1
Lecture15
Homework, Rate of Convergence of
CG, preconditioning, FRGC, PRGC
Homework (Due April 17)
• 5.1
• 5.9
•
Proof for Theorem 5.5 (see the slides)
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2
Theorem 5.4
If
A
has only
r
distinct eigenvalues, then the CG iteration
will terminate at the solution in at most
r
iterations.
Theorem 5.5
2
*
0
2
1
1
2
*
1




A
k
n
k
n
A
k
x
x
x
x

+

≤



+
l
l
l
l
If
A
has eigenvalues
we have
n
l
l
l
≤
≤
≤
K
2
1
n
k
n
k
n
l
l
l
l
,
,
,
,
,
1
1
K
K
+


Eigenvalues