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16 Some Fundamental Numerical Methods
where T,; indicates a degree-l (linear) extrapolation. To extrapolate trapezoid
widths t~ zero, set h equal to zero in (2.24) and simplify the result by using
(2.21):
T .= 3To,i+I+ T O,i+I-To,i
1,1 22_ 1
The linear extrapolation in (2.25) is rewritten in a form for later use:
T· -T
T -T + 0,1+1 OJ
I,i- O,i+1 2 2 -1
(2.25)
(2.26)
This has an error from the true integral value proportional to h;, and is thus a
more accurate estimate than the individual trapezoidal estimates,
Again note that the linear extrapolation is versus h 2 and not simply versus
h: the reasons for this choice and the following general formula are explained
by McCalla (1967) and, in more detail, by Bauer et al. (1963). Briefly, the
trapezoid rule estimate may be expanded as a finite Taylor series with a
remainder in the variable h, the true value being the constant term. Since the
error is of the order h', the remainder term is proportional to f"Wh', with ~
somewhere on the interval h. McCalla (1967, p. 289) argues that f"(~) should
be about equal over h; and its subdivided intervals h;+ \ = hJ2. This leads
directly to (2.25), thus justifying the linear extrapolation to zero of successive
trapezoidal estimates. in the variable hI
The scheme is simply this: one, two, and then four trapezoids in the range
of integration enable two linear extrapolations, as described. The two extrapolated
results can then be extrapolated again for a new estimate. McCalla
(\967) shows that repeating linear extrapolations once is equivalent to quadratic
(second-degree) extrapolation. The concept of estimating performance
at a limit, here at h=O, is known as Richardson extrapolation; it will appear
again in Chapter Five.
Using this rationale, a general expression for Romberg integration IS
obtained from (2.26):
k +j = i, (2.27)
{ j=i-I, i-2, ..., I,D.
Index k is the order of extrapolation, and there are i bisections of the
integration interval (b - a). The compactness of (2.27) makes it ideal for
programming. The table in Figure 2.5 illustrates the Romberg extrapolation
process. The step length, or trapezoid width, is shown in the left-hand column.
The brackets indicate pairs of lower-order estimates that produce an estimate
of next higher order by linear extrapolation to step length h = O. Then the
better estimates are similarly paired for extrapolation. Accuracy is ultimately
limited, because the estimates are the result of the subtraction of two numbers.
Eventually, the significant digits will diminish on a finite word-length computer,
and the process should be terminated.