0.1
Newton’s method and the Mean Value
Theorem
Newton’s method for computing the zeros of functions is a good example of
the practical application of the Mean Value Theorem.
Let
f
(
x
) be a real
valued function on the real line that has two continuous derivatives.
We
are looking for a
root
of
f
, i.e., a point ˆ
x
such that
f
(ˆ
x
) = 0. In Newton’s
method, which is geometrical, we consider the curve
y
=
f
(
x
).
Then the
curve crosses the
x
axis at the point (ˆ
x,f
(ˆ
x
)).
Let
x
0
be an initial guess
for the root. To improve on the guess we construct the tangent line to the
curve
y
=
f
(
x
) that passes through the point (
x
0
,f
(
x
0
)) on the curve. This
tangent line satisfies the equation
y
−
f
(
x
0
) =
f
′
(
x
0
)(
x
−
x
0
)
.
The tangent line crosses the
x
axis at the point
x
1
=
x
0
−
f
(
x
0
)
f
′
(
x
0
)
,
and we take
x
1
as our improved estimate of the root ˆ
x
. Now we repeat this
procedure with
x
1
to get an improved estimate
x
2
, and so on. Thus we have
a sequence
{
x
n
}
such that
x
n
+1
=
x
n
−
f
(
x
n
)
f
′
(
x
n
)
,
n
= 0
,
1
,
· · ·
.
We need to give conditions that will guarantee that the sequence will converge
to a root of
f
(
x
), and will provide information about the rate of convergence.
To analyze this procedure we define an updating function
T
(
x
) by
T
(
x
) =
x
−
f
(
x
)
f
′
(
x
)
.
We will not yet fix the domain
D
of this function, but it is clear that we
must require
f
′
(
x
)
negationslash
= 0 for all
x
∈
D
.
Then ˆ
x
will be a fixed point of
T
,
(
T
(ˆ
x
) = ˆ
x
) if and only if
f
(ˆ
x
) = 0. To get the growth rate for the iteration
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 MING
 Calculus, Intermediate Value Theorem, Mean Value Theorem, Continuous function, 10 digits, 6 digits, 0.1 Newton, 24 digits

Click to edit the document details