I. Interpolation from Mathematical Tables

A. Direct Interpolation.

1. Interpolation by First Differences.—The simplest cases are those in which the first difference in u is constant, or nearly so. For example:—

Example 1.—(u = log10x).    Example 2.—(u = log10x).
x. u. 1st Diff.    x. u. 1st Diff.
+    +
4.341 .6375898    7.40 .86923
1000    59
4.342 .6376898    7.41 .86982
1000    58
4.343 .6377898    7.42 .87040
1000    59
4.344 .6378898    7.43 .87099
1000    58
4.345 .6379898    7.44 .87157

In Example 1 the first difference of u corresponding to a difference of h ≡ .001 in x is .0001000; but, since we are working throughout to seven places of decimals, it is more convenient to write it 1000. This system of ignoring the decimal point in dealing with differences will be adopted throughout this article. To find u for an intermediate value of x we assume the principle of proportional parts, i.e. we assume that the difference in u is proportional to the difference in x. Thus for x = 4.342945 the difference in u is .945 of 1000 = 945, so that u is .6376898 + .0000945 = .6377843. For x = 4.34294482 the difference in u would be 944.82, so that the value of u would apparently be .6376898 + .000094482 = .637784282. This, however, would be incorrect. It must be remembered that the values of u are only given “correct to seven places of decimals,” i.e. each tabulated value differs from the corresponding true value by a tabular error which may have any value up to ± ½ of .0000001; and we cannot therefore by interpolation obtain a result which is correct to nine places. If the interpolated value of u has to be used in calculations for which it is important that this value should be as accurate as possible, it may be convenient to retain it temporarily in the form .6376898 + 944 82 = .6377842 82 or .6376898 + 94482 = .637784282; but we must ultimately return to the seven-place arrangement and write it as .6377843. The result of interpolation by first difference is thus usually subject to two inaccuracies, the first being the tabular error of u itself, and the second being due to the necessity of adjusting the final figure of the added (proportional) difference. If the tabulated values are correct to seven places of decimals, the interpolated value, with the final figure adjusted, will be within .0000001 of its true value.

In Example 2 the differences do not at first sight appear to run regularly, but this is only due to the fact that the final figure in each value of u represents, as explained in the last paragraph, an approximation to the true value. The general principle on which we proceed is the same; but we use the actual difference corresponding to the interval in which the value of x lies. Thus for x = 7.41373 we should have u = .86982 + (.373 of 58) = .87004; this result being correct within .00001.

2. Interpolation by Second Differences.—If the consecutive first differences of u are not approximately equal, we must take account of the next order of differences. For example:—

Example 3.—(u = log 10x).

x.u.1st Diff.2nd Diff.
6.0.77815
+718
6.1.78533 −12
+706
6.2.79239 −11
+695
6.3.79934 −11
+684
6.4.80618 −11
+673
6.5.81291

In such a case the advancing-difference formula is generally used. The notation is as follows. The series of values of x and of u are respectively x0, x1, x2, ... and u0, u1, u2, ... ; and the successive differences of u are denoted by Δu, Δ2u, ... Thus Δu0 denotes u1 − u0, and Δ2u0 denotes Δu1 - Δu0 = u2 − 2u1 + u0. The value of x for which u is sought is supposed to lie between x0 and x1. If we write it equal to x0 + θ(x1 − x0) = x0 + θh, so that θ lies between 0 and 1, we may denote it by xθ, and the corresponding value of u by uθ. We have then