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CS205  Class 4
QR Continued (Readings Heath pp120137)
1.
Transition to Householder.
So far, we’ve seen two different ways of finding a least squares
solution, namely the normal equations and the (modified) GramSchmidt method.
Using the
normal equations squares the condition number of the A matrix, which could be bad to begin
with.
The GramSchmidt method does not suffer from the same numerical problems, but can
still be unstable.
If the columns of A are “nearly linearlydependent,” then this method can
suffer from numerical instabilities. The QR decomposition is exactly what the GramSchmidt
procedure does couched in the language of matrix factorization.
There is an algorithm to
performs the QR factorization that does not suffer from the numerical instabilities mentioned
above and this is the one that is quite often used by practitioners.
2.
A
Householder transform
is defined by
T
T
vv
v
v
I
H
2
−
=
for some vector
0
v
≠
.
Note that
1
T
HH H
−
==
, and thus H is orthogonal.
a.
For a vector
a
, we define
k
H
using
2
ˆˆ
()
kk
k
va
S
a
a
e
=+
where
ˆ
(
0
,,
0
,,,)
T
km
aa
a
=
""
and
() 1
k
Sa
=±
is the sign function. Then
k
Ha
zeroes out the entries of
a
below
k
a
.
b.
Let
2
1
2
a
⎡⎤
⎢⎥
=
⎣⎦
. Then
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
+
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
+
=
2
1
5
0
0
1
9
2
2
1
2
ˆ
ˆ
1
2
1
1
S
e
a
a
S
a
v
.
c.
Note that
v
v
v
a
v
a
Ha
T
T
2
−
=
so we never need to form H explicitly, but instead only need
to find the vector
v
.
d.
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡−
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
×
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
0
0
3
2
1
5
30
15
2
2
1
2
2
1
5
2
,
1
,
5
2
,
1
,
5
2
,
1
,
2
2
,
1
,
5
2
2
1
2
2
1
2
1
T
T
H
.
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

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