6.2: Composite Rules

We now use our elementary formulas obtained for (6.2) to perform the integral given by (6.1)

6.2.1. Trapezoidal rule

We suppose that the function 𝑓⁡(𝑥) is known at the 𝑛 +1 points labeled as 𝑥0,𝑥1,…,𝑥𝑛, with the endpoints given by 𝑥0 =𝑎 and 𝑥𝑛 =𝑏. Define

𝑓𝑖=𝑓⁡(𝑥𝑖),ℎ𝑖=𝑥𝑖+1−𝑥𝑖

Then the integral of (6.1) may be decomposed as

∫𝑏𝑎𝑓⁡(𝑥)𝑑𝑥=𝑛−1∑𝑖=0∫𝑥𝑖+1𝑥𝑖𝑓⁡(𝑥)𝑑𝑥=𝑛−1∑𝑖=0∫ℎ𝑖0𝑓⁡(𝑥𝑖+𝑠)𝑑𝑠

where the last equality arises from the change-of-variables 𝑠 =𝑥 −𝑥𝑖. Applying the trapezoidal rule to the integral, we have

∫𝑏𝑎𝑓⁡(𝑥)𝑑𝑥=𝑛−1∑𝑖=0ℎ𝑖2⁢(𝑓𝑖+𝑓𝑖+1)

If the points are not evenly spaced, say because the data are experimental values, then the ℎ𝑖 may differ for each value of 𝑖 and (6.6) is to be used directly.

However, if the points are evenly spaced, say because 𝑓⁡(𝑥) can be computed, we have ℎ𝑖 =ℎ, independent of 𝑖. We can then define

𝑥𝑖=𝑎+𝑖⁢ℎ,𝑖=0,1,…,𝑛

and since the end point 𝑏 satisfies 𝑏 =𝑎 +𝑛⁢ℎ, we have

ℎ=𝑏−𝑎𝑛

The composite trapezoidal rule for evenly space points then becomes

∫𝑏𝑎𝑓⁡(𝑥)𝑑𝑥=ℎ2⁢𝑛−1∑𝑖=0(𝑓𝑖+𝑓𝑖+1)=ℎ2⁢(𝑓0+2⁢𝑓1+⋯+2⁢𝑓𝑛−1+𝑓𝑛)

The first and last terms have a multiple of one; all other terms have a multiple of two; and the entire sum is multiplied by ℎ⁡/2.

6.2.2. Simpson’s rule

We here consider the composite Simpson’s rule for evenly space points. We apply Simpson’s rule over intervals of 2⁢ℎ, starting from 𝑎 and ending at 𝑏 :

∫𝑏𝑎𝑓⁡(𝑥)𝑑𝑥=ℎ3⁢(𝑓0+4⁢𝑓1+𝑓2)+ℎ3⁢(𝑓2+4⁢𝑓3+𝑓4)+…+ℎ3⁢(𝑓𝑛−2+4⁢𝑓𝑛−1+𝑓𝑛)

Note that 𝑛 must be even for this scheme to work. Combining terms, we have

∫𝑏𝑎𝑓⁡(𝑥)𝑑𝑥=ℎ3⁢(𝑓0+4⁢𝑓1+2⁢𝑓2+4⁢𝑓3+2⁢𝑓4+⋯+4⁢𝑓𝑛−1+𝑓𝑛)

The first and last terms have a multiple of one; the even indexed terms have a multiple of 2; the odd indexed terms have a multiple of 4; and the entire sum is multiplied by ℎ⁡/3.