5. Numerical integration
5.1 Manual method
5.2 Trapezium rule
The approximation represented by
While the error for each step is
5.3 Mid-point rule
By evaluating the function
with the graphical interpretation shown in figure 15 . The difference between the Trapezium Rule and Mid-point Rule is greatly diminished in their compound forms. Comparison of equations ( 47 ) and ( 49 ) show the only difference is in the phase relationship between the points used and the domain, plus how the first and last intervals are calculated.
There are two further advantages of the Mid-point Rule over the Trapezium Rule. The first is that is requires one fewer function evaluations for a given number of subintervals, and the second that it can be used more effectively for determining the integral near an integrable singularity. The reasons for this are clear from figure 16 .
5.4 Simpson's rule
Whereas the error in the Trapezium rule was
and the corresponding error
5.5 Quadratic triangulation*
remembering that some increase the area while others decrease it relative to the initial trapezoidal (triangular) estimate. The overall estimate (ignoring linear measurement errors) will be
5.6 Romberg integration
By choosing the weighting factor
Comparison with equation ( 51 ) shows that this formula is precisely that for Simpson's Rule.
This same process may be carried out to higher orders using
A similar process may also be applied to the Compound Simpson's Rule.
5.7 Gauss quadrature
In general it is possible to choose
Equating the various terms reveals
the solution of which gives the positions stated in equation ( 56 ).
5.8 Example of numerical integration
which may be integrated numerically using any of the methods described in the previous sections. Table 2 gives the error in the numerical estimates for the Trapezium Rule, Midpoint Rule, Simpson's Rule and Gauss Quadrature. The results are presented in terms of the number of function evaluations required. The calculations were performed in double precision.
|No. intervals||Trapezium Rule||Midpoint Rule||Simpson's Rule||Gauss Quadrature|
5.8.1 Program for numerical integration*
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Stuart Dalziel, last page update: 17 February 1998
Так что полной тьмы быть не. Во-вторых, Стратмор гораздо лучше меня знает, что происходит в шифровалке в данный момент. Почему бы тебе не позвонить .