Calculus and Probability for Actuarial Students

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CHAPTER VIII
FINITE DIFFERENCES. DIVIDED DIFFERENCES

A simple method of interpolation is available, where the inbetween the given terms are unequal, by the method of Divided Diferences.

The application of the method rests upon the assumption, which, as has been shown, is the basis of all theorems for interpolaby means of Finite Differences, that

f (x)

is a rational integral function of x of the nth degree.

On this assumption, it can be shown that

f (x)

can be expressed in the form

A,,+ A, (x-a,)+ A,

(x-a,)(x-a,)+...

+A„(x-a,)(x-a,)...

(x-a„),

where A,,, A,, ... A

a„ a,, ... a

₇

are constants.

In order to apply this formula in practice, it is convenient to introduce a scheme of notation on the following lines, where the symbol of operation is denoted by A^'in order to distinguish it from the ordinary A.

Value of x	Value ofFunction	_A'	*D,2*	A,3
o	_f(o₎	f (^a,) -f (^o)	AI (%) - A'.f (0)	^A'².f (^a,) -'²f (o)
		a_l	a₂	a₃
a,as	f (^al)f (^as)	f (^as) -f (^a,)	*^A'f (a2) -* A'f (al)	A' ².f (^a2) - A'²f (^al)
		a₂—a,f (^a3) -f (^as)	a₃—a,o'.f (^a3) - ^A'.f_(^as)	A' ².f (^a2) - A'²f (^al)
		a₂—a,f (^a3) -f (^as)	a₃—a,o'.f (^a3) - ^A'.f_(^as)	a₄—a,
		_a3— _a2	_a4— _a2
a3	f (a₃)	f (^a4) ^—f(^as)
		a₄—a₃
^a4	f (^a4)

Generally, we have
^0'"f (^ar) — ^A-if _(ar+1    f (at^.)_.
J    ^an+r - a_r4—2

52    FINITE DIFFERENCES
Hence f (a,) = f (0) + a, 0' f (0),
and    , f (a₂) = f (a,) + (a₂ — a,) . f (a,) =.f(0)+a,D.f(0)+(a,—a,){D'.f(0)+a,A2f (⁰)}
=f(0)+a,0^' f (0)+a₂(a₂— a,)6.'²f(0).
By proceeding similarly for further terms, we find that we can write generally
f(x)=f(0)+x0'f(0)+x(x—a,)0'2f(0)
+x(x—a,)(x— a,) 6^'2f (0) + ... (1). This general form can be established by the method of in
By giving appropriate values to a„ a„ ... the ordinary formulas applicable to equal intervals can be at once deduced.

4. The general method of working will be shown more simply by an example.
Example. Find the value of log 4'0180, having given the foldata:

Number	Logarithm
4'0000	•6020600
4^.0127	'6034367
4^.0233	'6045824
4^.0298	•6052835
4^.0369	'6060480

Transposing the origin, we have the following scheme:

x	f (x)	A'		A^'2
		.°°1376' =	.108402	— •000317	_ .0136
0 0000	6020600	'0127		= '0233
		•0011457		— '000223
0^.0127	•6034367	^p106	'108085	O]i1 =	— 0130
0'0233	'6045824	'0007011₌	^.107862	-0136 0186 — _ .0137
		•0065
		•0007645
0^.0298	'6052835	— ₀₀ ₇₁=	107676
0^.0369	'6060480

DIVIDED DIFFERENCES 53

We have to find f (•0180), which is, by the above formula, f (0) + •0180.1 ' f (0) + •0180 x •00530^'2 f (0)
[the further terms will not affect the seventh place of decimals], where f (0) = •6020600, 0'f(0) = •108402, 0'² f (0) = — •0136.
Thus log 4^.0180 = •6020600 + '00195124 — 00000130 = •6040099 to seven decimal places, which agrees exactly with the true result.