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CSE/MATH 6643: Numerical Linear Algebra
Haesun Park
{
hpark
}
@cc.gatech.edu
School of Computational Science and Engineering
College of Computing
Georgia Institute of Technology
Atlanta, GA 30332, USA
Lecture 7
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5.3
Cholesky Decomposition in Banded Systems
Definition:
A
has
symmetric
bandwidth
p
if
a
ij
= 0
when

i
−
j

> p
.
A
has
upper
bandwidth
p
if
a
ij
= 0
when
j > i
+
p
.
A
has
lower
bandwidth
p
if
a
ij
= 0
when
i > j
+
p
A
is
tridiagonal
if
a
ij
= 0
when

i
−
j

>
1
If
A
is S.P.D with sym. bandwidth
p
, then its Cholesky decomposition is
A
=
GG
T
where
G
has lower bandwidth
p
.
CSE/MATH 6643: Numerical Linear Algebra – p.1/8
e.g.
p
= 1
,
A
=
2
6
6
6
6
6
6
6
6
4
a
1
c
1
c
1
a
2
c
2
c
2
a
3
.
.
.
.
.
.
.
.
.
c
n

1
c
n

1
a
n
3
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
4
g
11
g
21
g
22
g
32
g
33
.
.
.
.
.
.
g
n,n

1
g
nn
3
7
7
7
7
7
7
5
2
6
6
6
6
6
6
4
g
11
g
21
g
22
g
32
g
33
.
.
.
.
.
.
g
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This note was uploaded on 02/04/2012 for the course CS 8801 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Staff

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