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Homework 6 Foundations of Computational Math 1 Fall
2010
The solutions will be posted on Monday, 11/8/10
Problem 6.1
Suppose you are attempting to solve
Ax
=
b
using a linear stationary iterative method
deﬁned by
x
k
=
Gx
k

1
+
f
that is consistent with
Ax
=
b
.
Suppose the eigenvalues of
G
are real and such that

λ
1

>
1 and

λ
i

<
1 for 2
≤
i
≤
n
.
Also suppose that
G
has
n
linearly independent eigenvectors,
z
i
, 1
≤
i
≤
n
.
6.1.a
. Show that there exists an initial condition
x
0
such that
x
k
converges to
x
=
A

1
b
.
6.1.b
. Does your answer give a characterization of selecting
x
0
that could be used in
practice to create an algorithm that would ensure convergence?
Problem 6.2
Suppose you are attempting to solve
Ax
=
b
using a linear stationary iterative method
deﬁned by
x
k
=
M

1
Nx
k

1
+
M

1
b
where
A
=
M

N
. Suppose further that
M
=
D
+
F
where
D
=
diag
(
α
11
,...,α
nn
) and
F
is made up of any subset of the oﬀdiagonal elements of
A
. The matrix
N
is therefore the
remaining oﬀdiagonal elements of
A
after removing those in
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This note was uploaded on 07/25/2011 for the course MAD 5403 taught by Professor Gallivan during the Spring '11 term at University of Florida.
 Spring '11
 Gallivan

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