(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 ±½ρ 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 uθ, as calculated from the first r terms of the series in (1), to ½ρ is the sum of the first r terms of the series
1 + o + θ (1 − θ) + θ (1 − θ) (2 − θ) + 7⁄12θ (1 − θ) (2 − θ) (3 − θ) +
¼θ (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 δ2pu0 inclusive in (4) or up to δ2p u1 and o2p u0 in (9) (i.e. in effect, up to δ2p+1 u1/2) 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 | 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 ±½ρ. 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.