X

T

only show that , where is the iteration matrix for either Jacobi or

W “ W “

X

y

Gauss-Seidel. A proof by contradiction will be used to show that in fact . W “

y

70

¬ ²

Assume that is an eigenvalue of with . Then the matrix

© ©

W “ W “

0©

y

² ‚§

¬ ¬C¢ 0©

would be singular and, as a result, would also be singular. Since , © 0

¥

¤

y

§

it is clear that is also an irreducibly diagonally dominant matrix. This would contradict

¦

¤

Corollary 4.2.

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¢ ¡

£ ¡C

C @¡

¨§ £ "!

¡5 ) B 078 I 0A B E G¡5 © 0¤ CA 7F # ) B 3 A 5 Q0© F

B

6§¨

II

5BB '

5

3A

F

T

§

It is possible to show that when is Symmetric Positive De¬nite, then SOR will converge

X

# ©

for any in the open interval and for any initial guess . In fact, the reverse is also

™Y

˜(

true under certain assumptions.

cv ˜¤

¥¦ ¥¡

B9

A

' 1 #

§

If is symmetric with positive diagonal elements and for ,

Y ˜

© §

SOR converges for any if and only if is positive de¬nite.

£6¨

¢ § 73P6 #3

A5' 6E B P5 © ©46P0A F 7F ©' ) ©C8 8

E

3' AE5 B E

F 3

©

A number of properties which are related to the graph of a ¬nite difference matrix are

now de¬ned. The ¬rst of these properties is called Property A. A matrix has Property A

if its graph is bipartite. This means that the graph is two-colorable in the sense de¬ned in

Chapter 3: Its vertices can be partitioned in two sets in such a way that no two vertices in

the same set are connected by an edge. Note that, as usual, the self-connecting edges which

correspond to the diagonal elements are ignored.

¥ h¤

§ ' )$ 2¨) 2 ¤B9

¥A

)§

¦

A matrix has Property A if the vertices of its adjacency graph can be

£¦

partitioned in two sets and , so that any edge in the graph links a vertex of to a

¦ ¦

W W

£¦

vertex of .

In other words, nodes from the ¬rst set are connected only to nodes from the second set

and vice versa. This de¬nition is illustrated in Figure 4.5.

£¦

¦

W

´ AB@875¤4 2

963

Graph illustration of Property A.

An alternative de¬nition is that a matrix has Property A if it can be permuted into a

matrix with the following structure:

r±0 ¯ i ¯®

²

# 1

'

² ¬§ (

¤ W %1 £#

| ¢ Cv¥ ¡At ¢

| §| ¥ ¡C

C

¡¨

£#

where and are diagonal matrices. This structure can be obtained by ¬rst labeling

#

W ¤ ¬W¤ ¤ ¤

all the unknowns in from 1 to , in which and the rest from to . w

0¦ 0

¦

W W W W £ r( y #

Note that the Jacobi iteration matrix will have the same structure except that the #

W

blocks will be replaced by zero blocks. These Jacobi iteration matrices satisfy an important

property stated in the following proposition.

Q¤ ¡ v A£

£¦ DB9

CA

' ' ©' 2 )

§)

Let be a matrix with the following structure:

¢

a±a ¯ i ¯®

£

¢

"¢