Chapter 4 Numerical Differentiation And Integration

13.(W.G., p.202, 47)
∫ b
n∑
E(f) = f(x)dx − w k f(x k )
a
k=1
is the error of a quadrature rule having degree of exactness d. Show that none of
the Peano kernels K 0 , K 1 , ..., K d−1 can be definite.
14.(W.G., p.203, 50)
(a) Use the method of undetermined coefficients to construct a quadrature formula
of the type
∫ 1
0
f(x)dx = af(0) + bf(1) + cf ′′ (γ) + E(f)
having maximum degree of exactness d the variables being a, b, c and γ.
(b) Show that the Peano kernel K d for the formula obtained in (a) is definite and
hence express the remainder in the form
E(f) = e d+1 f (d+1) (ξ), 0 < ξ < 1.
15. Let f ∈ C 4 derive the three-point midpoint numerical differentiation formula for
f ′′ (c) with truncation error.
f ′′ (c) =
f(c + h) − 2f(c) + f(c − h)
h 2
− h2
12 f(4) (ξ)
16