Chapter 4 Numerical Differentiation And Integration

(b) Discuss how the formula in (a) can be used to approximate
∫ h
g(t) · t ln(1/t)dt for small h > 0
0
6.(W.G., p.195, 17) Let s be the function defined by
{
(x + 1)
3
if − 1 ≤ x ≤ 0
s(x) =
(1 − x) 3 if 0 ≤ x ≤ 1
(a) With △ denoting the subdivision of [−1, 1] into the two subintervals [−1, 0]
and [0, 1] to what class S k m(△) does the spline s belong ?
(b) Estimate the error of the composite trapezoidal rule applied to ∫ 1
−1 s(x)dx
when [−1, 1] is divided into n subintervals of equal length h = 2/n and n is
even.
(c) What is the error of the composite Simpson’s rule applied to ∫ 1
−1 s(x)dx with
the same subdivision of [−1, 1] as in (b).
(d) What is the error resulting from applying the 2-point Gauss-Legendra rule to
∫ 0
−1 s(x)dx and ∫ 1
0 s(x)dx separately and summing ?
7.(W.G., p.195, 18)(Gauss-Kronrod rule) Let π n (·;w) be the (monic) orthogonal
polynomial of degree n relative to a nonnegative weight function w on [a,b] and
t (n)
k its zeros. Use the theorem proved in class to determine conditions on w k , wk,
∗
t ∗ k for the quadrature rule
∫ b
a
f(t)w(t)dt =
n∑
k=1
w k f ( t (n)
k
to have degree of exactness at least 3n + 1.
Hind: Replace in Theorem 4.2 n by 2n + 1.
) n+1 ∑
+ wkf(t ∗ ∗ k) + E n (f)
k=1
8.(W.G., p.198, 29) Let π n (·;w) be the n-th degree orthogonal polynomial with respect
to the weight function w on [a,b], t 1 , t 2 , ..., t n its n zeros and w 1 , w 2 , ..., w n the n
Gauss weights.
(a) Assuming n > 1 show that the n polynomials π 0 , π 1 , ..., π n−1 are also orthogonal
with respect to the discrete inner product
n∑
(u,v) = w j u(t j )v(t j )
j=1
14