P1 If kxk
2
A
:= x
T
Ax denotes the A-norm of x (A SPD), verify that
q(x) = 1
2
x
T
Ax â b
T
x satisfies
q(x) = 1
2
kAx â bk
2
A
â1 â kbk
2
A
â1
1
2
kx â A
â1
bk
2
A â kA
â1
bk
2
A
(hence reaches its minimum â
1
2
kbk
2
A
â1 = â
1
2
kA
â1
bk
2
A
for x = A
â1
b).
P2 Consider the basic iteration with M =
1.52 â0.64
â0.64 2.48
applied to the system
2 â1
â1 2
x =
0
3
started with x0 =
1
0
.
a) Compare the iterates x1
and x2 with those obtained using SD.
b) Compare the iterate x3 obtained by both methods.
©2019 1/2
Not
S23 SD & CG acceleration - Problems II
P3 Consider the basic iteration with M =
58
35 â
4
7
â
41
35
31
14
applied to the system
2 â1
â1 2
x =
0
3
.
a) Compare the iterates x1
and x2 with those obtained using CG.
b) What happens if x0 is changed?
P4 Consider the system


2 â1
â1 2 â1
â1 2

 x =


1
2
3

.
a) Solve the system by hand using CG starting starting with x0 =


3
3
4

.
b) Repeat a) using MATLAB. Verify the relations d2 â¥A d0
and r2 ⥠r0.
c) Repeat a) using MATLAB and symmetric Gauss-Seidel preconditioning.