Civil Engineering Reference
In-Depth Information
Germinal Pierre Dandelin in 1826 and Karl Heinrich Gräffe in 1837.
The Graeffe's root squaring method is especially effective if all roots
are real. The derivation of this method proceeds by multiplying a poly-
nomial
f
(
x
) by
f
(-
x
) using the following polynomial equations in fac-
tored form:
()
=−
(
)
(
)
(
)
…−
(
)
fx xaxaxa xa
f x xaxaxa
−
−
1
2
3
n
−
()
=
()
−
(
)
(
)
(
)
(
)
1
−
−
……−
xa
1
2
3
n
(
)
(
)
(
)
…−
(
)
()
−
()
=
()
−
n
2
2
2
2
2
2
2
2
fxf x xaxaxa xa
1
−
−
1
2
3
n
For example, use a third degree polynomial with roots
x
1
,
x
2
, and
x
3
as
follows:
()
==+++
3
2
fx xaxaxa
0
1
2
3
A polynomial with roots −
x
1
,
−x
2
, and −
x
3
follows:
−
()
==−+ −+
3
2
f x xaxaxa
0
1
2
3
Multiplying the two equations together yields the following:
(
)
+− +
(
)
+
()
−
()
==−+ −
6
2
4
2
2
fxf x xaax a axa
0
2
2
1
2
2
13
3
Letting
y =
−
x
2
this equation may be written as follows:
(
)
+−
(
)
+
3
2
2
2
2
0
=+−
yaay a aya
2
2
1
2
2
13
3
This equation has roots of −
x
1
2
,
−
x
2
2
, and −
x
3
2
. If the procedure was
applied again, another polynomial would be derived with roots of −
x
1
4
,
−
x
2
4
, and −
x
3
4
. If computed a third time, they would be of −
x
1
8
,
−
x
2
8
, and
−
x
3
8
. The pattern of the roots becomes clear with continuing cycles. The
general process of an
n
th degree polynomial would be in the following
forms:
(
)
+− +
(
)
0
=+−
yaay a aay
n
2
2
n
−
1
2
2
2
n
−
2
1
2
2
13
4
(
)
++
+− +
aaa aa
2
2
2
−
2
y
n
−
3
a
2
3
24 15
6
n
