Homework 3 – Math 104B, Winter 2011
Due on Thursday, February 10th, 2011
Section 7.1:
3.d, 4.d, 5.a, 7, 9, and 11.
Section 7.2:
4.d, 8.a, 9, 10, 13, 14, 16, and 17.
Additional Problem 1:
Given a matrix
A
∈
R
n
×
n
, prove that
b
A
b
1
= max
1
≤
j
≤
n
n
s
i
=1

a
ij

.
Additional Problem 2:
Consider the tridiagonal matrix
A
= (
a
i,j
)
1
≤
i,j
≤
n
given by
a
i,j
=
−
1
h
2

i
−
j

= 1
2
h
2
i
=
j
0
Otherwise
(1)
obtained when the following ODE,
−
u
′′
(
x
) =
f
(
x
)
,
x
∈
[0
,
1]
u
(0) =
u
(1) = 0
,
(2)
is discretized using second order centered diFerences:
−
u
i
+1
−
2
u
i
+
u
i
−
1
h
2
=
f
(
x
i
)
i
= 1
,
2
, . . . , n,
(3)
where
h
= 1
/
(
n
+ 1).
1. ±or each
k
= 1
,
2
, . . . , n
, show that the vector
u
(
k
)
given by
u
(
k
)
i
= sin
p
πki
n
+ 1
P
, i
= 1
,
2
, . . . , n
(4)
is an eigenvector of the matrix
A
, and determine the corresponding
eigenvalue
λ
k
.
2. Set up the Jacobi iteration for system (3), and show that the vectors