MAT128A: Numerical Analysis, Homework 7

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1. Find a polynomial p of degree 3 such that
p(0) = 0, p(1) = 1, p(2) = 1, and p
0
(0) = 1.
2. (a) Show that the roots of
pN (x) = TN+1(x) − TN−1(x)
are
xj = cos ´ π
N
j
¯
, j = 0, 1, . . . , N.
(b) Use (a) to prove that
(x − x0)· · ·(x − xN ) = 2−N pN (x).
(c) Show that
|(x − x0)· · ·(x − xN )| ≤ 2
−N+1
for all x ∈ [−1, 1].
3. Suppose that f : [a, b] → R is (N + 1)-times continuously differentiable, and that x0, . . . , xN are
the (N + 1) nodes of the Chebyshev extrema grid on the interval [a, b] so that
xj =
b − a
2
cos ˆ
j
N
π
˙
+
b + a
2
for all j = 0, 1 . . . , N.
Also, let pN be the polynomial of degree N which interpolates f at the nodes x0, x1, . . . , xN . Show
that there exists ξ ∈ (a, b) such that
|f(x) − pk(x)| ≤ 2
−N+1 ˆ
b − a
2
˙N+1 ˇ
ˇ
ˇ
ˇ
ˇ
f
(N+1)(ξ)
(N + 1)!
ˇ
ˇ
ˇ
ˇ
ˇ
Hint: Let g(x) = f
`
b−a
2
x +
b+a
2
˘
and use 2(c) to develop an error bound for g.
4. Suppose that f(x) = cos(x), that N is a positive integer, and that x0, . . . , xN are the nodes of
the Chebyshev extrema grid on the interval [0, 1]. Also, let pN denote the polynomial of degree N
which interpolates f at the nodes x0, . . . , xN . Show that
|f(x) − pN (x)| ≤
2
−2N
(N + 1)!
for all −1 ≤ x ≤ 1.
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