Page 77 - Standard Handbook Of Petroleum & Natural Gas Engineering

66 Mathematics

Table 1-13
Central Difference Table with Base Line
Old x New x f(x) Af A2f A3f A9 AY
0 -2.5 0
0.5 -2.0 1 2
1 -1.5 2 1 -2
1.5 -1 .o 2 0 0 4
2 -0.5 2 1 2 5 2
3 +0.5 4 6 8 7.5 3
3.5 +1 .o 9 10 12.5 9
4 +1.5 14 18.5 17
4.5 +2.0 27.5 27
5 +2.5 41

and Bessel's formula (half line as base) as

f(x) = f(O)+x(6y1,) = (x2 - 3 (62yo) + X(X2 - +I

2! 3!
( x2 -t)( - 4) .( -+)( x2 - 4)

+ x2 (6 YO)+ x2 (65yo)+ . . .
4! 5!

Interpolation with nonequally spaced data may be accomplished by the use of
Lagrange Polynomials, defined as a set of nth degree polynomials such that each
one, PAX) (j = 0, 1, . . ., n), passes through zero at each of the data points except
one, xt, where k = j. For each polynomial in the set

where if

then