Matthias St. George
Numerical Analysis
Chapter 9 Homework
9.2.13
a)
A=
0 0
½ 0
0 ¼
0 0
2 4
0 0
0 0
1/8 0
Use Gerschgorin Circle Method:
Row 1
Row 2
Row 3
Row 4
=>
=>
=>
=>
0
0
0
0
=6
=½
=¼
 = 1/8
 < or = 6
All the circles fall inside of Row 1 which = 6.
b)
I tried to use the power method algorithm for this problem, but it would not run.
I calculated this by hand when I got to 30 iterations I quit. The dominant eigen-value kept moving position
Throughout the eigen-vectors. The dominant eigen-value is 0.697668497234
The associated eigen-vector is {1, 0.7166727, 0.2568099, 0.04601217}
I solved this using my TI-85.
c)
This algorithm will not work either. I think it is because this is has complex eigen-values. Once again,
I soved this part of the problem with my TI-85.
{0.697668497234, -0.230177594226 + 0.569658840151i, -0.230177594226 – 0.569658840151i,
-0.237313308781}
d) Using the eigen-values as roots, the characteristic polynomial is x4 – ¼ x – 1/16
e) The beetle population is decreasing and eventually will cease to exist. This is because A is converging.
A5 is getting small, A20 is even smaller, and A40 the values in A are practically zero.