Civil Engineering Reference
In-Depth Information
Table 1.19.
Example 1.15 Graeffe's root squaring method—real and
equal roots
m
x
3
a
1
x
2
a
2
x
1
a
3
x
0
cycle
1
1
3
0
1
−4
1
9
0
16
0
24
2
1
9
24
16
2
1
81
576
256
−48
−288
4
1
33
288
256
3
1
1089
82944
65536
−576
−16896
8
1
513
66048
65536
4
1
2.632E+05
4.362E+09
4.295E+09
−1.321E+05
−6.724E+07
16
1
1.311E+05
4.295E+09
4.295E+09
5
1
1.718E+10
1.845E+19
1.845E+19
−8.590E+09
−1.126E+15
32
1
8.590E+09
1.845E+19
1.845E+19
6
1
7.379E+19
3.403E+38
3.403E+38
−3.689E+19
−3.169E+29
64
1
3.689E+19
3.403E+38
3.403E+38
7
1
1.361E+39
1.158E+77
1.158E+77
−6.806E+38
−2.511E+58
128
1
6.806E+38
1.158E+77
1.158E+77
8
1
4.632E+77
1.341E+154
1.341E+154
−2.316E+77
−1.576E+116
256
1
2.316E+77
1.341E+154
1.341E+154
9
r
2
2
1
Notice that the first term in the table for cycle 9 is ½ the square of the term
in the previous cycles and the following solution applies:
1
256
1
b
=
2 316 10
2
.
(
)
77
≅
m
1
r
=±
2 000
.
1
2
1
2 256
1
2
≅
()
=
(
)
154
r b
m
13
. 41 10
()
=±
2 000
.
=
r
2
2
1
1
1
256
≅
b
b
m
1 341 10
.
(
)
15
4
r
3
=
=±
1 000
.
3
154
1 341 10
.
(
)
2
