Exact Simpson’s Rule
Prove that Simpson’s Rule is exact (no error) when approximating the definite integral of a linear function and a quadratic function.
Three points integration rule derived using the method of undetermined coefficients.
Suppose that we add a quadrature point at the middle of the interval [a, b],
To avoid algebra,
substitute x = and define h =
so that the integral becomes
Since we have three unknowns,
we can make this formula exact for all quadratic functions;
so, let us use,
e.g., F ≡ 1, F ≡ u and F ≡ u 2 .
F ≡ 1 ⇒ = 2h = w1 + w2 + w3;
F ≡ u ⇒ = 0 = −hw1 + hw3 ⇒ w1 = w3;
F ≡ ⇒ = = w1 + w3 ⇒ w1 = w3 = ; w2 = 2h − w1 − w3 = 2h − h =
Hence we obtain the approximation
which translates into
This is called Simpson’s rule.
As this formula is exact for all quadratic functions,
we expect the error to be of the form KF ′′′(ξ).
However, if we examine F ≡ u 3 ,
= 0 and
Simpson’s rule gives
= 0 ⇒ K ≡ 0.
Simpson’s rule is exact for cubic functions as well.
Consequently, we try an error term of the form OF (iv)(ξ), ξ ∈ (−h, h).
To find the value of K use,
e.g., F(u) ≡ (with F (iv)(ξ) = 4!, independent of ξ).
Simpson’s rule is fifth-order accurate: the error varies as the fifth power of the width of the interval. Simpson’s rule is given by
f (ξ), ξ ∈ (a, b)