Page 71 - Standard Handbook Of Petroleum & Natural Gas Engineering

60 Mathematics

NUMERICAL METHODS
See References 1 and 9-22 for additional information.

Expansion in Series
If the value of a function f(x) can be expressed in the region close to x = a,
and if all derivatives of f(x) near a exist and are finite, then by the infinite
power series

(x - a)' (x - a)"
f(x)=f(a)+(x-a)f'(a)+- f"(a)+ . . . +- f"(a)+ . . .
2! n!
and f(x) is analytic near x = a. The preceding power series is called the Taylor
series expansion of f(x) near x = a. If for some value of x as [x - a] is increased,
then the series is no longer convergent, then that value of x is outside the radius
of convergence of the series.
The error due to truncation of the series is partially due to [x - a] and
partially due to the number of terms (n) to which the series is taken. The
quantities [x - a] and n can be controlled and the truncation error is said to
be of the order of (x - a)"+I or O(x - a)"".

Finite Difference Calculus

In the finite difference calculus, the fundamental rules of ordinary calculus
are employed, but Ax is treated as a small quantity, rather than infinitesimal.
Given a function f(x) which is analytic (i.e., can be expanded in a Taylor series)
in the region of a point x, where h = Ax, if f(x + h) is expanded about x, f'(x)
can be defined at x = xi as

f'(xi) = f; = (f,+, - fi)/h + O(h)
The first forward difference of f at xi may be written as

Afi = fi+l - fi

and then

f'(x) = (Af,)/h + O(h)
The first backward difference of f at xi is

Vf, = fi - fi-l

and f'(x) may also be written as

f'(x) = (Vfi)/h + O(h)