Page 91 - Standard Handbook Of Petroleum & Natural Gas Engineering

80 Mathematics

and the next higher degree approximation as
D, = c, + (C, - Ck_,)/63
The limit suggested by the sequence
T, + S, + C, -+ D, +

is known as Romberg Integration.
If a new notation Ti") is defined, where k is the order of the approximation
(n = 2k) and m is the level of the integration algorithm, then
m = 0 (trapezoidal rule)
T:") = T,
m = 1 (Simpson's rule)
TL') = S,
m = 2 (Cote's rule)
Ti2' = Ck
m=3
Tis) = D
etc.
The generalization of the preceding definitions leads to the Romberg equation

The procedure is to start with the one-panel trapezoidal rule
1
To 3 Tf) = -(b-a)(f, +f,)
2
and then increase the order (k) of the calculation by

k
0 T(0)
1 TlO) T!' )
2 TIO) TO) Tp
3 Tb T! ) Tf) Ti3)
4 TIO) Th' ) Tf) T13) T:")