Graphics Programs Reference
In-Depth Information
It is wastefultoform
P
i
and then carry out the matrix multiplication
P
i
AP
i
.We
note that
A
I
uu
T
H
A
u
H
u
T
A
Q
A
−
A
−
vu
T
=
−
=
=
where
A
u
H
=
v
(9.43)
Therefore,
I
A
−
uu
T
H
vu
T
=
uu
T
H
A
−
vu
T
QA
Q
A
−
vu
T
=
−
−
u
u
T
A
H
u
u
T
v
u
T
H
A
−
vu
T
=
−
+
A
−
vu
T
uv
T
2
g
uu
T
=
−
+
where
u
T
v
2
H
=
g
(9.44)
Letting
w
=
v
−
g
u
(9.45)
itcan beeasilyverified that the transformation can be writtenas
QA
Q
A
−
wu
T
uw
T
=
−
(9.46)
which gives us the following computational procedure which istobecarried out with
i
=
1
,
2
,...,
n
−
2:
1. Let
A
be the (
n
−
i
)
×
(
n
−
i
) lowerright-hand portion of
A
.
A
i
+
1
,
i
A
i
+
2
,
i
A
n
,
i
T
2. Let
x
=
···
(the column of length
n
−
i
just to the left of
A
).
3. Compute
|
|
= |
|
if
x
1
>
=−|
|
if
x
1
<
x
.Let
k
x
0 and
k
x
0 (this choice of sign mini-
mizes the roundoff error).
4.Let
u
k
x
n
−
i
T
.
=
+
x
1
x
2
x
3
···
2
5. Compute
H
= |
u
|
/
2.
A
u
6. Compute
v
=
/
H
.
u
T
v
7. Compute
g
=
/
(2
H
).
8. Compute
w
g
u
.
9. Compute the transformation
A
=
v
−
A
−
w
T
u
u
T
w
.
←
−
10.Set
A
i
,
i
+
1
=
A
i
+
1
,
i
=−
k
.
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