Homework 3 Solution
1. Determine the approximated root of the following equation:
f
(
x
) = 5
x
3

3
x
2
+ 6
x

2
(a) [1pt] Using three iterations of the NewtonRaphson method with the initial guess,
x
0
= 1.
(b) [1pt] Using three iterations of the secant method with the initial guesses,
x

1
= 0 and
x
0
= 1.
Solution:
(a) NewtonRaphson iteration
First order derivative:
f
0
(
x
) = 15
x
2

6
x
+ 6
x
1
=
x
0

f
(
x
0
)
f
0
(
x
0
)
= 1

6
15
= 0
.
6
x
2
=
x
1

f
(
x
1
)
f
0
(
x
1
)
= 0
.
6

1
.
6
7
.
8
= 0
.
3949
x
3
=
x
2

f
(
x
2
)
f
0
(
x
2
)
= 0
.
3949

0
.
2095
5
.
9698
= 0
.
3598
, f
(0
.
3598) = 0
.
0033
(b) secant method iteration
x
1
=
x
0

f
(
x
0
)(
x

1

x
0
)
f
(
x

1
)

f
(
x
0
)
= 1

6
×
(0

1)

2

6
= 0
.
25
x
2
=
x
1

f
(
x
1
)(
x
0

x
1
)
f
(
x
0
)

f
(
x
1
)
= 0
.
25


0
.
6094
×
(1

0
.
25)
6 + 0
.
6094
= 0
.
3192
x
3
=
x
2

f
(
x
2
)(
x
1

x
2
)
f
(
x
1
)

f
(
x
2
)
= 0
.
3192


0
.
2281
×
(0
.
25

0
.
3192)

0
.
6094 + 0
.
2281
= 0
.
3606
2. For a given equation,
A~x
=
~
b
where A is an nbyn matrix and
~x
and
~
b
are nby1 vectors,
wellconditioned
systems have ‘det
A
6
= 0’ and
illconditioned
systems have ‘det
A
≈
0’,
i.e., the determinant is not zero but close to zero. Given the equations
0
.
5
x
1

x
2
=

9
.
5
,
1
.
02
x
1

2
x
2
=

18
.
8
,
(a) [0
.
5pt]Compute the determinant of A. On the basis of (a) and (b), what would you
expect regarding the system’s condition?
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 Spring '11
 LEE
 Numerical Analysis, Gaussian Elimination, Secant method, Rootfinding algorithm, detA

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