Page 88 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 77

Another possible form is the exponential function

F(t) = aeP‘

and although partial differentiation will produce two equations in two unknowns,
they will be nonlinear and cannot be written in matrix form. However, a change
in variable form may produce a model that is linear, for example, for the
preceding equation

In (F) = In (a) + Pt

and if X is defined to be t, Y to be In (F), a, = In (a), and a, = j3, the equation
becomes

Y(x) = a,, + a,X

and linear least squares analysis may be applied.
In order to determine the quality (or the validity) of fit of a particular
function to the data points given, a comparison of the deviation of the curve
from the data to the size of the experimental error can be made. The deviations
(i.e., the scatter off the curve) should be of the same order of magnitude as
the experimental error, so that the quantity “chi-squared’’ is defined as

where y,’ = is the fitted function and y, is the measured value of y at xl, so that
Ay, is the magnitude of the error of y,’. The sum is over n points and if the
number of parameters in the model function is g,, then if 0 (X’) > 0 (n - g),
the approximating function is a poor fit, while if 0 (X2) < 0 (n - g), the function
may be overfit, representing noise [lo].
The simplest form of approximation to a continuous function is some polynomial.
Continuous functions may be approximated in order to provide a “simpler form”
than the original function. Truncated power series representations (such as the
Taylor series) are one class of polynomial approximations.
The Chebyshev polynomials, T,(x) or Tn, which exist on the interval -1 I x I 1,
form the series

To = 1
T, = x

(see Table 1-14 for a more complete list). If these polynomials are inverted,
powers of x are given in terms of T,(x) (Table 1-15). Any finite interval a I y I b
can be mapped onto the interval -1 I x I 1 by the formula

x = (2y - b - a)/(b - a)