Civil Engineering Reference
In-Depth Information
Example 2.16 shows the process of determining the characteristic polyno-
mial and the eigenvalues by the Faddeev-Leverrier method.
Example 2.16
Faddeev-Leverrier method
Determine the eigenvalues for the following set of equations using the
Faddeev-Leverrier method.
x
++=
−++=
−++=
230
10 020
2480
x x
xx x
1
2
3
1
2
3
x
x
x
1
2
3
The matrix operations are shown in Table 2.19.
The characteristic polynomial is found from the trace values.
(
)
=
()
−−−
3
(
)
−
3
2
1
ll l
9
26
24
0
−+ −+=
lll
3
9
2
26 40
The solution to the cubic equation can be found by many of the methods
from Chapter 1 and represent the eigenvalues
l
= 2, 3, and 4.
2.15
POWER MEtHOD OR itERAtiOn MEtHOD
The power method is an iterative method used when only the smallest or
largest eigenvalues and eigenvectors are desired. It may also be used to
find intermediate eigenvalues and eigenvectors using a sweeping tech-
nique. The sweeping technique can be found in “Applied Numerical
Methods for Digital Computations,” By M.L. James, G.M. Smith, and
J.C. Wolford. The largest eigenvalue is found by iterating on the equation
[
A
][
x
]=
l
[
x
].
The steps of procedure are as follows:
1. Assume values for the components of the eigenvector [
x
]=1.
2. Multiply the coefficient matrix times the vector [
A
][
x
].
3.
Normalize the right hand side of the equation as follows
l
[
x
]:
