Seleted solutions
1 The oeÆient matrix of the red-blak ordering is
A=
D1 B
B D2
4
D =
I; i = 1; 2
h2
i
T
B is a sparse matrix in whih eah row has at most 4 non-zero entries. The size of the matrix is
(n 1)2 by (n 1)2 with non-zero entries of O(5n2 ). The SOR(!) at k-th iteration an be written
as
i; j = 1; 2; ; n 1:
!
+1
u +1 = (1 !)u + u +11 + u +1
+u
u
k +1
ij
=u ;
k
ij
k
k
ij
ij
k
4
k
i
;j
i
;j
k +1
i;j
1 +u
h2 f
k +1
i;j +1
ij
; i; j = 1; 2; ; n 1:
The iterative method does not depend on the ordering of the equations and unknowns, but does
depend on the index i and j .
2 (a) The results of one iteration are
0
1
1=3
x
1=2 A ;
1=2
x
J
0
GS
1
0
1=3
= 1=2 A
5=4
x
SOR
(b) and (), The iteration matries are:
2
0 1=3
6
R =6
40 0
J
0 1=2
Sine kR k1 = 2=3 < 1 and kR
R
GS
0
:
2
0 1=3
6
=6
40 0
0
3
7
1=2 7
5:
1=3
1=4
0
k1 = 2=3 < 1, both iterative methods onverge.
For the rst matrix, the eigenvalues of R are the diagonals. Notie that ja j < 1 for i = 1; 2; 3; 4.
We just need to hek ja j = 1 sin(). Note that 0 < sin x < 1 if 0 < x < and sin x is a
J
3
3
7
1=2 7
5;
1=3
1
0
5=4 A
(1 5) =
31=16
GS
ii
55
periodi funtion of 2. Thus, if 2k < < 2k + 1, then the iterative method onverges, where k
is an integer.
For the seond matrix we have kRk1 = 0:9999 < 1, the iterative method onverges.
4 (a) The iteration matries are:
2
0 0
6
R =6
40 0
J
0 2
Sine (R ) = 2 > 1 and (R
J
GS
3
7
2 7
5;
0
0
R
G
S
2
0 0
6
=6
40 0
0 0
3
7
2 7
5:
0
4
) = 4 > 1, both iterative methods diverge.
(b) The matrix is weakly diagonally dominant and irreduible. Both Jaobi and Gauss-Seidel
iterative methods onverge.