Civil Engineering Reference
In-Depth Information
Substituting the Enke roots into the original equations yields
x
1
=
3
.
000
and
x
2
=
−
2
.
000
.
1
256
1
2
()
()
138
=
b
b
m
1 3652 10
4 012 10
.
R
4
=
=±
1 414
.
=
± 2
99
.
2
Using the fact that
R
2
= u
2
+ v
2
, the following is used to find
u
and
v
:
(
)
=− −+
(
)
∴=−
1
==−++
a
x
x u uu
2
322
1
1
1
2
2
2
2
2
RuvvRu
=+∴= −=−=
211
Thus results are
x
3
=
−
1
+ i
and
x
4
=
−
1
−
i.
1.13
bAiRStOW'S MEtHOD
Bairstow's method was first published by Leonard Bairstow in 1920
(Bairstow 1920). If we divided a polynomial of
n
th degree by a quadratic
equation, the result will be a polynomial of
n
−2 degree plus some remain-
der. This remainder can be used to give a closer approximation of the root
quadratic equation. When the remainder is zero, the quadratic is a root
equation. Bairstow's method involves using the remainders from double
synthetic division to approximate the error in an assumed quadratic root
equation of a polynomial. The derivation is omitted from this text, but
may be found in “Applied Numerical Methods for Digital Computations,”
by James, Smith and Wolford (1977). Look at the process of factoring a
polynomial into a quadratic equation times a polynomial of two degrees
less than the original polynomial as follows:
n
n
−
1
n
−
2
n
−
3
1
xax ax
+
+
+
a x
+
+
a xa
+
=
0
1
2
3
n
−
1
n
(
)
(
)
=
x
2
+
ux vx bx
+
n
−
2
+
n
−
3
+ ++ ++
bx
n
−
4
bxb
1
remainder
0
1
2
n
−
3
n
−
2
The derived polynomial follows with the terms in the brackets being the
remainder:
(
)
=
[
]
n
−
2
n
−
3
n
−
4
1
x
+
b x
+ ++ +++
b x
b xb bb
0
1
2
n
−
3
n
−
2
n
−
1
n
Divide the resulting polynomial by the quadratic equation in order to
derive an equation that has something to do with the derivative of the
original equation. This will be a polynomial four degrees less than the
