]> 25.2 Simpson's Rule

## 25.2 Simpson's Rule

In the notation of the last section the actual area under the function $f$ in the interval between $− d 2$ and $d 2$ will be

$f ( 0 ) d + b d 3 12 + e d 5 80 + ⋯$

The trapezoid rule that we have described, on the other hand, gives the following proposed answer for this area

$f ( 0 ) d + b d 3 4 + e d 5 16 + ⋯$

while the "midpoint rule" approximates the area as $f ( 0 ) d$ .

This means that if we mix two parts of the midpoint rule and one part of the trapezoid rule, we will get the quadratic $b$ term exactly right, and the leading term in the error will come from the $e$ term which will be on the order of $d 4$ .

The rule for approsimating integrals just described is called Simpson's rule, and it takes the following form in the interval between $− d 2$ and $d 2$

$( f ( − d 2 ) + 4 f ( 0 ) + f ( d 2 ) ) d 6$

Notice that in this formula $f$ is evaluated at intervals $d 2$ apart . If we give the parameter $d 2$ a new name here, calling it $h$ , the Simpson's rule formula in terms of $h$ becomes

$( f ( − h ) + 4 f ( 0 ) + f ( h ) ) h 3$

and it represents an approximation to the area under the curve defined by $f$ in the interval from $− h$ to $h$ ; an approximation whose error is of fourth order in $h$ .

It is worthwhile noticing what these various rules look like when applied to a number of small subintervals in a row.

The trapezoid rule gives equal weight $d$ to evaluations at all intermediate points, since each is the left end and the right end of one subinterval and it gets $d 2$ weight from each end.

The end two evaluations on the other hand are ends of only one subinterval each, and these get weight $d 2$ .

The midpoint rule gives equal weight to the odd numbered evaluations of $d$ or $2 h$ .

Simpson's rule gives weights that form the pattern $14242 … 41$ multiplied by $h 3$ , since the midpoints get weight $2 3$ and the trapezoid rule divided by 3 accounts for the rest.

 endpoint $a$ midpoint first interval $a + h$ end first interval $a + 2 h$ midpoint second interval $a + 3 h$ end second interval $a + 4 h$ etc. Trapezoid rule terms $h f ( a )$ 0 $2 h f ( a + 2 h )$ 0 $2 h f ( a + 4 h )$ etc. Midpoint rule terms 0 $2 h f ( a + h )$ 0 $2 h f ( a + 3 h )$ 0 etc. Simpson's rule terms $h f ( a ) 3$ $4 h f ( a + h ) 3$ $2 h f ( a + 2 h ) 3$ $4 h f ( a + 3 h ) 3$ $2 h f ( a + 4 h ) 3$ etc.

Notice that the leading term in the power series for $f$ that produces an error in Simpson's rule is the $e x 4$ term, and that produces an error which goes down by a factor of 16 if we divide our intervals in half.