CS 257
Numerical Methods  Homework 3
September 20, 2006
1.
[1pt]
Section 3.2 #15
Solution:
The first iteration of Newton’s method is:
x
1
=
x
0

f
(
x
0
)
f
(
x
0
)
In this case
x
0
= 1,
f
(
x
0
) = 1, and
f
(
x
0
) = 3
x
2
0

1 = 2. Therefore
x
1
= 1
/
2.
2.
[1pt]
Section 3.2 #28 (use MATLAB if you wish)
Solution:
Beginning with
x
0
= 1 we have the following sequence
x
1
=
1

f
(1)
f
(1)
= 1


8

4
=

1
x
2
=

1

f
(

1)
f
(

1)
=

1

16

8
= 1
x
3
=
1

f
(1)
f
(1)
= 1


8

4
=

1
In MATLAB one could solve the problem as follows (using the UP key to scroll through previous commands):
>> x = 1
x =
1
>> x = x  (3*x^3 + x^2 15*x + 3)/(9*x^2 + 2*x  15)
x =
1
>> x = x  (3*x^3 + x^2 15*x + 3)/(9*x^2 + 2*x  15)
x =
1
>> x = x  (3*x^3 + x^2 15*x + 3)/(9*x^2 + 2*x  15)
x =
1
>> x = x  (3*x^3 + x^2 15*x + 3)/(9*x^2 + 2*x  15)
x =
1
1
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3.
[1pt]
Using the MATLAB implementation of
Newton’s method
(see page 106) on the course webpage or
one of your own, solve the following problems.
Compute a zero of
f
(
x
) =
x
4

5
x
3
+ 9
x
2

7
x
+ 2 with
x
0
= 2
.
2, call this value ˆ
x
. Now create a semilog plot (not to be turned in) with blue lines showing the error
e
i
=

ˆ
x

x
i

at each iteration.
Solution:
See next problem
4.
[1pt]
Repeat the previous problem, but change
x
0
= 0
.
8 and the line color to red. Create a single semilog
plot titled
Plot 1
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 Fall '05
 ThomasKerkhoven
 Numerical Analysis, Secant method, Rootfinding algorithm

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