If x1, x2, ..., xn
are all approximations of a value x, then the average
is also an approximation of x. Less apparent is that, given the same approximations, the sum
Is also an approximation of x, so long as the denominator
is not zero. The first approximation is a special case of the
second where a1 = a2 = ⋅⋅⋅ = an = 1.
One example of this will be Romberg integration where we find
two approximations R0,0 and R1,0 and define
a better approximation by evaluating (4 R0,0 − R1,0)/3.
In this examle, a1 = 4 and a2 = -1.
The average value of an integrable function f(x) on an interval [a, b] is
given by the formula
If we partition the interval [a, b] into n equal subintervals,
we may therefore approximate this integral by the Riemann sum
where h = (b - a)/n and
xi* ∈ [a + (i - 1)h, a + ih].
Now, by noting that b - a = nh, it we may rewrite this
Consider the average value function f(x) = e-xsin(x) on the
interval [1.2, 1.8]. Using 10 decimal digits of precision, this is equal to 0.2220376327.
Next, partition the interval into two, three, and six subintervals, and in each case
choose the midpoint of each interval to represent xi*:
1/2⋅(f(1.35) + f(1.65)) = 0.2221973076
1/3⋅(f(1.3) + f(1.5) + f(1.7)) = 0.2221107594
1/6⋅(f(1.25) + f(1.35) + f(1.45) + f(1.55) + f(1.65) + f(1.75)) = 0.2220562383
A plot of the function and the various partitions is shown in
Figure 1. The function f(x) = e-xsin(x) on [1.2, 1.8].
Hence, we see that even by averaging the values of just two points on
the interval that we get a reasonable approximation to the mean value.
In many cases when we are doing error analysis, we will end up
with a sum of the mth derivative evaluated at a number of points along an interval. To
simplify our writing of the error term, we will usually use the
where the average is assumed to be along the appropriate interval.
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.