# Approximations

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.

# Definition

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 approximation as:

# Example

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.

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.

# Application

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 approximation

where the average is assumed to be along the appropriate interval.