CHAPTER II
FINITE DIFFERENCES. DEFINITIONS
- 1.
The subject or calculus of Finite Differences deals with the changes in the values of a function (the dependent variable) arising from finite changes in the value of the independent variable (see Chapter I, 1).
Many questions arise which can be dealt with on systematic lines, but probably the most important problems which require to be solved in actual practice, and with which we are concerned at this stage of the subject, are the summation of series, and the insertion of missing terms in a series of which only certain terms are given.
It will be convenient to proceed in the first place to some eleconceptions and definitions.
- 2.
If we have a series consisting of a number of values of a function, corresponding to equidistant values of the independent variable, and from each term of the series we subtract the algebraic value of the immediately preceding term, we shall obtain a further series of equidistant terms. The process is known as differencing the terms of the series, and the terms of the new series are known as the first differences of the original terms. By repeating the prowith the terms forming the first differences, we shall obtain a further series forming the second differences of the original function, and so on. Thus if we have f (x) for the first term of the series and f (x + h) for the second term, the first difference of f(x) is f (x + h) — f (x) and is designated f (x). The second difference of f (x) is Of (x + h) — f (x) and is designated AQ f (x). This may be set out as in the following scheme :
| Function ' |
First Differences |
Second Differences |
Third Differences |
| f(x) |
f(x+h)—f(x) |
f(x+2h)—2f(x+h)+f(x) |
f(x+3h)—3f(x+2h)+3f(x+h)-.f(x) |
| f(x+h)f(x+2h)f(x+3h)f (x+ 4h) |
f(x+2h)—f(x+h)f(x+3h)—f(x+2h)f(x+4h)—f(x+31e) |
f(x+3h)—2f(x+2h)+.f (x+h)f(x+4h)—2f(x+3h)+f (x+2h) |
f(x+4h)—3f(x+3h)+3f (x+2h) — f (x+h) |
10 FINITE DIFFERENCES
The first term of the series is known as the leading term and the terms in the top line of differences are known as the leading difof the series.
It must be clearly understood at the outset that A is merely a symbol representing the operation of differencing f (x) once; it is in no sense a coefficient by which f (x) is multiplied. This point is dealt with again in § 5.
- 3.
An examination of the character of the series which ultimately results from the process of differencing repeatedly, leads to the development of certain important theorems. Before proceeding further, it will be helpful to give a practical example.
Example 1. Obtain the differences of the series given by f (x) = x3, where x has all integral values from 1 to 6.
|
x |
f (x) |
FirstDifferences |
SecondDifferences |
ThirdDifferences |
FourthDifferences |
|
1 |
1 |
7 |
12 |
6 |
0 |
|
2 |
8 |
19 |
18 |
6 |
0 |
|
3 |
27 |
37 |
24 |
6 |
|
|
4 |
64 |
61 |
30 |
|
|
|
5 |
125 |
91 |
|
|
|
|
6 |
216 |
|
|
|
|
It will be observed in the above example that the fourth and, therefore, all higher differences are zero ; it will be seen later that this would equally have been the case had more terms of the series been taken. We can therefore construct all the remaining terms of the series by a process of continuous addition.
- 4.
Although most functions with which the actuary has to deal are not of the simple character of that shown above, yet it will usually be found that the differences of the function for which further values are required tend to the value zero and are susto treatment by methods which will be developed subse
The student should obtain confirmation of this fact and insight into the character of certain series by taking out the differences of tabulated functions such as logarithms, annuity-values, etc.
DEFINITIONS 11
- 5.Before proceeding to the consideration of the various problems which arise, it is necessary to develop certain fundamental formulas.
In g 2 0 has already been defined as the symbol of the operation by means of which the value of f (x + h) — f (x) is obtained. Similarly, it is customary to use the symbol E as representing the operation by which the value of f (x) is changed to the value f (x + It), so that
Ef(x)=f(x+h)=f(x)+D.f(x).
It must be carefully remembered that these symbols represent operations only and must be interpreted accordingly. Thus E2x2 is clearly not the equivalent of (Ex)2; the former expresses the result of operating twice upon the function x2 in the manner indicated above, giving a value (x + 2h)2, whereas in the latter case the operation is applied once to the function x and the resulting term (x + h) is squared.
6.
If these symbols are found to obey the ordinary algebraical laws, they can be dealt with algebraically provided always that the results are interpreted symbolically in relation to the function which is the subject of the operation. This principle is known as that of Separation of Symbols or Calculus of Operations.
The algebraic laws referred to above comprise :
- (1)The Law of Distribution.
The Law of Indices.The Law of Commutation.Taking these laws in succession :
- (1)The symbol 0 is distributive in its operation, for
0[f (x)+ f2(x)+f3 (x) +...] _ [I; (x+h) + f2 (x +h) +f3 (x+h) +...]
- [ f 1 (x) + f 2 (x) +A(x) + ... ]=[fi(x+h)—fi(x)]+[f2(x+h)—f2(x)]+[f3(x+h) f3 (x)] +...=0.f1(x)+O.f2(x)+Ofs(x)+....Similarly the symbol E is distributive, forL
E[f1(x)+f2(x)+f3(x)+...]=[f1(x+h)+f2(x+h)+f3(x+h)+...]
=Ef1(x)+Ef2(x)+E.f3(x)+....
- The symbol 0 obeys the law of indices, for in the case of positive integers the symbol Am f (x) represents ,the operation, repeated m times, of differencing f (x).
12 FINITE DIFFERENCES
Thus At. f (x) = (AAA ... m times) f (x),
f (x) = (AAA ... ii. times) (AAA ... m times) f (x) =(AAA ... (m+n) times) f (x)
=An+mf(x)
Similarly it may be shown that the symbol E obeys the law of indices.
(3) The symbol A is commutative in its operation as regards constants, for, if c be a constant,
A[cf(x)]=cf(x+h)—cf(x)
c[f(x+h)—f(x)]
=cAf(x).
The like result can be deduced as regards E.
7. It follows that, since Ef(x)=(1+A)f(x),
therefore E = 1+A
and A=E—1.
The two operators are thus connected by a simple relation, which will be found later to lead to important results.
- As an example of the manner in which the relationship between the operations represented by E and A can be utilised in the solution of problems, we may take the following:
Example 2. Prove thatf(0)+xf(1)+22' f(2)+3 f(3)+...= ex [ f (0) + xAf (0) +f (0) + ...] . Sincef (1) = Ef (o) = (1 +A).f(o); f(2)=E2.f(0)=(1+A), f(0),etc.,we have f(0)+xf(1)+22f(2)+31 f(3)+..._ [1+x(1+A)+2i(1+A)'+...] f(o)= [e"(' +')].f (o)= ex [exo] .f (o)A!2=ex [1+x0+x2i+...if(o)=ex [f(0)+xAf(0)+22.0'f(0)+...~.