Civil Engineering Reference
In-Depth Information
Note the pattern of the powers and the relationship for any other variations
of two roots may be found. Derive one more case for a triple root. If the
first term was
1
/
3
the square of the term in the previous cycle, it would indi-
cate a triple root or
r
1
= r
2
= r
3
. If we consider only the dominant terms in
each expression, the following relationships occur:
b
= +++=++∴≅
=
r
mmmmmmm m
r
r
r
r
r
r br
3
1
1
2
3
4
1
1
1
1
1
mm mm m
mmmmmmm
m m
brr
+
r r
+
r
r
+
r r
+
r r
+
r r
=
r r
4
2
3
2
4
3
4
1
1
2
1
2
1
3
1
mm mm
+ ∴≅
2
m
+
rr rr br
3
11 11 2
1
mmm mmm mmm mmm mmm
b rrr rrr rrr rrr rrr b
= ∴≅
r
m
1
3
=
+
+
+
3
1 23 124
1 34 234 111 3
These become the following:
1
≅∴≅
b
m
br
3
m
r
1
1
1
1
3
1
2
b
≅∴=
==
m
2
m
br
3
r
2
rr
2
1
1
2
3
2
1
3
=∴=
()
==
3
m
m
br
≅
br
r b rr
m
3
1
23
1
3
2
3
After the multiple roots have been passed, the rest of the terms have the
regular solution relationship and will appear as follows:
1
≅∴=
b
b
b
b
m
m
n
n
r
r
n
n
n
−
1
n
−
1
Just like with the regular solution for real and distinct roots, the Enke roots
only lack the proper sign and either + or - must be checked.
Example 1.15 Graeffe's root squaring method—real and equal
roots
Find the root of the following polynomial using Graeffe's root squaring
method.
3
2
fx
()==+−
0
x
3
x
4
Refer to Table 1.17 for the basic procedure for the root squaring. Table 1.19
shows the process for this polynomial.
