Page 90 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 79

under a complicated curve may be approximated by the trapezoidal rule. The area
is subdivided into n panels and

where Axn = (b - a)/n and f, is the value of the function at each xi. If the number
of panels n = 2', an alternate form of the trapezoidal can be given, where

1 "-1
I=T, =-Tk_,+Axkxf(a+iAxk)
2 ,=I
I add
where Ax, = (b - a)/2', To = 1/2(fa + fJ(b - a), and the equation for T, is
repeatedly applied for k = 1,2, . . . until sufficient accuracy has been obtained.
If the function f(x) is approximated by parabolas, Simpson's Rule is obtained,
by which (the number of panels n being even)
I "-1 n-2 1
r
I=S, =-Ax f,+4xf,+2xfL+f,
3 ;:id I eYe"
i=2

where E is the dominant error term involving the fourth derivative off, so that
it is impractical to attempt to provide error correction by approximating this
term. Instead Simpson's rule with end correction (sixth order rather than fourth
order) may be applied where

+Ax[f'(a)-f'(b)]

The original Simpson's formula without end correction may be generalized in a
similar way as the trapezoidal formula for n = 2' panels, using A3 = (b - a)/2, and
increasing k until sufficient accuracy is achieved, where
1

n-2
i=l i=2
i odd i even
For the next higher level of integration algorithm, f(x) over segments of [a,b]
can be approximated by a cubic and if this kth order result is C,, then Cote's
rule can be given as

= 'k k + ( -
'