Page:EB1911 - Volume 14.djvu/740

which may similarly be regarded as an extension of the theorem that, if n is a positive integer,

un = u0 + nδu1/2 +n (n − 1)/2!δ2 u0 +(n + 1)n(n − 1)/3!δ3 u1/2 + . . .

(16).

There are other central-difference formulae besides those mentioned above; the general symbolical expression is

uθ = (cosh θhD + sinh θhD) u0

(17).

where

cosh 1/2hD = μ, sinh 1/2hD = 1/2δ

(18).

8. Comparative Accuracy.—Central-difference formulae are usually more accurate than advancing-difference formulae, whether we consider the inaccuracy due to omission of the “remainder” mentioned in the last paragraph or the error due to the approximative character of the tabulated values. The latter is the more important. If each tabulated value of u is within ±1/2ρ of the corresponding true value, and if the differences used in the formulae are the tabular differences, i.e. the actual successive differences of the tabulated values of u, then the ratio of the limit of error of u_θ, as calculated from the first r terms of the series in (1), to 1/2ρ is the sum of the first r terms of the series

1 + 0 + θ (1 − θ) + θ (1 − θ) (2 − θ) + 7/12θ (1 − θ) (2 − θ) (3 − θ) +
   1/4θ (1 − θ) (2 − θ) (3 − θ) (4 − θ) + 31/360θ (1 − θ) . . . (5 − θ) + . . .,

while the corresponding ratio for the use of differences up to δ^2pu₀ inclusive in (4) or up to δ^2p u₁ and o^2p u₀ in (9) (i.e. in effect, up to δ^2p+1 u_1/2) is the sum of the first p + 1 terms of the series

1 +	θ (1 − θ)	+	(1 + θ) θ (1 − θ) (2 − θ)	+	(2 + θ) (1 + θ) θ (1 − θ) (2 − θ) (3 − θ)	+ . . .,
	1.1		(2!)2		(3!)2

it being supposed in each case that θ lies between 0 and 1. The following table gives a comparison of the respective limits of error; the lines I. and II. give the errors due to the advancing-difference and the central-difference formulae, and the coefficient ρ is omitted throughout.

Table 4.

		Error due to use of Differences up to and  including
		1st.	2nd.	3rd.	4th.	5th.	6th.	7th.
.5${\Big \{}$	I. . .	.500	.625	.813	1.086	1.497	2.132	3.147
	II. . .	.500	.625	.625	.696	.696	.745	.745
.2${\Big \{}$	I. . .	.500	.580	.724	.960	1.343	1.976	3.042
	II. . .	.500	.580	.580	.624	.624	.653	.653
.4${\Big \{}$	I. . .	.500	.620	.812	1.104	1.553	2.265	3.422
	II. . .	.500	.620	.620	.688	.688	.734	.734
.6${\Big \{}$	I. . .	.500	.620	.788	1.024	1.366	1.886	2.700
	II. . .	.500	.620	.620	.688	.688	.734	.734
.8${\Big \{}$	I. . .	.500	.580	.676	.800	.969	1.213	1.582
	II. . .	.500	.580	.580	.624	.624	.653	.653

In some cases the differences tabulated are not the tabular differences, but the corrected differences; i.e. each difference, like each value of u, is correct within ±1/2ρ. It does not follow that these differences should be used for interpolation. Whatever formula is employed, the first difference should always be the tabular first difference, not the corrected first difference; and, further, if a central-difference formula is used, each difference of odd order should be the tabular difference of the corrected differences of the next lower order. (This last result is indirectly achieved if Everett’s formula is used.) With these precautions (i.) the central-difference formula is slightly improved by using corrected instead of tabular differences, and (ii.) the advancing-difference formula is greatly improved, being better than the central-difference formula with tabular differences, but still not so good as the latter with corrected differences. For θ = .5, for instance, supposing we have to go to fifth differences, the limits ±1.497 and ±.696, as given above, become ±.627 and ±.575 respectively.

9. Completion of Table of Differences.—If no values of u outside the range within which we have to interpolate are given, the series of differences will be incomplete at both ends. It may be continued in each direction by treating as constant the extreme difference of the highest order involved; and central-difference formulae can then be employed uniformly throughout the whole range.

Suppose, for instance, that the values of tan x in § 6 extended only from x = 60° to x = 80°, we could then complete the table of differences by making the entries shown in italics below.

Example 7.

x.	u = tan x.	δu.	δ2u.	δ3u.	δ4u.	δ5u.	δ6u.
		+	+	+	+	+	+
		6775		34
60°	1.73205		425		9
		7200		43
61°	1.80405		468		9
		7668		52
62°	1.88073		520		9
		8188		61
63°	1.96261		581		10
		8769		71
64°	2.05030	.	652	.	9
.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.
75°	3.73205	.	3409	.	187	.	18
		27873		788		73
76°	4.01078		4197		260		51
		32070		1048		124
77°	4.33148		5245		384		64
		37315		1432		188
78°	4.70463		6677		572		64
		43992		2004		252
79°	5.14455		8681		824		64
		52673		2828		316
80°	5.67128		11509		1140		64
		64182		3968		380

For interpolating between x = 60° and x = 61° we should obtain the same result by applying Everett’s formula to this table as by using the advancing-difference formula; and similarly at the other end for the receding differences.

Interpolation by Substituted Tabulation.

10. The relation of u to x may be such that the successive differences of u increase rapidly, so that interpolation-formulae cannot be employed directly. Other methods have then to be used. The best method is to replace u by some expression v which is a function of u such that (i.) the value of v or of u can be determined for any given value of u or of v, and (ii.) when v is tabulated in terms of x the differences decrease rapidly. We can then calculate v, and thence u, for any intermediate value of x.

If, for instance, we require tan x for a value of x which is nearly 90°, it will be found that the table of tangents is not suitable for interpolation. We can, however, convert it into a table of cotangents to about the same number of significant figures; from this we can easily calculate cot x, and thence tan x.

11. This method is specially suitable for statistical data, where the successive values of u represent the area of a figure of frequency up to successive ordinates. We have first to determine, by inspection, a curve which bears a general similarity to the unknown curve of frequency, and whose area and abscissa are so related that either can be readily calculated when the other is known. This may be called the auxiliary curve. Denoting by ξ the abscissa of this curve which corresponds to area u, we find the value of ξ corresponding to each of the given values of u. Then, tabulating ξ in terms of x, we have a table in which, if the auxiliary curve has been well chosen, differences of ξ after the first or second are negligible. We can therefore find ξ, and thence u, for any intermediate value of x.

Extensions.

12. Construction of Formulae.—Any difference of u of the r th order involves r + 1 consecutive values of u, and it might be expressed by the suffixes which indicate these values. Thus we might write the table of differences

x.	u.	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·
·	·	(−1, 0)	·	(−2, −1, 0, 1)	·
x0	u0		(−1, 0, 1)		(−2, −1, 0, 1, 2)
		(0, 1)		(−1, 0, 1, 2)
x1	u1		(0, 1, 2)		(−1, 0, 1, 2, 3)
		(1, 2)		(0, 1, 2, 3)
x2	u2		(1, 2, 3)		(0, 1, 2, 3, 4)
·	·	(2, 3)	·	(1, 2, 3, 4)	·
·	·	·	·	·	·
·	·	·	·	·	·
·	·	·	·	·	·