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University of Houston
Numerical Analysis I
Dr. Ronald H.W. Hoppe
Numerical Analysis I
(5th Homework Assignment)
Exercise 17
(
Reformulation of linear least squares problems
)
Let
A
∈
R
m
×
n
, m > n,
rank
A
=
n, b
∈
R
m
. The linear least squares problem
(
*
)
k
Ax

b
k
2
= min
can be formulated as the linear algebraic system
(
**
)
ˆ
I
m
A
A
T
0
! ˆ
r
x
!
=
ˆ
b
0
!
,
where
I
m
stands for the
m
×
m
unit matrix and
r
:=
b

Ax
∈
R
m
is the residual.
(i) Using the normal equations, show that the component
x
of the solution of
(
**
) solves the linear least squares prolem (
*
).
(ii) Given a decomposition of
A
according to
(
†
)
A
=
Q
ˆ
R
0
!
with an orthogonal matrix
Q
∈
R
m
×
m
and a regular upper triangular matrix
R
∈
R
n
×
n
show that by orthogonal row and column transformations the linear
system
ˆ
I
m
A
A
T
0
! ˆ
p
z
!
=
ˆ
f
g
!
can be transformed to the form
I
n
0
R
0
I
m

n
0
R
T
0
0
h
d
z
=
f
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 Spring '12
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