Graphics Programs Reference
In-Depth Information
−
×
−
(
n
1) isconstructedusing Eqs. (9.36)-(9.38).Referring to Eq. (9.39), we
see that the transformationreduces the first column of
A
to
1)
(
n
⎡
⎣
⎤
⎦
A
11
−
A
11
Qx
k
0
.
0
=
The transformation
A
11
(
Qx
)
T
A
←
P
1
AP
1
=
(9.41)
QA
Q
Qx
thustridiagonalizes the first rowas well as the first column of
A
. Here is a diagram of
the transformation fora4
×
4 matrix:
1
00 0
A
11
A
12
A
13
A
14
1
00 0
0
0
A
21
A
31
0
0
·
·
A
Q
Q
0
A
41
0
A
11
−
k
0
0
−
k
0
=
QA
Q
0
The second row and column of
A
are reducednext by applying the transformation to
the 3
×
3 lowerright portion of the matrix. Thistransformation can beexpressedas
A
←
P
2
AP
2
, where now
I
2
0
T
0Q
P
2
=
(9.42)
InEq. (9.42)
I
2
is a2
×
2identitymatrix and
Q
is a (
n
−
2)
×
(
n
−
2) matrix constructed
by choosing for
x
the bottom
n
−
2 elements of the second column of
A
. It takes a total
of
n
−
2 transformations with
I
i
0
T
0Q
,
P
i
=
i
=
1
,
2
,...,
n
−
2
to attain the tridiagonalform.
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