Page 78 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 67

and the linear combination of Pj(x) may be formed

It can be seen that for any xi, p,(xi) = f(xi).
Interpolation of this type may be extremely unreliable toward the center of
the region where the independent variable is widely spaced. If it is possible to
select the values of x for which values of f(x) will be obtained, the maximum
error can be minimized by the proper choices. In this particular case Chebyshev
polynomials can be computed and interpolated [ll].
Neville’s algorithm constructs the same unique interpolating polynomial and
improves the straightforward Lagrange implementation by the addition of an
error estimate.
If Pi(i = 1, . . .,n) is defined as the value at x of the unique polynomial of
degree zero passing through the point (xi,yi) and Pi. (i = 1, . . ., n - 1, j = 2, . . ., n)
the polynomial of degree one passing through 60th (xi,yi) and (x.,~.), then the
higher-order polynomials may likewise be defined up to PIPB,,,n, which is the value
of the unique interpolating polynomial passing through all n points. A table
may be constructed, e.g., if n = 3

Neville’s algorithm recursively calculates the preceding columns from left to
right as

In addition the differences between the columns may be calculated as

and is equal to the sum of any yi plus a set of C’s and/or D’s that lead to
the rightmost member of the table [22].
Functions with localized strong inflections or poles may be approximated by
rational functions of the general form