This preview shows pages 1–3. Sign up to view the full content.
20. Newton’s Method
P. K. Lamm
11/01/11 (14:36)
p. 1 / 6
Lecture Notes:
20. Newton’s Method
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of deﬁnitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. Newton’s Method
Newton’s Method uses the idea of linear approximation in a very important way. Suppose we are
given
y
=
f
(
x
) where we know there is some
c
for which
f
(
c
) = 0;
that is, the point
x
=
c
solves the equation
f
(
x
) = 0
,
or
c
is a root
of
f
(
x
). The goal is to actually estimate
c
.
We’ll start with a guess
x
0
for
c
; hopefully the initial guess is reasonably close to
c
. If we then
construct the tangent line to the curve at (
x
0
,f
(
x
0
)) we have the line
y

f
(
x
0
) =
f
0
(
x
0
)(
x

x
0
)
,
or
y
=
f
(
x
0
) +
f
0
(
x
0
)(
x

x
0
)
.
Now for
x
near
x
0
, we know the
y
value of the curve is wellapproximated by the
y
value of the
tangent line. That is,
f
(
x
)
≈
f
(
x
0
) +
f
0
(
x
0
)(
x

x
0
)
,
for
x
≈
x
0
.
Since our goal is to ﬁnd
c
such that
y
=
f
(
x
) crosses the
x
axis at
c
, we could try to approximate
c
by ﬁnding where this tangent line to the curve at
x
0
crosses the
x
axis. That is, we set
0 =
f
(
x
0
) +
f
0
(
x
0
)(
x

x
0
)
and solve for
x
. Then
f
0
(
x
0
)(
x

x
0
) =

f
(
x
0
)
x

x
0
=

f
(
x
0
)
f
0
(
x
0
)
x
=
x
0

f
(
x
0
)
f
0
(
x
0
)
,
provided
f
0
(
x
0
)
6
= 0.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document20. Newton’s Method
P. K. Lamm
11/01/11 (14:36)
p. 2 / 6
Hopefully this point where the tangent line crosses the
x
axis is closer to
c
than our initial guess
x
0
was. We will call
x
1
this new
x
value,
x
1
=
x
0

f
(
x
0
)
f
0
(
x
0
)
.
But now we can repeat the process with our new initial guess
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '10
 KIHYUNHYUN
 Calculus

Click to edit the document details