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
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- Fall '08
- MING
- Calculus, Intermediate Value Theorem, Mean Value Theorem, Continuous function, 10 digits, 6 digits, 0.1 Newton, 24 digits
-
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