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and of all succeeding orders vanish. Hence if in any other function such differences become very small, that function may, quite irrespectively of its form, be approximately represented by a function which is rational and integral. Of course it is supposed that the value of x for which that of the function is required is not very remote from those, or from some of those, values for which the values of the function are given. The same assumption as to the form of the unknown function and the same condition of limitation as to the use of that form flow in an equally obvious manner from the expansion in Taylor's theorem.
2. The problem of interpolation assumes different forms, according as the values given are equidistant, i. e. correspondent to equidifferent values of the independent variable, or not. But the solution of all its cases rests upon the same principle. The most obvious mode in which that principle can be applied is the following. If for n values a, b, ... of an independent variable x the corresponding value uq, Un, ... of an unknown function of x represented by ur, are given, then, assuming as the approximate general expression of Uzi Un= A + Bx + Cx2 ... + Exn-1.
........(1), a form which is rational and integral and involves n arbitrary coefficients, the data in succession give
Ug = A + Ba + Ca? ... + Ea"-1,
a system of n linear equations which determine A, B... E. To avoid the solving of these equations other but equivalent modes of procedure are employed, all such being in effect reducible to the two following, viz. either to an application of that property of the rational and integral function in the second member of (1) which is expressed by the equation A"un = 0, or to the substitution of a different but equivalent form for the rational and integral function. These methods will be respectively illustrated in Prop. 1 and its deductions, and in Prop. 2, of the following sections.
PROP. 1. Given n consecutive equidistant values Uo, Wy, Um-, of a function Uz, to find its approximate general expression.
By Chap. II. Art. 10,
m (m – 1) A’ux + &c.
Uzim = Uz+mAu+
Hence, substituting 0 for x, and wc for m, we have
ac (oc — 1)
1.2 But on the assumption that the proposed expression is rational and integral and of the degree n-1, we have A"ur=0, and therefore Allu, = 0. Hence
(2), the expression required. It will be observed that the second member is really a rational and integral function of ac of the degree n-1, while the coefficients are made determinate by the data.
In applying this theorem the value of x may be conceived to express the distance of the term sought from the first term in the series, the common distance of the terms given being taken as unity.
Ex. Given log 3:14=.4969296, log 3.15 =·4983106, log 3:16 = .4996871, log 3:17 = 5010593; required an approximate value of log 3:14159.
Here, omitting the decimal point, we have the following table of numbers and differences:
The first column gives the values of u, and its differences up to Aug. Now the common difference of 3.14, 3.15, &c. being taken as unity, the value of x which corresponds to 3:14159 will be .159. Hence we have
(159) (-159 - 1) Uz = 4969296 +159 13810 +
1.2 ('159) (-159 – 1) (-159 - 2)
1.2.3 Effecting the calculations we find ur='4971495, which is true to the last place of decimals. Had the first difference only been employed, which is equivalent to the ordinary rule of proportional parts, there would have been an error of 3 in the last decimal."
3. When the values given and that sought constitute a series of equidistant terms, whatever may be the position of the value sought in that series, it is better to proceed as follows.
Let Vo? Uz, Ug, ... Un be the series. Then since, according to the principle of the method, Aou, = 0, we have by Chap. II. Art. 10,
n (n-1) Un - nun-1
+ U m--...+(-1)" u, = 0...... (3),
1.2 an equation from which any one of the quantities
be found in terms of the others. Thus, to interpolate a term midway between two others
ut Uq Uy – 2u, + u= 0; i. U,
2 Here the middle term is only the arithmetical mean. To supply the middle term in a series of five, we have
U, — 4u, + 6u, – 4u, tun=0;
4 (U, +ug) - (u, + un) .. Un =
Ex. Representing as is usual E An-do by r (n), it is required to complete the following table by finding approximately log I
12 Let the series of values of log T (n) be represented by U7, Ug, ... Uy, the value sought being that of Ug. Then proceeding as before, we find
8.7 8.7.6 U, - 8u, +
1.2.34, + &c. = 0,
Uz + Ug -8 (ug + uc) + 28 (uz +wn) – 56 (un + ua) + 70u= 0; whence
56 (U, + wc) — 28 (uz +U-)+8(u,+w) - (U+ u) UG
.(6). 70 Substituting for U,, U,, &c., their values from the table,
the true value being •24858.
To shew the gradual closing of the approximation as the number of the values given is increased, the following results are added :
Calculated value of uge и, и.
•25610, Ug, U Ugg Ug
24820, U2, U3, U U, U, Uz
.24865, Uy, U,, Ug, u, Ug, Wg, Ug, Ug............ 24853. . 4. By an extension of the same method, we may treat any case in which the terms given and sought are terms, but not consecutive terms, of a series. Thus, if u,, U, Uy were given and ug sought, the equations A%u, = 0, Aøu, = 0 would give
u, - 3u, +3u, -u, = 0,
U, - 3u, + 3ug - u, = 0, from which, eliminating wg, we have 3u; - 8u, + 6u, - ,= 0
Oʻ........... ..(7), and hence u, can be found. But it is better to apply at once the general method of the following Proposition.
PROP, 2. Given n values of a function which are not consecutive and equidistant, to find any other value whose place is given.
Let U as Up, Wo, ... ux be the given values, corresponding to a, b, c ...
... k respectively as values of x, and let it be required to determine an approximate general expression for Ux.
We shall assume this expression rational and integral, Art. 1.
Now there being n conditions to be satisfied, viz. that for i = a, x=b ... x =k, it shall assume the respective values Ua, Ug ... Ux, the expression must contain n constants, whose values those conditions determine. We might therefore assume Uz = A + Bx + Caca
(8), and determine A, B, C by the linear system of equations formed by making a = a, b ... k, in succession.
The substitution of another but equivalent form for (8) enables us to dispense with the solution of the linear system.