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On Direct and Inverse Interpolation by Divided Differences

Published online by Cambridge University Press:  20 January 2009

Glenny Smeal
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The general interpolation series, originated by Newton, has been studied mainly for its algebraic interest, only the special case of equidistant data being developed on the practical side. This is justified by the simplicity of this case, and by the numerous problems for which it suffices, but it may lead to undue simplification of data and to restrictions on experimental and computative methods. Thus tables of functions which are not in common use, or which are carried to many places, must often be limited to relatively few entries, and these might conceivably be not in arithmetical progression, with advantage both of easier tabulation and of more accurate interpolation. Data from experiment or statistics, again, are often fitted to a parabolic curve of arbitrarily chosen degree, and on rather inadequate grounds. The formation of a difference-table not only avoids the suppression of the original data, but supplies at a glance a useful analysis of them—indicating their consistency and regularity, showing with what accuracy a parabolic curve can represent them, and supplying its expression with minimum labour. For direct interpolation to new points Lagrange's formula, the usual alternative, fails in this respect and, when applied to unfamiliar data, is very apt to mislead. It is wasteful of labour and more liable to error, and cannot easily be extended to include fresh terms.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , February 1919 , pp. 34 - 50
Copyright
Copyright © Edinburgh Mathematical Society 1919

References

* Cf. the rules of Tchebichev, Gauss, etc., on ohoioe of points for interpolation and integration. Also Professor Steggall's suggestions for economy of entries in ordinary tables (Napier Tercentenary Memorial Volume (1915), p. 319).

† Teubner (1909). Some early suggestions were contained in Gauss' Lectures, published by Encke (Berliner Astron. Jahrb. (1830); Abbhandlungen I.); these are reproduced in the Encycl. des Sc. Maths. (T. 1, Vol. 4 pp. 130–7) See also Encyc. Brit., Interpolation.

* This form of the theory is due to Ampère (Ann. de Gergonne 16 (1826), p. 329), whose work was extended by Cauchy (C.R. 11 (1840), p. 775…; Œuvres (1), 5). A similar notation was used at the same time by Legendre—Traité des functions Elliptiques, Vol. 2 (1826), p. 36. Ampère uses the term “interpolation-funotions,” but “divided-differences” seems more suitable for praotioal applications and is used by Oppermann (J. Inst. Act. 15 (1869), p. 145), Merrifleld (Brit. Assn. Report, 1880) and Thiele. For references to other work on the properties of “interpolation functions” see B. Pascal—Calcole delle differtnze finite, or his Repertorium der höheren Mathematik.

* This expression of the divided difference shows the relation between the Newton and Lagrange formulæ. Comparison of coefficients of fr yields a Newton series for the Lagrange term which might serve for a synthetic calculation, viz., for any order of the points beginning at xr we have

.

† An alternative form of difference-table waa suggested by E. McClintook (Amer. J. Mth. 2 (1879), p. 307) in which every difference bears reference to the initial point x 0 ; the first column contains (01), (02), (03)…, the second (012), (013)… The calculation is claimed to be simpler, but the table is available only for the one order of the data, that of tabulation, or at most for xr, x 0x 1x 2 ….

* Compare D. C. Fraser's diagram of formulæ for the equidistant table—J. Int. Act. 43 (1909), pp. 235, 442Google Scholar. Also Sheppard, W. F.–J. I. A. 50 (1916), p. 85Google Scholar; R. Todhunter—do., p. 133.

* A valuable discussion of the ordinary table will be found in two papers by Sheppard, W. F.—Proc L. Mth. Soc, 4 (1906), p. 320; 10 (1911), p. 139.Google Scholar

† A graphical analysis of these data is given in Lipka, J.—Graphical and Mechanical Compulation (Wiley 1918), p. 146Google Scholar. By plotting first differences of θ as a straight line he obtains the formula θ=141·4+0·620x+0·0130x 2, while the tables suggest as mostsuitable θ=a+bθ+cθ2+dθ3. The accuracy of the data is not sufficient for proper comparison of methods, but the difference in the value of the derivative is considerable.

* Veltmann, W.–Ztilschr. f. Mth. u. Phys., 44 (1899), p. 303Google Scholar—suggests a notation analogous to a continued fraction,

the double line denoting multiplication in place of division.

† Horner's algorithm for solution of numerical equations is a special case when the data are all coincident. The procedure for interpolation is found in Legendre—loc. cit. in § 2.