In our discussion of approximating the definite integral
we assumed that f(x) is continuous on [a,b]. However, the bound
on the error of the Trapezoidal method of approximation, given in
Theorem 1.1, requires that 
exists on
[a,b], and for Simpson's Rule, the bound for the error, given in
Theorem 1.2, requires that 
exists on [a,b].
Compared to continuity on [a,b], the existence of a derivative
on [a,b] is a rather strong requirement. In fact, a function that
is randomly selected from the set of all functions that are continuous
on [a,b] will have probability 0 that it also has a derivative on
[a,b]. Thus we would not expect to encounter any function that has
a derivative on a given interval (although we do so because the
functions we normally deal with are not selected at random). Because
of this, we need to find a way of approximating a definite integral
to a given degree of accuracy without any foreknowledge of the error
involved in the approximation.
When an appropriate number of derivatives can be found, then we have seen that the estimate of the error has the form
for each the Trapezoidal Rule and Simpson's Rule. Here k is a constant
which may vary with the function f(x), the interval [a,b], and the
approximation method, and r is a real number: r = 2 for the
Trapezoidal Rule and r = 4 for Simpson's Rule. If we denote 
to be either the sum 
from the Trapezoidal Rule or the sum 
from Simpson's Rule, then the expression for the error (7)
can be rewritten as
Now let a be a positive integer, and replace n with a n in (8). Then
so that
Note that this can then be compared with (8),
from which we obtain
Let us denote
Then 
is an alternative estimate of the definite integral, its
accuracy depending on the assumption of (7) for the estimate
of the error. It requires one to find both and 
, sums
that require finding n + 1 and a n + 1 values of f(x),
respectively, although the a n + 1 values of f(x) involved in
evaluating include the n + 1 values that make up . For
the simplest case, we take a = 2 to obtain
This is known as Richardson's extrapolation formula, and it is
generally a more accurate approximation to the definite integral
than is 
, depending upon the validity of (7).
We can use the approximation to estimate the error in
:
From (9), we obtain
Therefore, for a = 2 we have
This is known as Richardson's error estimate. Note that this is an
estimate of the error in , and not for the error in 
.
Example 15 In Example 4 we used the
Trapezoidal Rule to find the approximations , for 
,
of the definite integral
In particular, we have 
and 
,
with 
. We can incorporate this information to find
the Richardson's approximation 
with regard to the Trapezoidal
Rule. For the Trapezoidal Rule, the value r = 2. The Richardson's
approximation is then given by
This is a much better approximation of 
.
Richardson's error estimate gives
Example 16 In Example 12 we incorporated
Simpson's Rule with n = 4 to approximate the value of 
.
Recall that
and we obtained the value 
, with error
.
We can use the data given in Example 12 to apply Simpson's Rule with n = 2 to obtain
This information can be utilized to find the Richardson's
approximation 
with regard to Simpson's Rule. For Simpson's Rule,
the value r = 4. The Richardson's extrapolation value is then
given by
while Richardson's error estimate gives
Example 17 Let us apply the Richardson's extrapolation formula to the Trapezoidal approximation of the definite integral
Recall that in this case our value r = 2. We form a table to
illustrate the rate of convergence of 
for n = 1, 2, 4, 8, and 16. We also show the Richardson's estimate
of the error 
for the Trapezoidal approximation,
, along with the actual error
for the Richardson's method.
Example 18 We shall now apply the Richardson's extrapolation formula to Simpson's approximation of the definite integral
For Simpson's Rule, r = 4. Just as for Example 17, we
illustrate the rate of convergence of 
for n = 2, 4, and 8, along with the Richardson's estimate of the
error for the approximation due to Simpson's Rule,
, and the actual error
for the Richardson's method.
Example 19 Let us apply the Richardson's extrapolation formula to the Trapezoidal approximation of the definite integral
Once again our value r = 2. We form a table of values of
for n = 1, 2, 4, 8, 16, 32, 64, 128, 256,
and 512, illustrating the Richardson's estimate of the error
for the Trapezoidal approximation along with the actual error
for the Richardson's method.
