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Interpolation without Printed Differences: Jordan’s and Aitken’s Formulae

Published online by Cambridge University Press:  03 November 2016

J. Wishart ,
A. C. Aitken  and
G. J Lidstone
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Extract

I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.

Type
Research Article
Information
The Mathematical Gazette , Volume 16 , Issue 217 , February 1932 , pp. 14 - 25
Copyright
Copyright © Mathematical Association 1932

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References

page no 14 note † For further details, see Sheppard, W. F., Gazette, XV, 1930, 232–249 CrossRefGoogle Scholar.

page no 14 note ‡ Interpolation, 1927 (The Williams and Wilkins Co., Baltimore).

page no 15 note * Thompson, A. J., Tracts for Computers, No. v (Cambridge)Google Scholar; E. Chappell, Interpolation Coefficients, 1929 (The author; 41 Westcomb Park Road, S.E. 3); J. W Glover, Tables of Applied Mathematics in Statistics, 1924 (Wahr).

page no 15 note † Mathematical Tables, Vol. I, 1931. (British Association.)

page no 15 note ‡ Metron, vii, 3 (1928), 47-51.

page no 15 note § See the papers following this one, pp. 18, 22.

page no 16 note * G 11 = θ(1 – θ)/2!; G IV = θ(1 – θ 2)(2 – θ)/4!; G VI = θ(1 – θ 2)(4 – θ 2)(3 – θ)/6!, etc.

page no 17 note * British Association, loc. cit.

page no 17 note † See p. 18.

page no 17 note ‡ Journal of the Institute of Actuaries, lx. (1929), 349-352.

page no 17 note § See p. 22,

page no 22 note * Cf. the Newton-Bessel formula, but for the present purpose it is better to go back to first principles.

page no 22 note † This is the Newton-Bessel formula to the first difference.