+∞ using an electronic computer*, IBM J. Res. Develop.**1**(1957), 110–115. MR**0090146****[5]**Kurt Spielberg,*Efficient continued fraction approximations to elementary functions.*, Math. Comp.**15**(1961), 409–417. MR**0134842**, 10.1090/S0025-5718-1961-0134842-0**[6]**Nat. Bur. Standards, ``Tables of the exponential function ,''*Appl. Math. Ser.*14, Department of Commerce, Washington, D. C., 1951.**[7]**British Association for the Advancement of Science, Comittee on Mathematical Tables, Vol. X,*Bessel Functions, Part II*, Cambridge University Press, 1952, p. 220-237.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1963-0156449-3

Article copyright:
© Copyright 1963
American Mathematical Society