Mathematical Tools for Physics

11—Numerical Analysis 326

This is known as Simpson’s rule.

Simpson’s Rule
Before applying this last result, I’ll go back and derive it in a more systematic way, putting it into the form you’ll
see most often.
Integrate Taylor’s expansion over a symmetric domain to simplify the algebra:
∫h

−h

dxf(x) = 2hf(0) +

2

6

h^3 f′′(0) +

2

120

h^5 f′′′′(0) +···.

I’ll try to approximate this by a three point formulaα(−h) +βf(0) +γf(h)whereα,β, andγ, are unknown.
Because of the symmetry of the problem, you can anticipate thatα=γ, but let that go for now and it will come
out of the algebra.

αf(−h) +βf(0) +γf(h) =

α

[

f(0)−hf′(0) +

1

2

h^2 f′′(0)−

1

6

h^3 f′′′(0) +

1

24

h^4 f′′′′(0) +···

]

+βf(0)

+γ

[

f(0) +hf′(0) +

1

2

h^2 f′′(0) +

1

6

h^3 f′′′(0) +

1

24

h^4 f′′′′(0) +···

]

You now determine the three constants by requiring that the two series for the same integral agree to as
high an order as is possible for any f.

2 h=α+β+γ

0 =−αh+γh

1

3

h^3 =

1

2

(α+γ)h^2

=⇒ α=γ=h/ 3 , β= 4h/ 3

and so,

∫h

−h

dxf(x)≈

h

3

[

f(−h) + 4f(0) +f(h)

]

. (21)