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14 Some Fundamental Numerical Methods
f(a)
f(x}
f(a t hl
flbl
I
I
I
------1-----
I
I
------,----1----
~h ,I h4
Figure 2.3.
,+ h
b
x
The trapezoid rule for numerical integration.
methods, such as Gaussian quadrature, based on weighted-sample schemes,
but calculation of the weights consumes time and memory. The latter meth;)ds
pose difficulties in recursive calculation of estimates of increasing order, thus
limiting their use as computer aids.
The Romberg integration method first approximates the integral as the area
of just one trapezoid in the range of integration, then two, continuing for i
evenly spaced trapezoids until a larger i does not change the answer significantly.
The other main feature of the Romberg method is deciding how many
trapezoids are enough. The width of each trapezoidal area starts at h = b - a,
then h/2. The areas found for these values are linearly extrapolated versus h'
to h=O; when the estimate using width h/4 is found, the extrapolation to
h = 0 is quadratic, and this is tested against the linearly extrapolated answer
for convergence. There is a sequence of estimates for decreasing trapezoid
widths and increasing degrees of extrapolation until either convergence or a
state of numerical noise is obtained. The Romberg method is very efficient,
stahle, and especially suitable for digital computing. However, the integrand
must be computed from an equation, as opposed to using measured data.
In the next four sections it will be shown how the formulas for trapezoidal
integration, repeated linear interpolation, and the Romberg recursion are
obtained. A BASIC language program will then be described, and an example
will be considered. Finally, a once-repeated trapezoid rule will be shown to
yield Simpson's rule for integration; this will be used in Chapter Four.
2.3.1. Trapezoida/1ntegration. The integration problem is to find the value
of the integral T given the integrand f(x) and the limits of integration a and b:
T(a,b)= ff(X)dx.
(2.17)
Summing the two trapezoidal areas in Figure 2.3 yields
T
= (f,+f'+h fHh+fh)
h 2 + 2 '
(2.18)