+----------+------------------------------------------------+
| | Error due to use of Differences up to and |
| | including |
| +------+------+------+------+------+------+------+
| | 1st. | 2nd. | 3rd. | 4th. | 5th. | 6th. | 7th. |
+----------+------+------+------+------+------+------+------+
|.5 / I. | .500 | .625 | .813 |1.086 |1.497 |2.132 |3.147 |
| \ II. | .500 | .625 | .625 | .696 | .696 | .745 | .745 |
|.2 / I. | .500 | .580 | .724 | .960 |1.343 |1.976 |3.042 |
| \ II. | .500 | .580 | .580 | .624 | .624 | .653 | .653 |
|.4 / I. | .500 | .620 | .812 |1.104 |1.553 |2.265 |3.422 |
| \ II. | .500 | .620 | .620 | .688 | .688 | .734 | .734 |
|.6 / I. | .500 | .620 | .788 |1.024 |1.366 |1.886 |2.700 |
| \ II. | .500 | .620 | .620 | .688 | .688 | .734 | .734 |
|.8 / 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 [+-](1/2)[rho]. 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 [theta] = .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 S 6 extended only
from x = 60 deg. to x = 80 deg., we could then complete the table of
differences by making the entries shown in italics below.
_Example 7._
+--------+---------+---------+-----------+-----------+-----------+-----------+-----------+
| x. | tan x. |[delta]u.|[delta]^2u.|[delta]^3u.|[delta]^4u.|[delta]^5u.|[delta]^6u.|
+--------+---------+---------+-----------+-----------+-----------+-----------+-----------+
| | | + | + | + | + | + | + |
| | | _6775_ | | _34_ | | | |
| 60 deg.| 1.73205 | | _425_ | | _9_ | | |
| | | 7200 | | _43_ | | | |
| 61 deg.| 1.80405 | | 468 | | _9_ | | |
| | | 7668 | | 52 | | | |
| 62 deg.| 1.88073 | | 520 | | 9 | | |
| | | 8188 | | 61 | | | |
| 63 deg.| 1.96261 | | 581 | | 10 | | |
| | | 8769 | | 71 | | | |
| 64 deg.| 2.05030 | . | 652 | . | 9 | | |
| . | . | . | . | . | . | . | . |
| . | . | . | . | . | . | . | . |
| . | . | . | . | . | . | . | . |
| 75 deg.| 3.73205 | . | 3409 | . | 187 | . | 18 |
| | | 27873 | | 788 | | 73 | |
| 76 deg.| 4.01078 | | 4197 | | 260 | | 51 |
| | | 32070 | | 1048 | | 124 | |
| 77 deg.| 4.33148 | | 5245 | | 384 | | 64 |
| | | 37315 | | 1432 | | 188 | |
| 78 deg.| 4.70463 | | 6677 | | 572 | | _64_ |
| | | 43992 | | 2004 | | _252_ | |
| 79 deg.| 5.14455 | | 8681 | | _824_ | | _64_ |
| | | 52673 | | _2828_ | | _316_ | |
| 80 deg.| 5.67128 | | _11509_ | | _1140_ | | _64_ |
| | | _64182_ | | _3968_ | | _380_ | |
+--------+---------+---------+-----------+-----------+-----------+-----------+-----------+
For interpolating between x = 60 deg. and x = 61 deg. 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
deg., 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 [xi] the abscissa of this curve which
corresponds to area u, we find the value of [xi] corresponding to each
of the given values of u. Then, tabulating [xi] in terms of x, we have
a table in which, if the auxiliary curve has been well chosen,
differences of [xi] after the first or second are negligible. We can
therefore find [xi], and thence u, for any intermediate value of x.
_Extensions._
12. _Construction of Formulae._--Any difference of u of the rth 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)| . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
| . | . | . | . | . | . |
+----+----+---------+----------+--------------+-----------------+