Graphics Programs Reference
In-Depth Information
a bandedmatrix is
⎡
⎣
⎤
⎦
XX000
XXX00
0 XXX0
00XXX
000XX
A
=
where X's denote the nonzero elements thatform the populated band (some of these
elements may bezero).All the elements lying outside the band arezero. The matrix
shown abovehas a bandwidth of three, since there are atmost three nonzero elements
in each row (or column).Such amatrix iscalled
tridiagonal
.
If a bandedmatrix is decomposedinthe form
A
LU
, both
L
and
U
will retain
the banded structureof
A
.For example, if we decomposed the matrix shown above,
we wouldget
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
X 0000
XX000
0 XX00
00XX0
000XX
XX000
0 XX00
00XX0
000XX
0000X
L
=
U
=
The banded structureofa coefficient matrix can beexploited to save storage and
computation time. If the coefficient matrix is also symmetric, further economies are
possible. In this article we show how the methodsofsolutiondiscussedpreviously
can be adapted for banded and symmetric coefficient matrices.
Tridiagonal Coefficient Matrix
Consider the solution of
Ax
=
b
by Doolittle's decomposition, where
A
is the
n
×
n
tridiagonal matrix
⎡
⎣
⎤
⎦
d
1
e
1
00
···
0
c
1
d
2
e
2
0
···
0
0
c
2
d
3
e
3
···
0
A
=
00
c
3
d
4
···
0
.
.
.
.
.
.
.
.
00
...
0
c
n
−
1
d
n
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