Hermite Interpolating Polynomials and Gauss-Legendre Quadrature

Integrating L(x) over the interval a ≤ x ≤ b yields the trapazoidal rule
b
a
f(x) dx ≈
b
b
f(a) (x − b) dx/(a − b) + f(b) (x − a) dx/(b − a)
a
a
= f(a) ·
b − a
2
+ f(b) · b − a
2 ,
Here Q = 2, the quadrature points are x1 = a, x2 = b, and the quadrature weights are
w1 = w2 = b−a
2 .
Error Analysis for Lagrange Quadrature.
If L(x) is the polynomial of degree Q − 1 which interpolates f(x) at {xi} Q
i=1 and if
f ∈ C Q [a, b], then it can be shown that the pointwise approximation error is given by
where
f(x) − L(x) = 1 d
Q!
Qf dxQ (ξx) π(x), for some ξx ∈ (a, b), (7)
Q
π(x) ≡ (x − xk) = (x − x1)(x − x2) · · · (x − xQ). (8)
k=1
By integrating (7) we obtain the quadrature error bounds
b
Q
f(x)dx − f(xq)wq
a
q=0
=
1
b
d
Q! a
Qf dxQ (ξx)
≤
π(x) dx
1
Q! max
d
a≤ξ≤b
Q ≤
f b
(ξ) |π(x)| dx
dxQ a
(9)
1
Q! max
d
a≤ξ≤b
Q
f
(ξ)
dxQ max
a≤x≤b |π(x)|
b
dx
a
≤ const (b − a) Q+1 . (10)
Defining the interval length h = b − a, we obtain O(h Q+1 ) accuracy. The quadrature rule is
exact (i.e., there is no quadrature error) for all polynomials of degree ≤ Q − 1 because the
Qth derivative of these polynomials vanishes.
Continuation of Example 1: For the trapazoidal rule, Q = 2, so the method is exact for
polynomials of degree ≤ 1 and the accuracy is O(h 3 ) provided the function being integrated
is twice continuously differentiable.
Exercise 1. Derive Simpson’s rule, which is based on interpolation at points a, (a + b)/2,
and b, and show that the accuracy is O(h 4 ), where h = b − a.
Hermite Interpolation.
Given a differentiable function f defined on discrete points {x1, . . . , xQ}, the Hermite
interpolating polynomial is the unique polynomial H of degree 2Q − 1 that interpolates f
and its derivative,
H(xi) = f(xi),
dH
dx (xi) = df
dx (xi), i = 1, . . . , Q. (11)
2