MATH 128A, SUMMER 2010, HOMEWORK 4 SOLUTION
BENJAMIN JOHNSON
Homework 4: Due Wednesday, July 7
2.6; 1c, 4h
3.1; 1c, 3c, 12
section 2.6
1. Find the approximations to within 10

4
of all the real zeros of the following polynomials
using Newton’s method.
c.
f
(
x
) =
x
3

x

1
solution:
To apply Newton’s method, we need to do a fixedpoint iteration on the func
tion
g
(
x
) =
x

f
(
x
)
f
0
(
x
)
=
x

x
3

x

1
3
x
2

1
. It remains to choose the starting points. Let’s start
with
x
0
= 1. If we do this, we generate the sequence
h
p
n
i
=
h
1
,
1
.
5000
,
1
.
3478
,
1
.
3252
,
1
.
3247
i
;
and 1.3247 is correct to within 10

4
. [Ideally we should have made an effort to compute
all the values
f
(
p
n
) and
f
0
(
p
n
) using Horner’s method, but since this is such a simple
function, and my computer is fast, I did not bother to do that.]
The remaining roots are complex, and to get these we can either divide
f
(
x
) by (
x

1
.
3247) and use the quadratic formula, or we can continue using Newton’s method by
starting with a complex number. The latter strategy is easiest if we have already written
a function in MATLAB for Newton’s method.
In any case the additional complex roots are

0
.
6624 + 0
.
5623
i
and

0
.
6624 + 0
.
5623
i
.
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 Fall '10
 To
 Polynomials, Numerical Analysis, Approximation, Complex number, Newton’s method

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