Direct Interpolation Equal Intervals I

DIRECT INTERPOLATION (EQUAL INTERVALS) I. Notation and Arrangement.—We have a table showing the values of a quantity u corresponding to values of a quantity x. The values of x proceed by a constant increment h. We denote the values of x by . . . xo, xi, x2, . . ., so that xo-Fmh; and the corresponding values of u are . . . ui, u2. . . . We want to find us, which is the value of u corresponding to xe= x0-1-0h; for preciseness, we can say that 0 lies between —1 and + i.

The first step, usually, is to arrange the u's in a vertical column, followed by columns containing the successive differences. The differences of the u's are the first differences, those of the first differences are the second differences, and so on. These are also called differences of the first, second • • • order.

The differences, like the u's, have to be distinguished by suf fixes. There are three systems, the advancing-difference, the receding-difference and the central-difference. On the advancing difference system, 24 is regarded as the first difference of uo, and is denoted by so that is on the receding difference system, which is less important, is the first difference of On the central-difference system, is a difference corresponding to the interval between and and is therefore denoted by but, so that is Puo; there is also a symbol /./. for the mean of any two adjacent entries in a column, so that and so on. The system is more fully set out in the article CALCULUS OF DIF FERENCES.

2. Fundamental Principle.---The

methods which we are considering are, in general, applicable when the result of taking successive differences is to bring us ultimately to differences of a certain order, say the kth, which are practically constant, so that the differences of the next order, the (k+ i)th, are zero or at any rate are negligible. The fundamental principle then is that we assume that interpolation can be performed as if these latter dif ferences were actually zero. The implication of this assumption will be considered further in §6. The assumption may, in the case of a mathematical table, be made as a matter of course; for the table is constructed so as to be used in this way. In other cases it must be justified by the facts.

3. Interpolation by First Difference.

In Table I. we see that, as x increases, u increases steadily. We therefore use the principle of proportional parts, i.e., we assume that for any increase in x there is a proportional increase in u. Thus we can say that the logarithm of 2.725 is -4346+8 (where "+8" denotes the addition of 8 to the last figure of the tabulated entry, i.e., means and similarly log 2.7225= .435o.

Algebraically, we express the relation between x and u by saying that u is of the form p+qx. It follows that the formula for uo is (1) Graphically, this means that we take the upper boundary of the graph, between and to be a straight line joining the tops of the corresponding ordinates.

To find log 2.724 we should have to add -4 of 16= 6.4. But it would be incorrect to write the result as •43524; for this would imply that we could find the logarithm to five decimal places, which the table does not enable us to do. We can do either of two things. If we only want the best value, to four places, that we can get, we must take .4 of 16 to be 6, so that our answer is -4352. But a better method, if we are going to use log 2.724 for further calculation, is to retain an extra two or three figures in our work. We therefore, for the time being, write the logarithm in the form -4352 40 or any extra figures retained in this way are dropped at the conclusion of our work, the preceding figure being adjusted if necessary. This principle applies through out the whole of our work.

As stated above, the table shows that as x increases u increases steadily. But this does not mean that the second differences of u are absolutely zero; it only means that when we are working to four places of decimals they are negligible. That they do exist, can be seen either by taking more correct values, as in Table II., or by taking values of x at greater intervals, as in Table III.

Similarly we could detect the existence of differences of the third or a higher order.

The sequence of differences, of whatever order, will present some irregularities. These are due to the fact that the tabulated values are necessarily not exact. Thus in Table I. the occurrence of the difference 15 immediately following a series of i6's is due to the entry for 2-74 being slightly too large (as will be seen from Table II.), while the entry for 2.75 is slightly too small. A similar explanation applies to the 57 in Table II.: the entry for 2.73 is

too small, and that for 2.74 is too large.

4. Interpolation with Second Differences.—If we have to use second differences, but do not need third differences, the formula is ue= uo-FeAuo-F20(0— , (2) in which may be either or It is usually best to take the one which lies nearest in the table (see Table II. or Table III.) ; thus if 0 lies between —2 and +1 we should use but if it lies between +1 and +z we should use In most cases in which this formula has to be used, is small: a convenient form for calculation is then 6. Basis of Formulae.—Formulae (I) and (2) may be re garded as particular cases of (4), differences after the first in the one case, and after the second in the other, being negligible. But it is more instructive to consider them separately.

(i.) For first-difference interpolation we take the upper boundary of the graph to be a straight line; and this straight line must obviously pass through the tops of the ordinates representing and u1. Or, conversely, taking Table I. for an illustration, let us suppose that we have the graph drawn with absolute accuracy. Then, if we are unable to measure the magni tude of and and intermediate ordinates to a greater ac curacy than is represented by four significant figures, we are unable to see that it is anything but a straight line. The equa tion to the graph must therefore be taken to be ue = (ii.) Now suppose that (see Table II.) we are able to measure to a degree of accuracy represented by six or seven significant figures. Then the boundary of the graph will appear to be a parabola, its equation being of the form To determine p, q, and r we require three ordinates. If we take these to be ul, and u2, the equation to the graph becomes uo if we take them to be and the equation is ue = uo-I-Muo+20(0— . The connec tion between the equation being of the form and the formula not going beyond second differences is that, if u is of this form, third differences of the u's are o. Table IV. illustrates this for the particular case of u- (iii.) Finally, suppose that differences beyond those of order k are negligible, and let us, for simplicity, take k to be even and = 2g. Then u is a polynomial in x, of degree 2g. If we want to use advancing differences, we find the constants in this poly nomial by passing a curve of degree le through the tops of the If, on the other hand, we pass the curve through the tops of the ordinates u-0-1-1, • • • , uo, ui, • • • , we get the central-difference formula for in terms of uo, iuOuo, Puo, • • •,5"uo.

(iv.) An important fact is that, viewed algebraically, there is no essential difference between the advancing-difference formula and the central-difference formula. If, as in (iii.), we pass a curve of degree 2g through the tops of the ordinate • , the formula for u given by this curve is a difference formula for interpolation between and but it is also an advancing-difference formula for interpolation be tween and and a receding-difference formula for interpola tion between and 7. Central-difference Formulae.—A central-difference f or mula, in the general sense, is one which involves differences, or pairs of differences, lying on or close to a horizontal line in the table. There are several of these, but they are all equivalent either to (6) or to a formula constructed on the same principle but involving Aui, oui , • • Three of these, known to Newton but subsequently discovered by other writers, may be mentioned.

(a) The Newton-Gauss formula is a zigzag formula involving , , • - (b) The Newton-Stirling formula, easily obtained from the above, involves , , • • • , and becomes (6) if the coefficients of 0, • • • are collected.

(c) The Newton-Bessel formula involves , , • • • • If the coefficients of 0 , (0 (0 , • in this formula are collected, we get an alternative formula to (6), for use throughout an interval in the table.

8. Sub-tabulation.--When the values of u have been tabu lated for values of x, proceeding by a difference h, we may want to deduce a table in which the differences of x are h/n, where n is an integer. For the common case of n= io, we can proceed by two steps, first halving the interval, and then dividing the new interval into fifths.

(i.) For halving the interval we have