Civil Engineering Reference
In-Depth Information
If we consider only the dominant terms in each expression, the following
occurs:
1
=∴=
()
m
br r b
m
1
1
1
1
1
= ∴=
b
b
m
mm m
brr r
=
r
2
2
1
2
1 2
2
1
=
(
)
m
m
2
cosm
q
b
r rR
3
1 2
=
(
)
br rR
2
4
1 2
Dividing the second and fourth equations:
1
2
b
b
RR
b
=∴≅
m
2
m
4
4
b
2
2
Using the fact that
R
2
= u
2
+ v
2
the following is used to find
u
and
v
:
(
)
a
=− +++
x xxx
1
1
2
3
4
(
)
(
)
+−
(
)
a
=− +++
x xuvi u i
1
1
2
(
)
a
=− ++
x
x u
2
1
1
2
Use
b
1
and
b
2
to find
r
1
and
r
2
, then
x
1
and
x
2
. Use
b
4
and
b
2
to find
R
then
use
a
1
to find
u
and
R
to find
v.
The Enke roots, once again, only lack the
proper sign and either + or - may be correct.
Example 1.16
Graeffe's root squaring method—real and
complex roots
Find the root of the following polynomial using Graeffe's root squaring
method.
fx xx x
()==+−−−
0
4
3
6
2
14
x
12
Refer to Table 1.17 for the basic procedure for root squaring. Table 1.20
shows the process for this polynomial.
The third term has a sign fluctuation thus the previously derived rela-
tionships apply and the following is the solution:
