Simpson’s Rule
January 31, 2006
1 Simpson’s rule
Simpson’s rule is a numerical integration technique the belongs to the family of
Newton-Cotes formulas. Observe the following pattern:
1. The Left End rule integrates constant functions exactly. It converges to
thetruesolutionas1/N . The stencil use fi.
2. The Trapezoid rule integrates linear functions exactly. In converges to
thetruesolutionattherateof1/N 2. The stencil uses fiand fi+1 .
This pattern can extended to any order n.
3. The n-th order Newton-Cotes formula integrates polynomials up to order
nexactly. It converges at least as fast as 1/N n+1 — sometimes faster as we
shall see. (So perhaps I should call it the n+1order formula, but let’s stick to
"n-th".) The stencil uses fi,.., fi+n.
Secon order Newton-Cotes formula is called Simpson’s Rule.
Here’s a good way to derive Simpson’s formula. Formally, we should consider
thenodepointsxi,xi+1 =xi+h,andxi+2 =xi+2hand the corresponding
values of the function fi,fi+1,andfi+2. But to simplify the algebra, let us
instead consider x=0,1,2and label the corresponding values of the function
f1,f2,andf3. The stencil looks like this
c1f1+c2f2+c3f3
The coefficients c1,c2,andc3are determined from the conditions that the above
stencil inegrates 1. constant, 2. linear, and 3. quadratic functions exactly:
f(x)=1 : 1c1+1c2+1c3=R2
01dx =2
f(x)=x:0c1+1c2+2c3=R2
0xdx =2
f(x)=x2:0c1+1
2c2+2
2c3=R2
0x2dx =8
3
We get the following system
⎡
⎣
111
012
014
⎤
⎦⎡
⎣
c1
c2
c3
⎤
⎦=⎡
⎣
2
2
8
3
⎤
⎦
1
