A. Direct Interpolation.
1. _Interpolation by First Differences._--The simplest cases are those
in which the first difference in u is constant, or nearly so. For
example:--
_Example_ 1.--(u = log_10 x). _Example_ 2.--(u = log_10 x).
+-------+--------+---------+ +------+--------+---------+
| x. | u. |1st Diff.| | x. | u. |1st Diff.|
+-------+--------+---------+ +------+--------+---------+
| | | + | | | | + |
| 4.341 |.6375898| | | 7.40 | .86923 | |
| | | 1000 | | | | 59 |
| 4.342 |.6376898| | | 7.41 | .86982 | |
| | | 1000 | | | | 58 |
| 4.343 |.6377898| | | 7.42 | .87040 | |
| | | 1000 | | | | 59 |
| 4.344 |.6378898| | | 7.43 | .87099 | |
| | | 1000 | | | | 58 |
| 4.345 |.6379898| | | 7.44 | .87157 | |
+-------+--------+---------+ +------+--------+---------+
In Example 1 the first difference of u corresponding to a difference
of h [equivalent] .001 in x is .0001000; but, since we are working
throughout to seven places of decimals, it is more convenient to write
it 1000. This system of ignoring the decimal point in dealing with
differences will be adopted throughout this article. To find u for an
intermediate value of x we assume the principle of proportional parts,
i.e. we assume that the difference in u is proportional to the
difference in x. Thus for x = 4.342945 the difference in u is .945 of
1000 = 945, so that u is .6376898 + .0000945 = .6377843. For x =
4.34294482 the difference in u would be 944.82, so that the value of u
would apparently be .6376898 + .000094482 = .637784282. This, however,
would be incorrect. It must be remembered that the values of u are
only given "correct to seven places of decimals," i.e. each 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 + 944^82 = .6377842^82;
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_10 x).
+-----+-------+---------+---------+
| 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 [Delta]u, [Delta]^2u, ... Thus
[Delta]u0 denotes u1 - u0, and [Delta]^2u0 denotes [Delta]u1 -
[Delta]u0 = 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 +
[theta](x1 - x0) = x0 + [theta]h, so that [theta] lies between 0 and
1, we may denote it by x_([theta]), and the corresponding value of u
by u_([theta]). We have then
[theta] (1 - [theta])
u[theta] = u0 + [theta][Delta]u0 - --------------------- [Delta]^2u0
2!
[theta] (1 - [theta]) (2 - [theta])
+ ----------------------------------- [Delta]^3u0 - ... (1).
3!
Tables of the values of the coefficients of [Delta]^2u0 and [Delta]^3u0
to three places of decimals for various values of [theta] 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 [Delta]^2u0,
or if [Delta]^2u0 itself contains more than two significant figures.
To apply the formula to Example 3 for x = 6.277, we have [theta] =
.77, so that u_([theta]) = .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
[theta]^2 [theta]^3
u_([theta]) = u0 + c1[theta] + c2 --------- + c3 --------- + ... (2),
2! 3!
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
[mu][delta]u0 = (1/2)([Delta]u0 + [Delta]u_(-1)) \
[delta]^2u0 = [Delta]^2u_(-1) | (3),
[mu][delta]^3u0 = (1/2)([Delta]^3u_(-1) + [Delta]^3u_(-2)) |
&c. /
so that, if (as in SS 1 and 2) each difference is placed opposite the
space between the two quantities of which it is the difference, the
expressions [delta]^2u0, [delta]^4u0, ... denote the differences of
even order in a horizontal line with u0, and [mu][delta]u0,
[mu][delta]^3u0, ... 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 = [mu][delta]u0 - (1/6)[mu][delta]^3u0 + (1/30)[mu][delta]^5u0 \
- (1/140)[mu][delta]^7u0 + ... |
c2 = [delta]^2u0 - (1/12)[delta]^4u0 + (1/90)[delta]^6u0 - (1/560)[delta]^8u0 + ... |
c3 = [mu][delta]^3u0 - (1/4)[mu][delta]^5u0 + (7/120)[mu][delta]^7u0 - ... |
c4 = [delta]^4u0 - (1/6)[delta]^6u0 + (7/240)[delta]^8u0 - ... |
c5 = [mu][delta]^5u0 - (1/3)[mu][delta]^7u0 + ... | (4).
c6 = [delta]^6u0 - (1/4)[delta]^8u0 + ... |
. . |
. . |
. . /
If a calculating machine is used, the formula (2) is most conveniently
written
u_([theta]) = u0 + P1[theta] \
P1 = c1 + (1/2)P2[theta] |
P2 = c2 + (1/3)P3[theta] | (5).
. . |
. . |
. . /
Using [theta] as the multiplicand in each case, the successive
expressions ... P3, P2, P1, u_([theta]) 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 deg. 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 deg.| 2.14451| | 732 | | 16 |
| | | 10153 | | 96 | |
| 66 deg.| 2.24604| | 828 | | 19 |
| | | 10981 | | 115 | |
| 67 deg.| 2.35585| | 943 | | 18 |
+--------+--------+---------+---------+---------+---------+
To find u for x = 66 deg. 23', we have [theta] = 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. [mu][delta]u0. [delta]^2u0. [mu][delta]^3u0. [delta]^4u0.
2.24604 +10567^00 +828^00 +105^50 +19^00
- 17^58 - 1^58
--------- ------- --------- ------
c1 = +10549^42 c2 = +826^42 c3 = +105^50 c4 = +19^00
P1[theta] = +4105^67 (1/2)P2[theta] = + 161^02 (1/3)P3[theta] = + 13^71 (1/8)c4[theta] = + 1^82
-------- --------- ------- -------
u_[theta] = 2.28710 P1 = +10710^44 P2 = +840^13 P3 = +107^32
The value 2.2870967, obtained by retaining the extra figures, is
correct within .7 of .00001 (S 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
[mu][delta]^3u0 in calculating c1 and c3, we ought also to use
[delta]^4u0 for calculating c2 and c4, even though the term due to
[delta]^4u0 would be negligible if [delta]^4u0 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
[+-](1/2)[rho], 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 + Cx^2; 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 + Cx^2 + Dx^3 + ... 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 ...