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**Unformatted text preview: **Math 2940 Solutions, Fall 2011
Section 9.3 3) I − A = −1
1
1 −1 7) In Gauss-Seidel S −1 has −2 as an eigenvector. Since | − 2| > 1 the iteration diverges. T= Here |λmax | = 1/9.
For Jacobi, S −1 T= 30
03 30
−1 3
−1 −1 01
10 01
00
= =
1/3 0
1/3 0 1/3
0
1/9 1/3 01
00 = 0 1/3
.
0 1/9 so |λmax | = 1/3. 2 −1
1
2
2 −1
2
5
=
. Then u2 =
=
and
−1
2
0
−1
−1
2
−1
−4
2 −1
5
14
u3 =
=
.
−1
2
−4
−13
The eigenvalues of our matrix are 1 and 3. An eigenvector for the larger eigenvalue is
1
. That is the direction to which we are converging.
−1 16) u1 = ...

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