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 − b
T
x satisfies
q(x) = 1
2

kAx − bk
2
A
−1 − kbk
2
A
−1

1
2

kx − A
−1
bk
2
A − kA
−1
bk
2
A

(hence reaches its minimum −
1
2
kbk
2
A
−1 = −
1
2
kA
−1
bk
2
A
for x = A
−1
b).
P2 Consider the basic iteration with M =

1.52 −0.64
−0.64 2.48
applied to the system

2 −1
−1 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 −
4
7
−
41
35
31
14

applied to the system

2 −1
−1 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 −1
−1 2 −1
−1 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 ⊥A d0
and r2 ⊥ r0.
c) Repeat a) using MATLAB and symmetric Gauss-Seidel preconditioning.