Page 79 - Standard Handbook Of Petroleum & Natural Gas Engineering

68 Mathematics

as long as there are sufficient powers of x in the denominator to cancel any
nearby poles. Bulirsch and Stoer [16] give a Neville-type algorithm that performs
rational function extrapolation on tabulated data

starting with Ri = yi and returning an estimate of error, calculated by C and D
in a manner analogous with Neville’s algorithm for polynomial approximation.
In a high-order polynomial, the highly inflected character of the function
can more accurately be reproduced by the cubic spline function. Given a series
of xi(i = 0, 1, . . ., n) and corresponding f(xi), consider that for two arbitrary
and adjacent points xi and xi+j, the cubic fitting these points is

Fi(x) = a, + aix + a2x2 + a,x3
(x, I x I Xi+J

The approximating cubic spline function g(x) for the region (xo I I xn) is
x
constructed by matching the first and second derivatives (slope and curvature)
of Fi(x) to those of Fi-l(x), with special treatment (outlined below) at the end
points, so that g(x) is the set of cubics Fi(x), i = 0, 1, 2, . . ,, n - 1, and the
second derivative g“(x) is continuous over the region. The second derivative
varies linearly over [x,,xn] and at any x (x, 5 x I x,+,)

x - x,
g”( x) = g”( x, ) + - [g”(xi+l) - g”(x, )I
X~+I - Xi
Integrating twice and setting g(xi) = f(xi) and g(xCl) = f(xi+l), then using the
derivative matching conditions

Fi(xi) = F;+,(xi) and F:(xi) = FY-l(xi)
and applying the condition for i = [l, n - 11 finally yields a set of linear
simultaneous equations of the form