Graphics Programs Reference
In-Depth Information
As in the case of tridiagonal matrices, we store the nonzero elements in the three
vectors
⎡
⎣
⎤
⎦
d
1
d
2
.
d
n
−
2
d
n
−
1
d
n
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
e
1
e
2
.
e
n
−
2
e
n
−
1
f
1
f
2
.
f
n
−
2
d
=
e
=
f
=
b
by Doolittle's decomposi-
tion. The first stepistotransform
A
to upper triangular form by Gauss elimination. If
elimination has progressed to the stage where the
k
th row has become the pivot row,
wehave the following situation:
Let us now look at the solution of the equations
Ax
=
⎡
⎣
⎤
⎦
.
.
.
.
.
.
.
.
.
.
···
0
d
k
e
k
f
k
0
0
0
···
←
···
0
e
k
d
k
+
1
e
k
+
1
f
k
+
1
0
0
···
A
=
···
0
f
k
e
k
+
1
d
k
+
2
e
k
+
2
f
k
+
2
0
···
···
0
0
f
k
+
1
e
k
+
2
d
k
+
3
e
k
+
3
f
k
+
3
···
.
.
.
.
.
.
.
.
.
.
The elements
e
k
and
f
k
below the pivot rowareeliminatedbythe operations
row (
k
+
1)
←
row (
k
+
1)
−
(
e
k
/
d
k
)
×
row
k
row (
k
+
2)
←
row (
k
+
2)
−
(
f
k
/
d
k
)
×
row
k
The only terms (other than those being eliminated)that arechangedbythe above
operations are
d
k
+
1
←
d
k
+
1
−
(
e
k
/
d
k
)
e
k
e
k
+
1
←
e
k
+
1
−
(
e
k
/
d
k
)
f
k
(2.27a)
d
k
+
2
←
d
k
+
2
−
(
f
k
/
d
k
)
f
k
Storage of the multipliers in the
upper
triangular portion of the matrix results in
e
k
←
e
k
/
d
k
f
k
←
f
k
/
d
k
(2.27b)
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