Page 92 - Standard Handbook Of Petroleum & Natural Gas Engineering

Numerical Methods 81

Increasing accuracy may be obtained by stepping down or across the table,
while the most accurate approximation will be found on the lower vertex of the
diagonal. The Romberg procedure is terminated when the values along the
diagonal no longer change significantly, Le., when the relative convergence
criterion is less than some predetermined E. In higher-level approximations,
subtraction of like numbers occurs and the potential for round-off error
increases. In order to provide a means of detecting this problem, a value is defined

and since R:") should approach 1 as a limit, a satisfactory criterion of error is
if Rim) begins to differ significantly from 1.
An improper integral has one or more of the following qualities [38]:

1. Its integrand goes to finite limiting values at finite upper and lower limits,
but cannot be integrated right on one or both of these limits.
2. Its upper limit equals -, or its lower limit equals -00.
3. It has an integrable singularity at (a) either limit, (b) a known place
between its limits, or (c) an unknown place between its limits.

In the case of 3b, Gaussian quadrature can be used, choosing the weighting
function to remove the singularities from the desired integral. A variable step
size differential equation integration routine [38, Chapter 151 produces the only
practicable solution to 3c.
Improper integrals of the other types whose problems involve both limits are
handled by open formulas that do not require the integrand to be evaluated at
its endpoints. One such formula, the extended midpoint rule, is accurate to the
same order as the extended trapezoidal rule and is used when the limits of
integration are located halfway between tabulated abscissas:

Semi-open formulas are used when the problem exists at only one limit. At
the closed end of the integration, the weights from the standard closed-type
formulas are used and at the open end, the weights from open formulas are
used. (Weights for closed and open formulas of various orders of error may be
found in standard numerical methods texts.) Given a closed extended trapezoidal
rule of one order higher than the preceding formula, i.e.,

13
+f,)+-(f, +fn-,)+ Cfi
12 i=2

and the open extended formula of the same order of accuracy

7
+f,_l)+-(f,+f,-2)+Cfi
12
i=4