Page:EB1911 - Volume 14.djvu/739

Rh We first find an approximate value for : then calculate P1, and find by (6) a more accurate value of ; then, if necessary, recalculate P1, and thence, and so on.

6. When the values of u have been tabulated for values of x proceeding by a difference h, it is often desirable to deduce a table in which the differences of x are h/n, where n is an integer.

If n is even it may be advisable to form an intermediate table in which the intervals are h. For this purpose we have

where

The following is an example; the data are the values of tan x to five places of decimals, the interval in x being 1°. The differences of odd order are omitted for convenience of printing.

Example 5.

If a new table is formed from these values, the intervals being °, it will be found that differences beyond the fourth are negligible.

To subdivide h into smaller intervals than h, various methods may be used. One is to calculate the sets of quantities which in the new table will be the successive differences, corresponding to u0, u1, and to find the intermediate terms by successive additions. A better method is to use a formula due to J. D. Everett. If we write = 1 −, Everett’s formula is, in its most symmetrical form,

For actual calculations a less symmetrical form may be used. Denoting

(10)

by undefinedV1, we have, for interpolation between u0 and u1,

uundefined = u0 + u0 + undefinedV1 + 1−V0 (11),

the successive values of being 1/n, 2/n ,  (n − 1)/n. For interpolation between u1 and u2 we have, with the same succession of values of ,

u1+ = u1 + undefinedV1, &emsp; V2 + 1−V1 (12).

The values of 1−V1 in (12) are exactly the same as those of undefinedV1 in (11), but in the reverse order. The process is therefore that (i.) we find the successive values of u0 + u0, &c., i.e. we construct a table, with the required intervals of x, as if we had only to take first differences into account; (ii.) we construct, in a parallel column, a table giving the values of undefinedV1, &c.; (iii.) we repeat these latter values, placing the set belonging to each interval h in the interval next following it, and writing the values in the reverse order; and (iv.) by adding horizontally we get the final values for the new table.

As an example, take the values of tan x by intervals of ° in x, as found above (Ex. 5). The first diagram below is a portion of this table, with the differences, and the second shows the calculation of the terms of (11) so as to get a table in which the intervals are 0.1 of 1°. The last column but one in the second diagram is introduced for convenience of calculation.

Example 6.

The following are the values of the coefficients of u1, 2u1, 4u1, and 6u1 in (9) for certain values of n. For calculating the four terms due to 2u1 in the case of n = 5 it should be noticed that the third term is twice the first, the fourth is the mean of the first and the third, and the second is the mean of the third and the fourth. In table 3, and in the last column of table 2, the coefficients are corrected in the last figure.

—n = 5.

—n = 10.

—n = 12.

7. Derivation of Formulae.—The advancing-difference formula (1) may be written, in the symbolical notation of finite differences,

uundefined = (1 + )undefined u0 = Eundefined u0 (13);

and it is an extension of the theorem that if n is a positive integer

(14),

the series being continued until the terms vanish. The formula (14) is identically true: the formula (13) or (1) is only formally true, but its applicability to concrete cases is due to the fact that the series in (1), when taken for a definite number of terms, differs from the true value of uundefined by a “remainder” which in most cases is very small when this definite number of terms is properly chosen.

Everett’s formula (9), and the central-difference formula obtained by substituting from (4) in (2), are modifications of a standard formula

(15),