Description
Problem 1. Indicate which of the following statements are true and which are false. You do not need
to justify your answers.
1. If f : r´1, 1s Ñ R is a C
k
functions and tanu are the Chebyshev coefficients of f, then |an| “
O
`
1
nk
˘
.
2. The condition number of evaluation of the function fpxq “ 1
x
goes to 8 as x Ñ 0
`.
3. The condition number of evaluation of the function fpxq “ cospxq goes to 8 as x Ñ π{2.
4. The quadrature rule
ż π
´π
fptq dt «
2π
n ` 1
ÿn
j“0
f
ˆ
´π `
2π
n ` 1
j
˙
,
is exact for the collection of functions
expp´iktq, k “ ´n, ´n ` 1 ´ n ` 2, . . . , ´1, 0, 1, 2, . . . , n ´ 1, n.
5. If p is a monic polynomial of degree n, then
max
´1ďxď1
|ppxq| ě 2
´n`1
.
Problem 2. Find the unique polynomial p of degree less than or equal to 3 such that
ppx0q “ fpx0q
p
1
px0q “ f
1
px0q
ppx1q “ fpx1q
p
1
px1q “ f
1
px1q,
where
fpxq “ sinpxq
and
x0 “ 0, and x1 “
π
2
.
Problem 3. Let fpxq “ cospxq and, for each positive integer N, let pN be the polynomial of degree
less than or equal to N which interpolates f at the nodes
xj “ ´1 `
2j
N
, j “ 0, . . . , N.
Show that
max
´1ďxď1
|fpxq ´ pN pxq| Ñ 0 as N Ñ 8.
Problem 4. Find a quadrature rule of the form
ż 1
´1
fpxq |x| dx « fp´1qw0 ` fp0qw1 ` fp1qw2
which is exact whenever f is a polynomial of degree less than or equal to 2.
Problem 5. Compute the Chebyshev coefficients of the function
fpxq “ a
1 ´ x
2.
Problem 6. Suppose that f : R Ñ R is smooth, and that h ą 0. Find coefficients a, b and c such that
afp´hq ` bfphq ` cfp2hq “ f
1
p0q ` O
`
h
2
˘
.
Problem 7. Let fpxq “ cospxq and, for each positive integer N, let pN be the polynomial of degree
less than or equal to N which interpolates f at the nodes
xj “ cos ˜
j `
1
2
N ` 1
¸
, j “ 0, . . . , N.
Show that
max
´1ďxď1
|fpxq ´ pN pxq| ď 2
´N
pN ` 1q!
.
Problem 8. Show that the Legendre Polynomial Pn of degree n satisfies the differential equation
p1 ´ x
2
qf
2
pxq ´ 2xf1
pxq ` npn ` 1qfpxq “ 0.
Bonus problem 1: Find a quadrature rule of the form
ż 1
´1
fpxq |x| dx « fpx0qw0 ` fpx1qw1 ` fpx2qw2
which is exact whenever f is a polynomial of degree less than or equal to 5.
Bonus problem 2: Find the nodes t0, t1, t2, t3 and weights w0, w1, w2, w3 of a quadrature rule
ż π
´π
fptq dt «
ÿ
3
j“0
fptj qwj
which is exact for the functions
texppintq : n “ ´3, ´2, ´1, 0, 1, 2, 3u .