Chapter II: Construction of Tables by Subdivision of Intervals

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 (1/2)h. For this purpose we have

u_(1/2) = (1/2)(U0 + U1) (7),

where

U = u - (1/8)[delta]^2u + (3/128)[delta]^4u - (5/1024)[delta]^6u + ...

= u - (1/8)[[delta]^2u - (3/16){[delta]^4u - (5/24)([delta]^6u - ...)}] (8).

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

_Example 5._

+--------+---------+-----------+-----------+-----------+------------+------------+-------------+
| | u [eq] | | | | |u = mean of | |
| x. | tan x. |[delta]^2u.|[delta]^4u.|[delta]^6u.| U. |values of U.| x. |
+--------+---------+-----------+-----------+-----------+------------+------------+-------------+
| | | + | + | + | | | |
| 73 deg.| 3.27085 | 2339 | 100 | 5 | 3.26794 95 | | |
| | | | | | | 3.37594 | 73(1/2) deg.|
| 74 deg.| 3.48741 | 2808 | 132 | 23 | 3.48392 98 | | |
| | | | | | | 3.60588 | 74(1/2) deg.|
| 75 deg.| 3.73205 | 3409 | 187 | 18 | 3.72783 17 | | |
| | | | | | | 3.86671 | 75(1/2) deg.|
| 76 deg.| 4.01078 | 4197 | 260 | 51 | 4.00559 22 | | |
| | | | | | | 4.16530 | 76(1/2) deg.|
| 77 deg.| 4.33148 | 5245 | 384 | 64 | 4.32501 07 | | |
+--------+---------+-----------+-----------+-----------+------------+------------+-------------+

If a new table is formed from these values, the intervals being 1/2
deg., it will be found that differences beyond the fourth are
negligible.

To subdivide h into smaller intervals than (1/2)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 [phi] = 1
- [theta], Everett's formula is, in its most symmetrical form,

([theta] + 1)[theta]([theta] - 1)
u_([theta]) = [theta]u1 + ---------------------------------[delta]^2u1
3!

([theta] + 2)([theta] + 1)[theta]([theta] - 1)([theta] - 2)
+ -----------------------------------------------------------[delta]^4u1 + ...

([phi] + 1)[phi]([phi] - 1)
+ [phi]u0 + ---------------------------[delta]^2u0
3!

([phi] + 2)([phi] + 1)[phi]([phi] - 1)([phi] - 2)
+ -------------------------------------------------[delta]^4u0 + ... (9).
5!

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

([theta] + 1)[theta]([theta] - 1)
---------------------------------[delta]^2u1
3!

([theta] + 2)([theta] + 1)[theta]([theta] - 1)([theta] - 2)
+ -----------------------------------------------------------[delta]^4u1 + ... (10)
5!

by _([theta])V1, we have, for interpolation between u0 and u1,

u_([theta]) = u0 + [theta][Delta]u0 + _([theta])V1 + _(1 - [theta])V0 (11),

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

u_(1+[theta]) = u1 + _([theta])V1, V2 + _(1-[theta])V1 (12).

The values of _(1-[theta])V1 in (12) are exactly the same as those of
([theta])V1 in (11), but in the reverse order. The process is
therefore that (i.) we find the successive values of u0 +
[theta][Delta]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 _([theta])V1, &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 1/2 deg. 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 deg. The last column but one in the second diagram is introduced
for convenience of calculation.

_Example 6._

+---------+-----------+---------+-----------+-----------+-----------+
| x. | u = tan x.|[delta]u.|[delta]^2u.|[delta]^3u.|[delta]^4u.|
+---------+-----------+---------+-----------+-----------+-----------+
| | | + | + | + | + |
| | | 11147 | | 62 | |
| 74 deg.0| 3.48741 | | 700 | | 8 |
| | | 11847 | | 70 | |
| 74 deg.5| 3.60588 | | 770 | | 9 |
| | | 12617 | | 79 | |
+---------+-----------+---------+-----------+-----------+-----------+

+----------+------------------+--------------+----------------+----------------+---------+
| | u0 + | | | _([theta])V1 + | |
| x. | [theta][Delta]u0.| _([theta])V1.| _(1-[theta])V0.| _(1-[theta])V0.| u. |
+----------+------------------+--------------+----------------+----------------+---------+
| 73 deg.6 | . | -22 35 | . | . | . |
| 73 deg.7 | . | -39 11 | . | . | . |
| 73 deg.8 | . | -44 71 | . | . | . |
| 73 deg.9 | . | -33 54 | . | . | . |
| 74 deg.0 | 3.48741 00 | | | | 3.48741 |
| 74 deg.1 | 3.51110 40 | -24 58 | -33 54 | -58 12 | 3.51052 |
| 74 deg.2 | 3.53479 80 | -43 02 | -44 71 | -87 73 | 3.53392 |
| 74 deg.3 | 3.55849 20 | -49 18 | -39 11 | -88 29 | 3.55761 |
| 74 deg.4 | 3.58218 60 | -36 89 | -22 35 | -59 24 | 3.58159 |
| 74 deg.5 | 3.60588 00 | | | | 3.60588 |
+----------+------------------+--------------+----------------+----------------+---------+

The following are the values of the coefficients of u1, [delta]^2u1,
[delta]^4u1, and [delta]^6u1 in (9) for certain values of n. For
calculating the four terms due to [delta]^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.

TABLE 1.--n = 5.

+------+---------------+---------------+--------------------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.| co. [delta]^6u. |
+------+---------------+---------------+--------------------------+
| + | - | + | - |
| .2 | .032 | .006336 | .00135168 = 1/740 approx.|
| .4 | .056 | .010752 | .00226304 = 1/442 " |
| .6 | .064 | .011648 | .00239616 = 1/417 " |
| .8 | .048 | .008064 | .00160512 = 1/623 " |
+------+---------------+---------------+--------------------------+

TABLE 2.--n = 10.

+------+---------------+---------------+---------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.|
+------+---------------+---------------+---------------+
| + | - | + | - |
| .1 | .0165 | .00329175 | .000704591 |
| .2 | .0320 | .00633600 | .001351680 |
| .3 | .0455 | .00889525 | .001887064 |
| .4 | .0560 | .01075200 | .002263040 |
| .5 | .0625 | .01171875 | .002441406 |
| .6 | .0640 | .01164800 | .002396160 |
| .7 | .0595 | .01044225 | .002115799 |
| .8 | .0480 | .00806400 | .001605120 |
| .9 | .0285 | .00454575 | .000886421 |
+------+---------------+---------------+---------------+

TABLE 3.--n = 12.

+------+---------------+---------------+---------------+
|co. u.|co. [delta]^2u.|co. [delta]^4u.|co. [delta]^6u.|
+------+---------------+---------------+---------------+
| + | - | + | - |
| 1/12 | .013792438 | .002753699 | .000589623 |
| 2/12 | .027006173 | .005363726 | .001145822 |
| 3/12 | .039062500 | .007690430 | .001636505 |
| 4/12 | .049382716 | .009602195 | .002032211 |
| 5/12 | .057388117 | .010979463 | .002307357 |
| 6/12 | .062500000 | .011718750 | .002441406 |
| 7/12 | .064139660 | .011736667 | .002419911 |
| 8/12 | .061728395 | .010973937 | .002235432 |
| 9/12 | .054687500 | .009399414 | .001888275 |
|10/12 | .042438272 | .007014103 | .001387048 |
|11/12 | .024402006 | .003855178 | .000748981 |
+------+---------------+---------------+---------------+