Math 571 – Fall 2007 – Homework #4
Due Thursday, Sept 20
You can use any written resources, but please work alone.
11. Given vectors x6= 0 and y6= 0 in
C
m, determine all vectors vsuch that the corresponding
Householder reflector Fsatisfies
F x =γy
for some γ6= 0.
12. Let
A=
1 1 + δ1
1 1 1 + δ
111
.
(a) Use Householder QR Factorization to reduce Ato an upper triangular matrix. (Algorithm
10.1)
(b) Let b= (2,2,2−δ)T. Compute Q∗bby Algorithm 10.2.
(c) Use the results from (a) and (b) to find the solution xto Ax =b.
13. Solve the full rank least squares problem
min
x||Ax −b||2
2
where
A=
1 1
1 0
0 1
and b=
1
0
0
using the QR factorization of A(either Gram-Schmidt or Householder).
14. For Householder QR we choose to use reflectors as our unitary matrices. We could have used
rotations.
(a) Determine values for cand s(representing cos(θ) and sin(θ)) such that c2+s2= 1 and
c−s
s c ! x1
x2!= α
0!.
(b) Given a vector x∈
C
m, describe how you could use such 2 ×2 rotation matrices to form
am×mrotation matrix Rsuch that Rx =αe1.
(c) Determine the flop count for constructing the rotation matrix Rand the flop count for
applying it to another vector y. Hint: you should keep Ras a set of 2 ×2 matrices rather
than constructing a large m×mmatrix explicitly.
1
