Page:EB1911 - Volume 14.djvu/738

tabulated value differs from the corresponding true value by a tabular error which may have any value up to ± 1/2 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).

x.	u.	1st Diff.	2nd Diff.
6.0	.77815
		+718
6.1	.78533		−12
		+706
6.2	.79239		−11
		+695
6.3	.79934		−11
		+684
6.4	.80618		−11
		+673
6.5	.81291

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 Δu, Δ²u, . . . Thus Δu₀ denotes u₁ − u₀, and Δ²u₀ denotes Δu₁ − Δu₀ = 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 + θ(x₁ − x₀) = x₀ + θh, so that θ lies between 0 and 1, we may denote it by x_θ, and the corresponding value of u by u_θ. We have then

uθ = u0 + θΔu0 −	θ (1 − θ)	Δ2 u0 +	θ (1 − θ) (2 − θ)	Δ3 u0 − . . .
	2!		3!

(1).

Tables of the values of the coefficients of Δ²u₀ and Δ³u₀ to three places of decimals for various values of θ 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 Δ²u₀, or if Δ²u₀ itself contains more than two significant figures.

To apply the formula to Example 3 for x = 6.277, we have θ = .77, so that u_θ = .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 Infinitesimal Calculus) gives the formula

uθ = u0 + c1θ + c2	θ2	+	θ3	+ . . .
	2!		3!

(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

μδu0	= 1/2 (Δu0 + Δu−1)	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
δ2u0	= Δ2u−1
μδ3u0	= 1/2 (Δ3u−1 + Δ3u−2)
&c.

(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 δ²u₀, δ⁴u₀, . . . denote the differences of even order in a horizontal line with u₀, and μδu₀, μδ³u₀, . . . denote the means of the differences of odd order immediately below and above this line, then (see Differences, Calculus of) the values of c₁, c₂, . . . are given by

c1 = μδu0 − 1/6μδ3u0 + 1/30μδ5u0 − 1/140μδ7u0 + . . . c2 = δ2u0 − 1/12δ4u0 + 1/90δ6u0 − 1/560δ8u0 + . . . c3 = μδ3u0 − 1/4μδ5u0 + 7/120μδ7u0 − . . . c4 = δ4u0 − 1/6δ6u0 + 7/240δ8u0 − . . . c5 = μδ5u0 − 1/3μδ7u0 + . . . c6 = δ6u0 − 1/4δ8u0 + . . .  .    .  .    .  .    .

$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$

(4). 

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

uθ = u0 + P1θ P1 = c1 + 1/2P2θ P2 = c2 + 1/3P3θ  .   .  .   .  .   .

$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$

(5). 

Using θ as the multiplicand in each case, the successive expressions ... P₃, P₂, P₁, u_θ 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).

x.	u.	1st Diff.	2nd Diff.	3rd Diff.	4th Diff.
		+	+	+	+
65°	2.14451		732		16
		10153		96
66°	2.24604		828		19
		10981		115
67°	2.35585		943		18

To find u for x = 66° 23′, we have θ = 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.

u0.	μδu0.	δ2u0.	μδ3u0.	δ4u0.
2.24604	+1056700	+82800	+10550	+1900
	−   1758	−   158
	———	———	———	———
	c1 = +1054942	c2 = +82642	c3 = +10550	c4 = +1900
P1θ = +410567	1/2P2θ = +⁠16102	1/3P3θ = + 1371	1/8c4θ = +   182
———	———	———	———
uθ = 2.28710	P1 = +1071044	P2 = +84013	P3 = +10732

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 μδ³u₀ in calculating c₁ and c₃, we ought also to use δ⁴u₀ for calculating c₂ and c₄, even though the term due to δ⁴u₀ would be negligible if δ⁴u₀ 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 ±1/2ρ, 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 θ when u_θ 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

θ = (uθ − u0) ÷ (c1 + c2	θ	+ . . .)
	2!

= (uθ − u0) ÷ P1

(6).