Page:EB1911 - Volume 14.djvu/739

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

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 1/2h. For this purpose we have

u1/2 = 1/2 (U0 + U1)

(7)

where

U = u − 1/8δ2u + 3/128δ4u − 5/1024δ6u + . . .   = u − 1/8[δ2u − 3/16 {δ4u − 5/24 (δ6u − . . .) } ]

(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°. The differences of odd order are omitted for convenience of printing.

Example 5.

x.	u ≡ tan x.	δ2u.	δ4u.	δ6u.	U.	u = mean of values of U.	x.
		+	+	+
73°	3.27085	2339	100	5	3.26794 95
						3.37594	731/2°
74°	3.48741	2808	132	23	3.48392 98
						3.60588	741/2°
75°	3.73205	3409	187	18	3.72783 17
						3.86671	751/2°
76°	4.01078	4197	260	51	4.00559 22
						4.16530	761/2°
77°	4.33148	5245	384	64	4.32501 07

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

To subdivide h into smaller intervals than 1/2h, 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 φ = 1 − θ, Everett’s formula is, in its most symmetrical form,

uθ = θu1 +	(θ + 1) θ (θ − 1)	δ2u1 +	(θ + 2) (θ + 1) θ (θ − 1) (θ − 2)	δ4u1 + . . .	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$
	3!		5!
+ φu0 +	(φ + 1) φ (φ − 1)	δ2u0 +	(φ + 2) (φ + 1) φ (φ − 1) (φ − 2)	δ4u0 + . . .
	3!		5!

  (9).

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

(θ + 1) θ (θ − 1)	δ2u1 +	(θ + 2) (θ + 1) θ (θ − 1) (θ − 2)	δ4u1 + . . .
3!		5!

(10)

by _θV₁, we have, for interpolation between u₀ and u₁,

u_θ = u₀ + θΔu₀ + _θV₁ + _1−θV₀

(11),

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

u_1+θ = u₁ + _θV₁,   V₂ + _1−θV₁

(12).

The values of _1−θV₁ in (12) are exactly the same as those of _θV₁ in (11), but in the reverse order. The process is therefore that (i.) we find the successive values of u₀ + θΔu₀, &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 _θV₁, &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° 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.

x.	u = tan x.	δu.	δ2u.	δ3u.	δ4u.
		+	+	+	+
		11147		62
74°.0	3.48741		700		8
		11847		70
74°.5	3.60588		770		9
		12617		79

x	u0 + θΔu0.	θV1.	1−θV0.	θV1 + 1−θV0.	u.
73°.6	·	−22 35	·	·	·
73°.7	·	−39 11	·	·	·
73°.8	·	−44 71	·	·	·
73°.9	·	−33 54	·	·	·
74°.0	3.48741 00				3.48741
74°.1	3.51110 40	−24 58	−33 54	−58 12	3.51052
74°.2	3.53479 80	−43 02	−44 71	−87 73	3.53392
74°.3	3.55849 20	−49 18	−39 11	−88 29	3.55761
74°.4	3.58218 60	−36 89	−22 35	−59 24	3.58159
74°.5	3.60588 00				3.60588

The following are the values of the coefficients of u₁, δ²u₁, δ⁴u₁, and δ⁶u₁ in (9) for certain values of n. For calculating the four terms due to δ²u₁ 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. δ2u.	co. δ4u.	co. δ6u.
+	−	+	−
.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. δ2u.	co. δ4u.	co. δ6u.
+	−	+	−
.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. δ2u.	co. δ4u.	co. δ6u.
+	−	+	−
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

III. General Observations

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

u_θ = (1 + Δ)^θ u₀ = E^θ u₀

(13);

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

un = u0 + nΔu0 +	n (n − 1)	Δ2 u0 + . . .
	2!

(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 u_θ 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

uθ = u0 + θδu1/2 +	θ (θ − 1)	δ2 u0 +	(θ + 1) θ (θ − 1)	δ3 u1/2 +	(θ + 1) θ (θ − 1) (θ − 2)	δ4 u0 + . . .
	2!		3!		4!

(15),