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AIMS Exercise Set # 5
Peter J. Olver
1.
Use the power method to find the dominant eigenvalue and associated
eigenvector of the following matrices: (
a
)
-
2
0
1
-
3
-
2
0
-
2
5
4
,
(
b
)
4
1
0
1
1
4
1
0
0
1
4
1
1
0
1
4
.
2.
Use Newton’s Method to find all points of intersection of the following pairs of
plane curves:
x
3
+
y
3
= 3
,
x
2
-
y
2
= 2
.
3.
The system
x
2
+
xz
= 2
, xy
-
z
2
=
-
1
, y
2
+
z
2
= 1
,
has a solution
x
?
= 1
,
y
?
= 0
, z
?
= 1. Consider a fixed point iteration scheme with
g
(
x, y, z
) =
(
x
+
α
(
x
2
+
xz
-
2)
, y
+
α
(
xy
-
z
2
+ 1)
, z
+
α
(
y
2
+
z
2
-
1)
)
T
,
where
α
is a constant. (
a
) For which values of
α
does the iterative scheme converge to
the solution when the initial guess is nearby? (
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- Fall '09
- Olver
- Numerical Analysis, Matrices, Iterated function, point iteration scheme, Peter J. Olver, Newton iteration scheme
-
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