1
COMPUTER SCIENCE 349A
Handout Number 12
MULTIPLE ROOTS AND THE MULTIPLICITY OF A ZERO
(Section 6.4 in 5
th
ed. or Section 6.5 in 6
th
ed.)
If Newton's method converges to a zero
t
x
of
)
(
x
f
, a necessary condition for quadratic
convergence
is that
0
)
(
≠
′
t
x
f
.
We now relate this condition on the derivative of
)
(
x
f
to
the multiplicity
of the zero
t
x
.
Definition
(not in the textbook)
If
t
x
is a zero of any analytic function
)
(
x
f
, then there exists a positive
integer
m
and a function
)
(
x
q
such that
0
)
(
lim
where
),
(
)
(
)
(
≠
−
=
→
x
q
x
q
x
x
x
f
t
x
x
m
t
.
(In particular, if
)
(
t
x
q
is defined, note that
0
)
(
≠
t
x
q
.)
The value
m
is called the
multiplicity
of the zero
t
x
.
If
1
=
m
, then
t
x
is called a
simple zero
of
)
(
x
f
.
Example 1
Consider
)
5
.
2
(
)
4
(
160
56
18
5
.
9
)
(
3
2
3
4
−
+
=
−
−
+
+
=
x
x
x
x
x
x
x
f
The zero at
)
0
)
4
(
and
5
.
2
)
(
here
(
3
has
4
≠
−
−
=
=
−
=
q
x
x
q
m
x
t
.
The zero at
)
0
)
5
.
2
(
and
)
4
(
)
(
here
(
1
has
5
.
2
3
≠
+
=
=
=
q
x
x
q
m
x
t
.
Example 2
Consider
1
)
(
−
−
=
x
e
x
f
x
.
Since
0
)
0
(
=
f
,
0
=
t
x
is a zero of
)
(
x
f
.
This zero has multiplicity
2
=
m
since
)
(
)
0
(
)
(
2
x
q
x
x
f
−
=
with
2
1
)
(
x
x
e
x
q
x
−
−
=
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