Page 93 - Standard Handbook Of Petroleum & Natural Gas Engineering

82 Mathematics

a semi-open formula can be constructed that, in this example, is closed on the
right and open on the left:

In order to eliminate the restriction of evenly spaced points, Gauss Quadrature
algorithms may be constructed. In these algorithms not only the function values
are weighted, but the position of the function evaluations as well as the set of
weight factors are left as parameters to be determined by optimizing the overall
accuracy. If the function is evaluated at points xl,, xI, . . .,xn, the procedure has
2n + 2 parameters to be determined (the xi, and the wi for each xi) and is
required to be accurate for any polynomial of degree N = 2n + 1 or less.
These algorithms are frequently stated in terms of integrals over [-1,1], termed
Gauss-Legendre quadrature, and the general formula then is

(Ilf(x)dx s w,f,, + w,f, + . . . + w,f,

For example, for n = 1,

For each choice of n (the number of points), the wk and the n zeros (6,) of
the nth degree Legendre polynomial must be determined by requiring that the
approximation be exact for polynomials of degree less than 2n + 1. These have
been determined for n = 2 through 95 and an abbreviated table for some n is
given in Table 1-16. The interval - 1 S 6 I 1 is transformed onto the interval
a 5 x S b by calculating for each xk (k = 1, . . ., n)

b+a b-a
'-61,
Xk = -
2 2
and an approximation to the integral is then

b-a
I=- 2 wkf(xk)
2 k=l
Some other typical Gaussian quadrature formulas are:

1
(-131) C hebyshev
(OF-) xce-x Laguerre (c = 0,1, . . .)
(-P) e-x2 Herrnite