Page:EB1911 - Volume 14.djvu/738

Rh tabulated value differs from the corresponding true value by a tabular error which may have any value up to ± 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).

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 u, 2u, Thus u0 denotes u1 − u0, and 2u0 denotes u1 − u0 = 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 + (x1 − x0) = x0 + h, so that lies between 0 and 1, we may denote it by xundefined, and the corresponding value of u by uundefined. We have then

(1).

Tables of the values of the coefficients of 2u0 and 3u0 to three places of decimals for various values of from 0 to 1 are given in the ordinary collections of mathematical tables; but the formula is not really convenient if we have to go beyond 2u0, or if 2u0 itself contains more than two significant figures.

To apply the formula to Example 3 for x = 6.277, we have = .77, so that uundefined = .79239 + (.77 of 695) − (.089 of −11) = .79239 + 535 15 + 0 98 = .79775.

Here, as elsewhere, we use two extra figures in the intermediate calculations, for the purpose of adjusting the final figure in the ultimate result.

3. Taylor’s Theorem.—Where differences beyond the second are involved, Taylor’s Theorem is useful. This theorem (see ) gives the formula

(2),

where, c1, c2, c3, are the values for x = x0 of the first, second, third, differential coefficients of u with regard to x. The values of c1, c2, can occasionally be calculated from the analytical expressions for the differential coefficients of u; but more generally they have to be calculated from the tabulated differences. For this purpose central-difference formulae are the best. If we write

(3),

so that, if (as in §§ 1 and 2) each difference is placed opposite the space between the two quantities of which it is the difference, the expressions 2u0, 4u0, denote the differences of even order in a horizontal line with u0, and u0, 3u0,  denote the means of the differences of odd order immediately below and above this line, then (see ) the values of c1, c2, are given by

If a calculating machine is used, the formula (2) is most conveniently written

Using as the multiplicand in each case, the successive expressions ... P3, P2, P1, uundefined are easily calculated.

As an example, take u = tan x to five places of decimals, the values of x proceeding by a difference of 1°. It will be found that the following is part of the table:—

Example 4.—(u = tan x).

To find u for x = 66° 23′, we have = 23/60 = .3833333. The following shows the full working: in actual practice it would be abbreviated. The operations commence on the right-hand side. It will be noticed that two extra figures are retained throughout.

The value 2.2870967, obtained by retaining the extra figures, is correct within .7 of .00001 (§ 8), so that 2.28710 is correct within .00001 1.

In applying this method to mathematical tables, it is desirable, on account of the tabular error, that the differences taken into account in (4) should end with a difference of even order. If, e.g. we use 3u0 in calculating c1 and c3, we ought also to use 4u0 for calculating c2 and c4, even though the term due to 4u0 would be negligible if 4u0 were known exactly.

4. Geometrical and Algebraical Interpretation.—In applying the principle of proportional parts, in such a case as that of Example 1, we in effect treat the graph of u as a straight line. We see that the extremities of a number of consecutive ordinates lie approximately in a straight line: i.e. that, if the values are correct within ±, a straight line passes through points which are within a corresponding distance of the actual extremities of the ordinates; and we assume that this is true for intermediate ordinates. Algebraically we treat u as being of the form A + Bx, where A and B are constants determined by the values of u at the extremities of the interval through which we interpolate. In using first and second differences we treat u as being of the form A + Bx + Cx2; i.e. we pass a parabola (with axis vertical) through the extremities of three consecutive ordinates, and consider that this is the graph of u, to the degree of accuracy given by the data. Similarly in using differences of a higher order we replace the graph by a curve whose equation is of the form u = A + Bx + Cx2 + Dx3 + The various forms that interpolation-formulae take are due to the various principles on which ordinates are selected for determining the values of A, B, C

B. Inverse Interpolation.

5. To find the value of x when u is given, i.e. to find the value of when uundefined is given, we use the same formula as for direct interpolation, but proceed (if differences beyond the first are involved) by successive approximation. Taylor’s Theorem, for instance, gives